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' ' ( waterloopkundig laboratorium delft hydraulics laboratory
non linea e.tion In two horizontal
Rl"Hill'Pta dimension rvey
W 301 part 2 IC)W tember 1980 toegepast onderzoek waterstaat Technische Hogeschool ) , Bihliotheek Afdeling: Civiele Techniek '· Stevinweg 1 ( postbus 50,;.3 2600 GA Delft
surf ace wave propagation over an uneven bottom
non linear wave propagation in two horizontal spatial dimensions, a literature survey
M. W. Dingemans
W 301 part 2
September 1980 toegepast onderzoek waterstaat - i -
CONTENTS page Introduction ...... •...... •...... ••.
2 General properties of non-linear wave propagation 5
3 The average Lagrangian approach •....••....•....•...... •....••...•. 12 3. 1 Introduction ...... 12 3. 2 Uniform media ...... 13 3.3 Non-uniform media ...... 17 3.4 Alternative formulation ...... •....•...... •...... 22 3.5 Some comments on wave action ...... ••...... •...•. 24
4 Application of the average variational principle to water waves ... 25
5 Multiple scale techniques applied to water waves ...... •...... 30 5. 1 Introduction ...... , ...... --;...... 30 5.2 Deep water ...... 33 5.3 Propagation in two horizontal spatial dimensions; constant depth ...... • ...... • ...... 41 5.4 Propagation in one horizontal spatial dimension; uneven bottom ...... 48 5.5 Propagation in two horizontal spatial dimensions; uneven bottom ...... 52
6 A discussion of the significance of the two methods .•...... 57
7 Fairly long waves :; :: ~ ...... 60
7. I Introduction ...... ii o ••••••••••••••• ". 60 7.2 Multiple scale expansion of Boussinesq-like equations ....•.. 62 7.3 Method of Shen and Keller ...... ••••••.•.••...... •...... •.... 67 7.4 Cnoidal wave refraction ...••.....•.....•...... ••.•..••.... 71 7.5 A direct numerical solution of the Boussinesq-like equations 77 7.6 Some comments on the generalized KdV equation and its solu-
tions . 0 ••••• ~ e • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 80 7.7 Discussion ...... 92
8 Long waves ...... , . 95
9 Some comments on non-linear irregular wave propagation ..•.....••.. 99 - ii - CONTENTS (continued) page 10 Conclusions and recommendations ...... 109
APPENDICES A The scaling in the NLS approximation 115
B Derivation of the NLS equation from Eqs. (5.3) and (5.4) ...... 117
C Some solutions of the non-linear Schrodinger equation 125
D Some properties of the Davey and Stewartson equations 132
E On the applicability of the Boussinesq-like equations 140
F Sketch of the derivation of Shen and Keller ...... 142
G The smallness of bottom slope in case of cnoidal wave refraction .. 145
H The mean energy flux for cnoidal waves ...... •..•...... 147
J Derivation of spectral evolution equation from the D&S equations .. 148
K Some comments on modulational instability 152
REFERENCES ...... • ...... 16 1 Introduction
As part of the Applied Research Programme of the Department of Public Works (Toegepast Onderzoek Waterstaat (TOW)) the Working Party Refraction and Diffraction of Water Waves commissioned the Delft Hydraulics Laboratory to perform a literature survey concerning methods to extend the existing pro grammes of diffraction and refraction or monochromatic waves with irregular and with non-linear wave-behaviour. In the first Report irregular waves were considered, whereas in this, second, Report the problem of non-linear wave propagation over an uneven bottom is considered. Generally stated, the present Report is concerned with non-linear wave propagation problems in a non-uniform and possibly also moving medium. Here, the non-uniformity of the ~ediurn is due to the fact that the still water depth his a function of the two horizontal coordinates x and x 1 2 (the bottom is uneven); a medium is said to be moving when currents are present.
In order to get a clear understanding of the effect of the non-uniformity of the medium in which wave propagation takes place, it is at least necessary to know about the properties of evolution of non-linear surface waves in a uniform medium (h = constant), The most important effect of non-linearity is that the frequency w is a function of the amplitude a, which results in a coupling of the ray equations with the amplitude equation. Moreover, energy exchange to other frequencie~ becomes possible.
Because of the direct solution of the complete set of governing equations is usually not feasible, one has t_o consider approximate methods. A common assumption in the study of non-linear wave propagation problems is that the wave system is slowly varying. This implies the condition that the water depth h(t) is slowly varying in~- One often tries to derive differential equations for parameters of a plane wave, such as amplitude, wave number and frequency. The equations for the wave parameters have to be solved now instead of the complete set of equations which govern the wave motion. The derived equations are valid in some asymptotic sense, just as it is for linear wave propagation problems. - 2 -
When looking for methods to solve non-linear wave propagation problems, some subdivision is in order. A first distinction can be made by considering dif ferent classes of water waves, defined for ranges of the parameter h/A, h water depth and A wave length. For h/A << l one has long waves and for h/A
Consider first the subdivision according to h/A. Variations in water depth are more strongly felt by long waves than by short waves (h/A Sea, it is also necessary to investigate waves which are shorter. In the first instance it seems to be not necessary to consider the behaviour of waves with h/A > I. However, it can be shown that, due to the non-linearity, such modulated waves give rise to non-linear wave groups which, in turn, generate very long waves which do feel the bottom. Such long waves could exert influence on the sand transport.
Consider now the subdivision between heuristic methods and more formal methods. The essence of an heuristic method is that it is in fact a simple extension of linear refraction methods: the wave ray equations are the same as in the linear case, only for the phase velocity c the formula is taken that corresponds to the non-linear stationary wave on a horizontal bottom, and thus, c is also a function of the wave amplitude; the energy flux is calculated from the non-linear wave and the wave ~s supposed to adjust itself when progressing to the new depth (e.g., a cnoidal wave remains cnoidal in shape), An example of an heuristic method is the method of Skovgoard and Petersen (1977) for refraction of cnoidal waves; see Section 7.4. Apart from cnoidal wave refraction, such heuristic methods can be applied also using other stationary wave theories, i.e., Stokes' waves (see Chu (1975)) or using Dean's stream function method (Dean (1974)). A wave theory which would - 3 -
yield the best results, in our opinion, is that of Cokelet (1977); to our knowledge this one has not yet been used for propagation over an uneven bottom.
An obvious advantage of heuristic methods is that they are easy to apply, that is, easy as compared to the more formal methods; however, the bottom slopes need to be very small, smaller than is necessary in some other methods, and often only very simple bottom geometries are allowed to keep the procedure indeed simple (e.g., parallel bottom contours).
For a formal description of general wave propagation problems at least two small parameters can be identified, E as a non-linearity parameter and o as a modulation parameter. The parameter Eis a measure for the amplitude of the wave; for Stokes' waves E = ka, k·= 2TI/A and a the amplitude, and for long waves E = a/h. In case of a varying bottom, o can be viewed as the rate A/A, where A is the distance over which h varies order one; A is also called the inhomogeneity scale. In case of a horizontal bottom A is the modulation scale, A= \/o. Several approaches are possible to derive the evolution equations for the wave parameters; the value of E/o is crucial here. For E/O << l the leading order terms of the asymptotic expansions give rise to linear wave propagation prob lems; E/o >> 1 corresponds to Whitham's theory and when E/o ~ I, the effects of non-linearity and modulation can balance each other. In the most simple case one obtains for s/o = 1 the non-linear Schrodinger (NLS) equation which is the canonical equation for weakly non-linear, strongly dispersive wave systems, just as the Korteweg-de Vries (KdV) equation is the canonical equa tion describing the evolution of weakly non-linear and weakly dispersive waves. Both the KdV and the NLS equation exhibit soliton behaviour and are solvable by the inverse spectral transform (IST) methods.
The plan of this Report is as follows. At first, in Chapter 2, a general discussion is given of properties of non-linear wave propagation. The approach based on an average Lagrangian density (Whitham's method) is discussed in Chapter 3 and an application of this variational principle is given in Chapter 4. One of the advantages of the approach based on an average Lagrangian is that the basic features of (linear as well as non-linear) wave propagation are obtained in a general form; for specific applications the evolution equa tions for the wave parameters are readily obtained by introducing the specific form of the average Lagrangian L. Whereas the method can be shown to be exact - 4 -
for linear waves, the inherent approximations for the non-linear case are difficult to recognize. Therefore, in Chapter 5, multiple scale techniques are discussed; this is only done for the case that£~ 6 because it can be shown that Whitham's technique is equivalent to a multiple scale expansion for the case that E/6 >> 1. A separate discussion of the significance of the previous methods is given in Chapter 6. The case of long waves (i.e., h/A << 1) is dealt with in Chapters 7 and 8. In Chapter 7 the "fairly" long waves are discussed, i.e., waves for which the 2 Stokes number (also called the Ursell number) (a/h)/(h/A) is of order one; in this case the governing equations can be reduced to a set of Boussinesq like equations. These Boussinesq-like equations then form the basis of the study of the propagation of the fairly long waves. Subsequently, in Chapter 8, truly long waves are discussed·, that is, waves for which h/A << l and the 2 Stokes number>> l (a/h >> (h/A) ); here slowly varying Riemann invariants are useful to describe the evolution of the wave system. Some comments on non-linear irregular wave propagation are made in Chapter 9. The conclusions and recommendations are given in Chapter 10.
The investigations were carried out by M.W. Dingemans, who also drew up this Report. The literature surveyed in this Report consists of those papers which came to our attention before mid-August 1979. - 5 -
2 General properties of non-linear wave propagation
The purpose of investigating problems of wave propagation is to be able to predict the main properties of a wave solution at some later time in terms of appropriate initial conditions. The problem to be stated is then one in which equations are given for the evolution of wave parameters such as amplitude, frequency, wave number, mean current and mean surface elevation. It is to be noted that such an approach is also followed in problems of propa gation of linear waves in non-uniform media (i.e., in the usual refraction cal culations). For this approach to be possible, the waves are supposed to be slowly varying. In order to see the principal difference between linear and non-linear wave propagation, a simple heuristic extension of linear wave propagation to the non-linear case is given below. This example is treated in more detail by Whitham (1974, Section 14.2) and also by Yuen and Lake (1978). (The latter authors considered the case where the modulation parameter is of the same order of magnitude as the non-linearity parameter, see further Chapter 5.)
The modulation of a linear wave train, with s = acosX, X = k.x. - wt, J J can be described, in absence of currents, by
aa2 a ;:;--t + -- (2. I) 0 ax. J
and
ok. ow at""1+--=0 ax. . (2.2) ].
An essential point of 'non-linearity is the dependence of the frequency won the amplitude a. For near-linear waves w may be written as
(2.3)
·using simply this expression for w, Eq. (2.2) results in - 6 -
elk. l. -- + (2.4) el t
2 The important term in (2.4) is w ela /elxi; it leads to a correction 2 of O(a) in the characteristic velocities. The other extra term 1.n 2 (2.4) merely gives an O(a ) correction of an existing term in elk./elx .. Substitution of (2.3) into Eq. (2.1) only results in extra J l. 2 terms of relative order O(a ). Therefore, in the first instance the effect of non-linearity for near-linear waves may be described by
ela2 --+ = 0 (2.5) elk. aw c)k. aa2 __1. + __o _J_ + w -- = 0. For the sake of simplicity now only waves propagating 1.n one spatial dimension x = x are considered. The set (2.S) then reduces 1 to, with w'(k) = dw /dk, 0 0 2 2 ela 2 elk ~+ w' (k) --+ w" (k) a = 0 el t 0 elx 0 elx (2.6) 2 ok ela + w' (k) elk + w2 (k) = o. at 0 elx ax The characteristics (or wave rays) of this set of equations are given by (2. 7) When w (k)•w~(k) > 0 then (2,7) yields ,two different characteristic 2 velocities; this is clearly a non-linear effect, because for vanishing a (the linear approximation) one obtains dx/dt = w' (k), 0 as follows directly from (2.1) and (2.2) when w = w . 0 When (k)•w~(k) < 0 the characteristic velocities (2.7) are com- w2 plex and the system (2.6) is elliptic. This means that small sub harmonic modulations will grow in time and that the wave system - 7 - is unstable in this sense. For waves on deep water the dispersion relation reads 2 2 2 w = gk(l + k a + ••• ), or I k2 2 w = /gk (l + - a + ••• ) • 2 512 Therefore, = /gk and w = .!.. g½k and w w" < 0 for all k. WO 2 2 2 O It is noted that a direct substitution of the dispersion relation (2.3) into Eqs. (2.1) and (2.2) results, in the case of propagation in one spatial direction only, in the characteristic velocities 2 1 12 4 dx = w'(k) + 1 w'(k) a2 + {w w" a + ( w + w w") a }½, dt o 2 2 2 0 4 2 2 2 from which real velocities result for For deep water waves this means that for about a/A> 0,05 the characteristic velocities are real (A= 2TI/k) and for smaller waves instabilities occur. It is noted in passing that a consistent 4 approximation including O(a ) terms is obtained by substituting the 2 4 expression w = w (k) + w (k) a + w (k) a into Eqs. (2.1) and (2.2) 0 2 4 resulting in characteristic velocities dx dt a 4}½ • 2 2 2 Because one has, in case of deep water, w = gk(I + k a + 1 4 4 r;- I 22 I 44 2 ka + ... ), or w = vgk (l + 2 k a + 8 k a), it is easily seen · that inclusion of w term in the condition {} > 0 for real veloci- 4 ties makes little difference in the value found for a/A (with w4 one finds a/A> 0.0500 instead of a/A> 0.0488). Furthermore this - 8 - procedure carried out here is only done in order to see easily some effects of non-linearity on the propagation of nearly uniform wave trains. A precise investigation for exactly periodic deep water waves is given by Lighthill (1978, pp. 460-462) based on an averaged Lagrangian expressed in terms of exact relations for the averaged kinetic and potential energy (see also Section 3.2); instability is then found for a/A> 0.054. Although the problem of instability of wave trains is not a subject in the present study, the occurrence of instability is often encountered and there fore some connnents on instability are collected in Appendix K. For the way in which evolution equations for the wave parameters are obtained several methods exist. These methods have in connnon that either explicitly or implicitly use is made of a multiple scale expansion with respect to the modulation parameter,µ, which is defined as the ratio of the wave length A and the scale A over which the wave properties vary appreciably,µ= A/A<< l. (Note that from Chapter 5 onwards the modulation parameter is denoted by o.) An essential feature of surface water waves is that they propagate in the (1,t) space, where i = (x ,x ) is directed horizontally, while the structure 1 2 of the solution with respect to the vertical coordinate z is altogether different. The physical space (the space of the independent variables 1, z and t) is the product of a propagation space (spanned by 1 and t) and a cross space (here consisting of the vertical coordinate z). The evolution equations for the wave parameters are derived under the condition that the wave is locally a plane wave (either linear or non-linear) in propagation space. This means that a phase function X(i,t) exists; the frequency wand the wave num ber k. are then defined as J w(;t, t) = -ax/at k. (~,t) = ax/ax. (j=l,2), (2.8) J J where tt and ware slowly varying with* and t. The free surface elevation of a slowly varying water wave can now be written as ~ • + ~ = ~{µx, µt, X(x,t)}. - 9 - Because the second derivatives of X to x. and tare supposed to be small J quantities, a slowly varying water wave is represented by 1?;; (*, t) = ~{µt,µt, t X(µ1,µt)} (2.9) ~ + I I ~ + k a~ a~ clx. j ae + µ ax. . (2. I 0) J J The variable 0 is considered as being an independent variable, after intro ducing relations as (2.10) into the original differential equations. After ~ + ~ + a solution of l;;(X,T,0), ~ + n + l;;(X,T,0) = µ ?;; (X,T,0) n n=o (2.11) ~ + n + These expansions are to be substituted into the governing equations (with independent variables l,z,T,0). It is then demanded that, because sand The governing equations for irrotational water waves are given by the field equation a2cp a2cp - < < + 0 - h(~) z = r; (x, t) , (2.12) dX, ax. + -2 - 1. 1. dZ and the boundary conditions ~+~5-=21 at z r;(x,t)+ (2. 13) at ax. dX. dZ = J J ~~+21 = 0 at z = -h(~) (2. 14) dX. dX, az J J 1 acp acp _ acp + r; +------0 at z = (2. 15) at g 2 ax. ax. r; (*, t)' J J where the bottom is given by z = -h(*) and (acp/ax , acp/ax , acp/az) = (u ,u jw) 1 2 1 2 are the velocity components. Equation (2.15) results from the condition that the pressure p is zero at the free surface. The governing equations (2.12)-(2.15) also follow from a variational principle for irrotational water waves as given by Luke (1967): off L di dt = O, (2.16) with s<*, t) { acp 1 acp acp 1 a Note that the integrand of (2.17) represents in effect the pressure within the fluid and that Lis a functional which is integrated over the cross space. It turns out (see Seliger and Whitham ( 1968)) that, for an "Eulerian" descrip tion of the dependent variables, the most simple variational principle in fluid dynamics is obtained when as the Lagrangian function the pressure is taken. In Hamilton's principle the Lagrangian is the kinetic minus the poten tial energy. When "Lagrangian" variables are used, Hamilton's principle is well suited but in Eulerian variables the side conditions have to be obtained - 11 - by using Lagrangian multipliers. For deep water waves Hamilton's principle is useful because the free surface elevation is in that case a suitable Lagrangian coordinate and the bottom condition (2.14) is replaced by IV~I + 0 with z + -oo (see Lighthill (1967)). There are now in effect two different ways to derive the desired evolution equations for the wave parameters of slowly varying wave trains, both depend ing on multiple scale expansions, The first method is to apply multiple scale expansions directly on the governing equations (2.12)-(2.15); this method is used in the methods described in Chapter 5. The second way is to apply multi ple scale expansion directly to the variational principle; this is done in a rather implicit way. It amounts to the substitution of a uniform wave solution into the Lagrangian function Land integrating this expression over one period of the phase, keeping the parameters fixed. When the period of the phase is normalized to 2TI, the result is 2TISv. The assumption is now made that off t., d1 dt = 0 can be used to derive the Euler-Lagrange equations which govern the evolution of the slowly varying wave parameters. This approach is due to Whitham and is treated in Chapter 3. - 12 - 3 The average Lagrangian approach 3.1 Introduction In this Chapter some comments are made about the stationary principle of the average Lagrangian, which can be used to obtain evolution equations for the parameters of slowly varying wave trains. At first only homogeneous media are treated. Because the methods of the average Lagrangian rests upon a multiple scale expansion, the resulting evolution equations are valid in some asymptotic sense. Because of the required gradual change of the wave parameters within a wave length and a wave period, the asymptotic approximation is often called an adiabatic approximation. We only consider conservative waves, that is, waves which obey equations derivable from a variational principle applied to a Lagrangian functional. In some instances it is possible to derive dissipative equations from a varia tional principle by using Prigogine's method devised for irreversible thermo dynamics, see, e.g., Jiminez and Whitham (1976); this is not investigated in this Report because we feel that a direct asymptotic approximation of the governing equations is in that case much simpler. It is now assumed that the governing equations for the waves under considera tion can be derived from the variational principle off L(q.,aq./3t,aq./ox.) d~ dt = O, (3. I) i i i J where o is the variational operator. The vector qi' i=l, •.• ,n, designates the set of quantities which determine the oscillations; i.e., the q, are the generalized coordinates. For water i waves, the q. are quantities such as the free surface elevations and the i velocity potential~. The principle (3.1) states that L dl, be stationary in small variations of qi(l,t) vanishing at end points t = t 1,t2 and +x = +x ,x+ . The equations. o f motion. become 1 2 - 13 - i=l ,2, ... ,n. (3.2) These equations are the usual Euler-Lagrange equations. The average Lagrangian approach (also called Whitham's method) is now based on the following ad hoe procedure. A solution of q. in the form of a plane wave (either linear or non- i linear) is substituted in the Lagrangian density Land subsequently Lis averaged over the phase of the wave, keeping the other parameters fixed; the result is an average Lagrangian g., depending only on the slowly varying para meters of the wave train. The variational principle o ff 8..- d~ dt = 0 is then used to derive Euler-Lagrange equations which govern the evolution of the parameters of the wave train. The method is considered further in Section 3,2. Before doing so, we mention the following relevant references. The method is due to Whitham (1965, 1967a, 1967b) and many details are given in his book, Whitham (1974). Discussions about the applicability of the method are given by Lighthill (1965, 1967) who gave also a clear discussion of the essential properties in his book, Lighthill (1978, pp. 455-462). A short general expose is given in Appendix A of Karpman (1975) and an introduction to the theory can be found in Whitham (1971). Furthermore, important papers on the subject are Hayes (1970, 1973) and an introduction to these is Hayes (1974). Application of a multiple scale expansion applied to the variational principle was considered by Whitham (1970) and was further elucidated by Gilbert (1974). 3.2 Uniform media For ease of exposition at first only one spatial coordinate x = x will be 1 considered, and the medium is taken to be uniform. For application to periodic waves it is convenient to take the integral over a single wave length and a single wave period; the variational principle remains valid if the variations of q. are constrained to be periodic with that i wave length and that period. Then one has the principle that the average Lagrangian~ (averaged over a wave length) is stationary. - 14 - In a periodic wave all generalized coordinates must take a form q . ( X' t) = f . (X) X = kx - wt. (3 .3) 1. 1. Then the quantity stationary for real wave forms can be written as 2TI 1 L(w,k,a.) = - L{f.(X), -wf!(X), k f!(X)} dX. (3.4) 2 1 1. TI O 1. 1. 1. ~is thus an average of the Lagrangian density with respect to the phase X. The actual solutions f.(X) bring in the amplitudes a. and therefore 8-is also 1. 1. a function of the a.'s. The average variational principle is thus 1. 8 ff &...(w,k,a.) dxdt = 0. (3.5) 1. It has to be remarked that wand k may not be varied independently of each 2 2 other because it follows from a X/8x8t = a X/atax that 8k/8t + aw/ax = O. The variation with respect to the a. gives 1. a&- aa. = O i=l, ... ,n, (3.6) 1. and the variation with respect to X yields (3. 7) ~t (~~ - ~x (~~ = O • Equation (3.7) can be recognized as being a conservation equation for the quantity as.;aw with as flux -at/ak. It is now possible to define the wave action density A as ag. A= dw . (3 .8) Note that A is the rate of change of &-with w, keeping k constant. It is also noted that in mechanical systems action is defined as the integral of the Lagrangian function L(q,q,t) between two instants, and the Euler-Lagrange equations follow then from Maupertius' principle of least action. - 15 - For systems where Lis the difference between kinetic and potential energy one obtains 3-..= WK - Wp, where WK and Wp are the average kinetic and potential energies per unit area (area, not distance, because for water waves energy is taken per unit length of crest). Only in linear systems the stationary value of 9.- is zero; for non-linear water waves one hast-= WK - Wp > 0. It is possible to relate the wave action density A to the average kinetic energy WK in the following way (see Lighthill (1978)). It follows from (3.4) that 8L c)f !] 8(8q./8x) .k aw 1. dX, l. where q. = 8q./8t = -w f! has been used (see (3.3)). l. l. l. Because~ is stationary with respect to any small changes 1.n wave form, it 1.s not necessary to allow for the small change in wave form which accompanies a small change in w. Thus, for fixed wave form f.(X) one has l. and thus, for waves of a given wave length, one has l f2Tf/W • • = - / q.•(aL/8q.) dt. (3 .9) 2Tf w O 1. 1. Because kinetic energy is an homogeneous function of the second degree, it 1 can be written (per unit area) as - q.•8L/8q. and therefore, it follows from 2 l. l. (3.9) that the action density A can be expressed in terms of the average kinetic energy WK as A = 89--/ aw (3. I O) It is to be stresses that this is only valid for stationary waves (also called permanent·waves) of a fixed wave length. For linear systems there is t> -I c.,.,= WK - Wp = 0 and. therefore one has 1.n the linear case A= W w , where W is the total averaged energy per unit area, W = WK + Wp. For waves of a given - 16 - wave length the excess~ of kinetic over potential energy rises as w increases above its linear value w and is obtained from (3.10) as 0 -1 2WK w dw. (3. 1 1) A relation as (3.11) can be used as a check on the accuracy of computation of waves from some wave theory. Furthermore it gives, at least for deep water waves (where the induced current is absent), the possibility to use the average Lagrangian &- for the case of strongly non-linear waves instead of being able to use it only for near-linear waves. This approach is followed by Peregrine and Thomas (1976) in problems of wave propagation upon non-uniform currents. For WK and Wp they used the values tabulated by Longuet-Higgins (1975) for deep water waves. So far the expressions are exact for waves (linear and non-linear) which are permanent and periodic. Indeed the substitution of (3.3) into the Euler Lagrange equations (3.2) and into the variational principle (3.5) where for g., the expression (3.4) is used, both lead to the same equations (3. 12) Before considering slowly varying waves, a few remarks are made concerning so-called potential variables and surface waves on water of finite depth. Usually the generalized coordinates qi' i=l, ••. ,n can be divided up into two groups: - one group is composed of those q. which enter into the Lagrangian function 1 L only through their derivatives; these are called potential variables and are denoted by., i=l, .•• ,m, m < n; 1 - the other group consists of the remaining genera,lized coordinates, denoted by r., i=l, ..• ,n-m. 1 Thus, { q I ' · · · ' qn} = { r I ' · · · ' r n-m' It is seen from the governing equations of irrotational water waves that the velocity potential ~(!,z,t) only occurs in the form of its derivatives. These - 17 - governing equations can also be obtained from Luke's (1967) variational prin ciple in which the Lagrangian functional Lis given by (3 .13) It is thus seen that the velocity potential ~(i,z,t) acts as a potential variable. The general solution for,,~, describing stationary periodic waves (i.e., the medium is uniform) is, because s, a~/ot and a~/ax, are periodic (there are no requirements of periodicity in z because periodicity is only measured in propagation space): = ri(X) (3. 14) s .X • - y t + cp (X ' z) ' J J where k., w, S., y are constant parameters. The solution is normalized so J J that cp is the periodic part of~ and the change of X in one wave period is 2TI. Furthermore, we have two other wave parameters, the mean surface level b + • and the wave amplitude a. The solution thus depends on (w,k,a) and (y,S,b), that is, on eight quantities. 3.3 Non-uniform media We consider now slowly varying wave trains in either uniform or non-uniform media. If parameters characterizing the waves vary sufficiently gradually, on a scale of wave lengths or wave periods, then locally the .waves must closely approximate to plane waves. Locally a phase function X(i,t) can then be defined; the frequency and wave number are then defined by w = -ax/at k. = oX/ax. j=l ,2. (3. 15) J J The frequency wand the wave number k. are now slowly varying functions oft J and t. It is noted that the fact that a wave train is slowly varying may be due to the non-uniformity of the medium (e.g., a sloping bottom) or to the boundary and/or initial conditions imposed on the wave train. - 18 - As was noted previously, the general idea is to start with the uniform wave train, either linear or non-linear, extend the solution to allow parameters a, k., wand any others that arise to be slowly varying and find ways of J obtaining evolution equations for these parameters. Whitham's method proceeds now as follows. Suppose that the Lagrangian function L(q., dq./dt, dq./dx.; x.,t) is known i i i J J for the problem at hand (i=J, ... ,n, j=l,2). The governing equations are thus derivable from the variational principle 6 ff L d;tdt = 0. A uniform solution for the q.(l,t) is substituted into Land the resulting expression is averaged i over phase, keeping the parameters fixed; the result is a function ~which depends only on the slowly varying parameters of the wave train. (Note that, for surface waves propagating on water of varying depth h(l), a uniform solu tion (either linear or non-linear) is obtained from the corresponding problem with horizontal bottom.) The evolution equations for the parameters of the wave train are obtained from the variational principle o f J 8-. d~dt = 0. It is this last step which is responsible for the approximate nature of the averaged variational principle because~ was calculated using the local plane wave solution for the q .. A measure for the gradual change of the waves (e.g., i due to the non-uniformity of the medium) is the modulation parameterµ;µ is defined as the ratio of the wave length A over the inhomogeneity scale A, µ=A/A, where A is the measure of length over which the medium varies appre ciably. Because of these two widely separated scales, A<< A, the application of a multiple scale expansion technique withµ as small parameter seems natural. Another kind of approximation often occurs in non-linear problems because it is then usually not possible to obtain an exact (non-linear) expression for the uniform solution; one has then to take recourse to some approximation technique to obtain a solution; usually one obtains a formal asymptotic solu tion with as small parameter the non-linearity parameters<< I (for Stokes' waves€= ka and for long waves s = a/h). In order to make the nature of the approximation more explicit, we carry out the multiple scale expansion applied to the variational principle itself. The generalized coordinates q.(x,t)-+ can be written, in the case of slowly i varying wave trains, as qi ( -+x,t ) = ~qi ( µx,-+ µt, µI X( µx,-+ µt ) ) , µ « 1. (3. 16) - 19 - The slow scales X= µi, T =µtare introduced now, together with the phase 0(X,T) = µ-l X(X,T). The frequency and the wave number are then given by + -+ w(X,T) = -cl0/clT k. cx,T)= cl0/clx .. (3. 17) J J It follows that clq. clq, clq. clq. clq. clq. 1 1 1 1 1 1 clx. = µ clX. + kj 88 at"=µaT-w~ (3. 18) J J After substitution of relations (3.16) and (3.18) into the Lagrangian L(q,, clq./clt, clq./clx.; -+x, t), 0 i·s considered as an independent variable; 1 1 1 J -+ once the solutions of q,(X,T,8) are obtained, the solution of the original i -I -+ problem is recovered for 8 = µ X(µx,µt). The original variational principle is given by (3.1), i.e., by o +ftf L(q,, clq./clt, clq,/clx.) d1dt = O. X 1 1 1 J Using (3.16) and (3.18) the Lagrangian Lis written as clq. clq. clq. clq' 1 l l L = L(qi' -w cl/+µ clT k. as"+µ clX.). (3. 19) J J Substitution of (3.19) into the Euler-Lagrange equations (3.2) results in clL cl cl clL cl cl clL F -(k. as+µ clX.) cl(k. clq./cl8) -(-was+µ clT) cl(-w clq./cl8) = O, qi J J J l l (3.20) which may be written as clq. clq. ~ cl l clL ~ cl 1 clL - L) + µ as cl clq' clL + µ = (3. 21) clT ccl/ cl(-w clq,/cl0)) o. J_ - 20 - These equations are also the Euler-Lagrange equations of the following variational principle: o +f J J 1 d8dtd~ = o. (3.22) X t 8 It is noted that (3.22) is still exact. Introducing the average Lagrangian L- by - f2TI L~ L = 2n o d8, (3.23) it is seen that (3.22) is an exact averaged variational principle. A first approximation is obtained by introducing the formal expansion ~ + m (m) q. (X,T,0) = ~ µ q. . (3. 24) 1. m=O 1. A first approximation to the Lagrangian (3.19) 1.s then cl (o) "oq, (o) = L(q~o), qi -w k. cl~ ). (3.25) 1. ae J Whitham's method 1.s now seen to be based on L(o); the average Lagrangian ~ l.S 8-., = 2n /2TI L (o) d8, (3. 26) and qi(o) is the plane wave solution. It has to be stresses that still no assumption is made about the magnitude 0 of the amplitude of q~ ); the only assumption made is that the waves are 1. slowly varying. When the waves were stationary, (3,.19) would reduce to the Lagrangian (3.4). Plane uniform irrotational water waves are described in general by Eqs. (3.14). For slowly varying waves wand k. are defined by (3.15). In a similar way y + J + and Scan be defined from a pseudo phase function w(x,t) as - 21 - y = -al)J/at s. = ol)J/ox. (j=l,2). (3. 27) J J The average Lagrangian S..,is now calculated by substituting the plane wave solution ~ i'._; (i, t) = n(X) (3. 28) ~ (Jt,z, t) = l)J(i,t) + ~(X,z), into the Lagrangian (3.13) and averaging the result over phase, keeping• w, ,+.k, y, S fixed: + ..!.. ,h2 2 'I' z + gz }dzdX. (3. 29) The evolution equations of the slowly varying wave parameters then follow from ..... • o /1f 8- (w, k, a;y, S, b) didt = 0. (3. 30) Because of (3.15), the variation of ff~ d1dt with respect tow and k can not -+ be carried out independently; similar for y and [:3. S.., is thus written as The evolution equations then follows from the variational principle (3.30) by considering the variations oX, oa, ol)J and ob (note that X and ljJ are potential variables): oa: ai 0 aa = ob: ~f = 0 (3 .31) . a a cSX· - ea&.) This set of equations is completed by the consistency conditions which follow directly from (3.15) and (3.27): elk. clw _J + -- = 0 j= I, 2 clt dX. J (3.32) as. cly __J + 0 j= 1, 2. clt clx. = J Note that these consistency conditions are needed because after obtaining Eqs. (3.31) there is reverted back tow and k. and toy and S .• Were this J J not done, then the last two of Eqs. (3.31) are second order partial differ- ential equations for X and~. The equation cl$./cla = 0 yields the dispersion relation and cl~clb = 0 yields a + Bernoulli-type of equation for the mean flow S, The variation oX gives rise to the conservation equation of wave action and the variation 8~ yields the equation of conservation of mass for the mean flow field. + Thus eight equations (3.31) and (3.32) are found for the eight unknowns w, k, a and y, /3, b. Once/)., is found, the evolution equations for the wave parameters are readily given. Because (3.29) is based on the use of the plane wave solu tion (3.28), and the representation (3.28) can be viewed as being the q~o) in i (3.25), it is seen that Eqs. (3.31) and (3.32) are in fact the lowest order asymptotic approximation in the modulation parameterµ. See also Gilbert (1974) and Whitham (1974, Section 14.4). It is noted that the evaluation of A by cl8./clw from the expression found for ~has to be done without substituting the dispersion relation resulting from clC-,/cla = 0 into the expression for8.-. Only after the evaluation of as.Jaw the dispersion relation cl~cla = 0 is substituted in the expression for aJVaw; this is a consequence of the fact that g_, has also to be defined for non solutions in order to keep the necessary freedom in' carrying out the necessary variations. 3.4 Alternative formulation Hayes (1973) uses a different procedure for obtaining the evolution equations. Consider the case that the average Lagrangian &..is only a function of w, k, a, - 23 - 8-..= &{w,k,a); the dependence of the mean motion quantities is dropped now because these can be treated in an analoguous way. The evolution equations are thus given now as aB. aA aki aw + __] = 0 and -- + -- = = 0, at dX. dt dX, o, J i where A is defined by A= a~aw and the wave action flux is given by B. = -at;ak .. J J The dependence of &..,on one of the variables may now be eliminated, so that~ becomes a function of less variables. Lighthill (1965) used the relation a~/aa = 0 to eliminate a so that the result is&_= g_(w,k). Hayes (1973) instead eliminates w by a differe·nt procedure; this amounts to the following. Given &-(w,k,a), calculate A = ac..;aw. Use this equation to eliminate a in favour of A; this results in 8.,(w,k,A). Calculate subsequently a8.;aA O; this yields the dispersion relation and w can thus be obtained in terms of k and A. The equation A= a~aw may be integrated and yields (\ • • o.. = Aw - 'tt.(k,A;x, t). (3.33) This equation is a partial contact transformation and'-t, which is a constant of integration here, is recognized as being a Hamiltonian. This Hamiltonian Mis found by substitution of the expression for w found from a~aA = 0 into the expression Aw-&-; the resulting expression is°M-(k,A;i,t). Instea d ofth e average Lagrangian' l)..W,k,a;x,t0 ( + + ) the average Hami·1 tonian. • ~(k,A;i, t) is obtained. As a measure of "wave intensity" the amplitude a is replaced by the wave action A and~ depends on one variable less. This dispersion relation, which was previously given by a[t/aa = 0 or a~aA = 0, is now given as • • w = a~aA = Q(k,A;x, t), (3. 34) and the wave action flux B.= -ag_;ak. follows now from ] ] Bj = ~~ (3.35) ] A non-linear group velocity •c is obtained from (3.34) as - 24 - 32l-4- clB. c. = = J (3.36) J 3Aclk. clA J Hayes (1973) calls this the basic non-linear group velocity and expects that this definition of group velocity is close to the velocity of the centre of a wave group or of the group as a whole. 3.5 Some comments on wave action It is seen from the definition (3.26) for 8-, that the wave action A= ac.;aw in Eq. (3.31) is in fact a first order asymptotic approximation (for µ+O) of the exact expression for the ~ave action which would be obtained as A . = cJL/clw (see (3.23)). Because wave action is one of the most funda- exact mental quantities in wave propagation problems it is necessary to consider its definition, It is noted that the wave action A is a measure of the wave intensity and that the conservation equation of wave action clA/clt + clB./clx. = 0 J J is always obtained in the non-linear water wave problems investigated in this Report. Wave action, however, can only be defined approximately in an Eulerian description of the flow field. The definition of A depends on the way "dis turbance" and "mean flow" are defined; in this respect it is noted that water waves are always regarded as some disturbance of a basic flow field. It is shown by Andrews and McIntyre (1978b) that wave action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when waves are defined respectively as departures of ensemble-, space- and time averaged flows. They obtained a wave action conservation law in a form which does not depend on the approximations of slow amplitude variation, lineari zation or conservative motion and it was not necessary to use variational principles, This was achieved by expressing A in terms of the particle dis placement field, and by using a generalized Lagrangian mean operation devel oped by Andrews and McIntyre (1978a). These papers by Andrews and McIntyre require further study in order to obtain descriptions for studying the evolution of essentially non-linear waves over a non-horizontal bottom geometry; also the inclusion of currents which are non-uniform in z and the inclusion of rotation in the wave field becomes possible then. - 25 - 4 Application of the average variational principle to water waves In this Chapter an example is given of functions&.. for the case of non-linear water waves. Stokes' waves are considered here; the medium is taken to be • non-uniform, i.e., the bottom is not horizontally, h = h(x), and slowly varying currents which are uniform with respect to the vertical coordinate may also be present. Once the average Lagrangian 3..- is found to the required order of approximation, the evolution equations for the parameters of the basic wave are readily found by variation of g,, see Chapter 3. Here the problem of obtaining a suitable ~ is investigated. It is known that the governing eq~ations for incompressible irrotational water waves follow from the variational principle as given by Luke (i967): o, ( 4. I) with L = -p (81J)2} + gJ dz, (4.2) 8z J where 1l(1,z,t) is the velocity potential, l;(i,t) the free surface and z = -h (i) the bottom. Because qi is a potential variable, the most general 0 form for a periodic wave train is given by r(i'.,z 't) = 1/J(ic,t) + qi(X,z) (4. 3) r,; (1, t) = n (X); where X(i,t) is the phase and the pseudo-phase 1/J(~,t) is given for uniform wave trains as • 1/J(x,t) = f3.x. - yt. (4.4) J J • For slowly varying wave trains, f3 and y are defined by f3 . 8\jJ/ax. y -a\jJ/at, (4.5) J J - 26 - + + that is, Bandy are defined in an analogous way ask and w. The physical + significance of Bis that it represents the mean current; y contributes to the mean pressure and is thus related to the mean level of the water. In the lowest order modulation approximation, the average Lagrangian is obtained by substituting the periodic solution (4.3) into the Lagrangian (4.2), and averaging subsequently over the phase X. One obtains by substi tution of (4.3) into (4.2): I 2 I 2 2 acp = (y - 2 S )(h +n) + g(h -n) + (w-8.k.) Jnh aX dz + 0 2 0 J J - 0 (4.6) n 2 2 {k2(ae) + (~u!) } dz, 2 J-h di'. __ 0 where B = lsl and k = Jkl. Because exact expressions for cp(X,z) and n(X) are not known, further progress is made by using approximate solutions. One could use now either the near linear expansions of Stokes, or the fairly long wave theory of Boussinesq. A derivation of .Lfor the case of Stokes' waves and a subsequent discussion of the modulation equations is given by Whitham (1974, Sections 16.6-16.13). We note here that, using the Stokes' expansion for the case of an horizontal bottom: 00 A cp (X, z) = E ~ cash nkz.sin nX n n=l (4.7) 00 n(X) = b + acos X + E a cos nX, n n=2 there can be obtained by substitution of (4.7) into (4.6) and subsequently integrating to X: 2 I 2 I 2 2 I (w-B.k.) p(y - 2 B )h + 2 pg(ho-b) + 2 E{gk,t~n~ kh - I}+ (4.8) 2 2 4 2 I k E {90 - 100 + 9 3 ------} + O(E ), 2 pg 804 where - 27 - 1 2 h = h + b, E 2 pga and 0 = tanh kh. ( 4. 9) 0 2 Because it is only necessary that the changes in S due to the waves are O(a ), •• not S itself, possible pre-existing slowly varying currents U(x,t) can be • incorporated in S (see Whitham (1974, p. 555)). The parameters of the wave • • • are now k, w, a and S, y, h, which are all slowly varying functions of x and t, Within the same order of approximation it is possible to replace the terms 2 o = tanh kh by o = tanh kh because bis of order a. 0 0 Equations (3.19) can now be worked out using the~ from Eq. (4.8). It is then readily seen that the dispersion relation follows from ac..;aa = O, or from ag/aE = O, as 2 (w-S. k.) 4 2 2 90 - 100 + 9 2 . . J J + (ka) . ------,,---- + 0 (E ) , (4.10) gk tanh kh 4 80 which is indeed the dispersion relation for second-order Stokes waves on water of finite depth. The wave action as.Jaw and the wave action flux -ag/ak. follow as J a~ E (w-S. k.) E "w = ----=-J--=-J o gk tanh kh w 0 and 2 = (S.+v.) + O(E ), - ak. w J J J 0 l where w = {gk tanh kh} 2 and v. = aw /elk. is the group velocity. 0 J O J The conservation equation for the wave action is thus a E cl E ~t (-) + -"- {- (S.+v.)} = o. o W oXj W J J 0 0 The mass conservation equation (cl/at)(a~ay) + (8/clx.)(8£./aS.) = 0 becomes J J a a k. at (ph) + ax. (ph Sj + E WJ) = 0. J 0 - 28 - -+ Introducing the mass-transport velocity U by Ek. J U. = B. + h ( 4. I I) J J p w 0 the mass conservation equation is simply written as a a at (ph) + clx. (ph U.) = 0. J J It is to be remembered that h(~,t) = h (1) + b(~,t). 0 It follows from cl~clb = 0 that the pseudo-frequency y is found as 2 y J__ 8 + gb + J__ E.!:_ (4. I 2) 2 2 p Resuming, with (4.8) fort we have found from (3.19) the relations (4.10) and (4.12) for wand y and the two conservation equations for wave action and mass E a (!__) + a {- (S.+v .) } = 0 at w clx. WO J J 0 J ( 4. 13) E a (ph) + --a {ph $. + - k.} 0. at clx. J WO J J The consistency relations (3.32) become upon substitution of wand y from ( 4 . I 0) and ( 4 . l 2) : elk. 4 2 i a 2 90 - 100 + +-- + w { I + (ka) . 9}!] = 0 F clx. r.k.J J 0 4 i 80 (4. 14) 2 as.i a 2 + -- [l- B2 + gb + -l gka l;a ] = 0. clx. 2 4 3t i Equations (4.13) and (4.14) constitute thus a set of six equations for the six unknowns a, b, k. and B, (i=l,2). For later use, the first of Equations i i . (4.14) is expanded. It was already noted that, within the same order of - 29 - approximation, the term a may be replaced by a tanh kh . It follows then 0 0 from (4.10) that w - (3.k. J J with D 0 Expanding for small kb, one obtains 1-02 1 2 1 0 l w - (3.k. + (ka) D + - kb -- (4. I 5) 16 0 2 a ' J J 0 and the influence of b on w is made more explicit. - 30 - 5 Multiple scale techniques applied to water waves 5.1 Introduction The method of the averaged Lagrangian as applied by Whitham is seen to be in effect the lowest order approximation of an asymptotic expansion with respect to the modulation parameter. Not the amplitude of the wave itself was supposed to be small, but its variation had to be small. Chu and Mei (1970) considered two small parameters, 8 as the small modulation rate of the wave train and E, characterizing the non-linear effect of finite amplitude. A multiple-scale technique was used to obtain asymptotic representations for slowly varying, weakly non-linear, dispersive wave problems for the case of E/O ~ 1. Remark that in Whitham's theory the case E/O >> 1 is considered and in linear ray theory one has E/o << 1. The effect of taking E/o ~ 1 instead of taking E/6 >> 1 is the appearance of some terms in the evolution equations additional to the results of Whitham, which terms contain second derivatives of the wave amplitude a to x. and t; these terms are diffraction-like terms and set a J bound on the growth of instabilities. The case E/0 = 0(1) can be compared with the case of taking the Stokes number 2 3 ((a/h)/(h/A) or ka/(kh) to be of order one in the case of fairly long waves (see Chapter 7). The relevance of inclusion of the diffraction-like terms is that greater bottom slopes can be considered than is the case without these terms. It is not the purpose of this Chapter to discuss all kinds of multiple-scale techniques which are relevant for water waves. In this Chapter we only discuss techniques which can be applied. to Stokes' waves and which are valid for E/0 ~ 1; in some instances the reduction of the evolution equations is given for the limit to shallow water (i.e., kh • O). Several approaches exist, which in fact differ only in detail. A common feature of these methods is that the equations reduce, for the case of deep water and almost monochromatic wave trains, to a non-linear Schrodinger equation for the complex amplitude A: 'aA 2 i - + Al = \) IAI A, (5. 1) dT 1 2 where T =Et,~= E(x-cgt): cg is the linear group velocity and A ,\! are 1 1 known coefficients. Equation (5.1) is the simplest type of equation describing - 31 - weakly non-linear waves with strong frequency dispersion, just as the Korte weg-de Vries (KdV) equation is the prototype equation for weakly non-linear waves with weak frequency dispersion. Equation (5.1) is nowadays known under the name of the non-linear Schrodinger (NLS) equation. It is remarked that the NLS equation describes the non-linear evolution of a wave group with carrier wave number k and frequency w . For Stokes' waves 0 0 the non-linearity parameter is ka. The modulation parameter o measures the width of the wave group ink-space. It is seen in Appendix A, following simple arguments of Asano (1974), which resemble the arguments used for showing that for stationary non-linear fairly long waves one should have the Stokes number 2 (a/h)(A/h) = 0(1), that one has in fact ~ = o(x-c t) T = OEt. (5.2) g Because o ~ E, one often puts simply o = E and ~,Tare then expressed in x,t as mentioned previously. The case of deep water waves is considered at first in Section 5.2; one spatial coordinate can be considered then. Although deep water waves do not "feel the bottom", the case of deep water waves is treated here because the effect of taking the modulation parameter o and the non-linearity parameter E of equal order of magnitude, E/o ~ I, instead of taking E/o >> I as is done in Whitham's theory, is seen most clearly. Furthermore, when restricting the variation of 2 the wave number to be O(E) on wave length scale, the non-linear evolution of wave groups is described by the NLS equation (5.1). The NLS equation has soliton behaviour and is solvable by the inverse spectral transform. Waves on water of finite depth hare considered in Sections 5.3-5.5. In Sec tion 5.3 the evolution of wave envelopes in two horizontal spatial dimensions is considered for the case of an horizontal bottom (i.e., h(x ,x) = constant); 1 2 the derivation of the so-called Davey and Stewartson (D&S) equations from the governing equations for irrotational wave motion i's sketched. The D&S equa tions are 2 aA a A i ~ + A -- + µ OL 1 (j~ 2 1 2 2 (gh-c) ~ + gh K g 0~2 1 - 32 - with 11 , µ , V , V , K known coefficients and ~ = o(x -c t), n = ox , 1 1 1 2 1 1 g 2 T = OEt, c being the linear group velocity of the carrier wave. The D&S g equations consist thus of a NLS equation for the complex amplitude A coupled to a Poisson-type equation for a mean flow quantity Q. For deep water (kh + 00) the D&S equations reduce to the NLS equation (5.1) and the limit for shallow water (kh + 0) is also given. Subsequently, in Section 5.4, the case of the non-linear evolution of a wave envelope in one spatial dimension x = x is considered for a varying depth 1 h = h(x). The resulting equation is the following variable coefficient NLS equation with an inhomogeneous term 2 in which the role of ~,T has been reversed (T = E{fx dx/c (~) - t}, ~ = E x) g because the group velocity is a function of x. In the shallow water limit this equation is seen to reduce to a variable coefficient KdV equation (see also Chapter 7). Finally, in Section 5.5, the configuration h = h(x ,x ) is considered and the 1 2 waves are taken to be slowly varying on wave length scale (i.e., the variation of k is allowed to be 0(1) on ;\/o scale and not O(E) as is supposed in Sections 5.3 and 5.4). The resulting equations, which were derived by Chu and Mei (1970), are the most general of this Chapter, because the other evolution equations can be obtained from them as special cases (in absence of effects of surface tension). It is remarked that the derivations of the resulting evolution equations, as sketched in Sections 5.3-5.5 all have, as their basis, the following expansion for the free surface elevations and a similar one for the velocity potential oo n + s(x,t) = r n=I where E = exp{iX(1,t)} and s - s* and an asterisk denotes the complex n,-m - nm' conjugate. Note that m counts the m-th harmonic and n the order of the approximation. - 33 - 5. 2 Deep water Davey (1972) gave a general derivation of a non-linear Schrodinger equation for the propagation of a weakly non- linear wave whose energy is concentrated in a narrow band of wave numbers and which wave may be both dispersive and dissipative. He also showed how to obtain, through the same method of deriva tion, the Korteweg-de Vries (KdV) equation and Burger's equation. Only one spatial coordinate x is considered . A more precise der ivation for finite depth (with h = constant) in which also slow variations perpendicular to the direc t i on of propagation are a llowe d i s gi ven by Davey and Stewartson (1 974), while the limit for long waves (i.e. , kh • 0) is given by Freeman and Davey (1975); see Section 5. 3 . Chu and Mei (1 970) derived a set of evol ution equations for the case of a non- horizontal bottom h(x ,x )(see Section 5 . 5) . For the case of deep water , 1 2 i . e . , kh • 00 , and one spa tial coordinate x , these equations become (see Chu and Me i (1971)) 2 a 2 a ( ~) + ( c ~) 0 a-I w ax g w / 0 0 (5. 3) ak dW aT + ax o, where w = Mand c = g/ (2w) and X and Tare stretched variables (i . e., 0 o g 0 the scale of X is \/o) . The frequency w is now given by w (5.4) where normalized quantities are used. i.e. , a, k = 0(1) and E << I. It 1.s noted that Whitham also obtained Equations ('5.3), however , with w given by (from (4.15) with S, = 0 and kh • 00) J 1 2 2 2 w = w {1 + E k a}. (5.5) 0 2 Chu and Mei (1971) gave the fol l owing elementary argument for the necessity 2 of including the additional dispersion term E (a/ w )T /(2w a). Consider the 0 1 0 - 34 - superposition of two sinusoidal waves of equal amplitude a and different 0 (w.,k.). The resulting wave is written as acosX with J J 6k t,,w = 2a cos( x t) /:,k = kl - k2 ia(x, t) 0 2 2 I I X(x,t) = 2(k1+k2)x - 2(w1+w2)t. The variation of a is slow because /:,k/k., 6w/w. << I. Introduce k and w by J J ~k(x,t) = I ~w(x,t) = -8X/8t = - (w +w ) • 2 l 2 It is seen now that if the individual waves satisfy the dispersion relation w. = f(k.), it is not true that also w = f(k), except for non-dispersive w~ves. I;deed, one has w = ½{f(k ) + f(k )} = 1{f(k- ~k) + f(k+ ~k)}; upon 1 2 expanding in a Taylor series (/:,k/k << I) one obtains 2 4 (6k) 11 (t,,k) (4) w = f(k) + 8 f (k) + 384 f (k) + ... -I Because (/:,k) represents the length scale of the modulation, it is concluded 2 that direct use of w = f(k) leads to an error of O(o ); the term 2 c (a/w )TT/(2w a) in (5.4) is in effect due to the inclusion of this second 0 0 order modulation effect (c = o). It is shown in Appendix B how the NLS equation can be obtained from Eqs. (5.3) and (5.4) by restricting the variation of k to O(s); it is noted that Eqs. (5.3) were derived under the condition that the variation of k was 0(1) (in the stretched variables). Here it is noted that writing k(X,T) = k + ck(X,T), 0 (l) = (l) + cw(X,T) and X = X + ex, XT = -w, X = k and substituting these o o X expressions into Eqs. (5.3), leads to a set of equ~tions which can be written ~ as the NLS equation in terms of the complex wave amplitude A= a exp(iX): w 2 .{aA + __l__-5?.. aA} -s w0 a A I 2 12 1 ------cw k IA A= O, (5.6) dT 2 k ax k2 ax2 2 o o 0 8 0 which becomes Eq. (5.l) after introduction of the moving coordinate frame s,T - 35 - with T = ET, t;, = (X-c T) and c = !~ = w /(2k). The coefficients A and g 2 g 2 0 0 0 1 \\ are then ;\ = -w /(8k) and v = w k 2/2. 1 0 0 1 0 0 Once A(t;,,T) is found from the NLS equation, the free surface elevation C(x,t) follows from 2 C(x,t) = iEA(t;,,T) exp{i(k x-w t)} +CC+ O(i:: ), (5. 7) 0 0 where CC denotes the complex conjugate. Yuen and Lake (1975) used the average Lagrangian technique to derive Eqs. (5.3) and (5.4) by taking into account the modulation term (5.4). They con sidered 1 2 . s = a(ox,ot)cosX + 2 ka (ox,ot)cos2X + •••• (5.8) which is the second-order Stokes' expansion for deep water waves; as the corresponding expansion for the velocity potential, correct to o(o,E) (E = ka), they took wa . X kz [ at X wax ( 1 k ) ] kz ~ = k sin .e + k cos + k2 - z cosX e + (5.9) 2 wa . 2kz + -2- sin 2x.e + ... The term with (1-kz) was found to be necessary to ensure that~ fulfils the Laplace equation to the order considered. The terms between the square brackets in (5.9) result from including the O(o) terms. The average Lagrangian 2 2 tthen becomes, retaining terms of O(o ,E ), 2 2 2 2 at wa a -p -18._, = _ W a + ~ + + X t 7;k 4 4k 4k2 (5. 1 O) 2 2 + -3 -w aa + _g_k a 4 • 8 k3 XX 8 2 2 Note that in deriving expression (5.10) for~ terms like a k are ignored in 2 2 X comparison with terms like a k because the variations of k and ware taken X to be much smaller than the variation of a in accordance with the nearly monochromatic wave train assumption on which the NLS equation approximation - 36 - is based; see also Appendix B. Furthermore it is noted that using Whitham's theory (in which 6 <<£and therefore also no terms with k ,k, etc. are X t present)cl becomes 2 -lo u.i2a I 2 I 2 4 -p ctv= - ~ + 4 ga + 8 gk a, ( 5. I 1) which expression follows by taking the deep-water limit of (4.8) together with the condition that no pre-existing current u is present, so that lsl is itself a small quantity; see also the text after equations (5.51). Variation of 8..,of (5.10) with respect to X yields '\_2 Od 0 (5. 12) at and variation with respect to a yields the dispersion relation 2 w2 = gk( I +k 2 a 2 ) + a- '{ a + ~ a + 2 wk axt }-, (5. I 3) tt k2 XX I w I 1 where, in the right-hand member, cg = k = (g/k) 2 has been used. 2 2 1 2 Using the leading order of (5.12) and substituting w = (gk) , cg = ½(g/k)½ in the right-hand side of (5.13), the dispersion relation can be written to the same order of approximation as 2 2 2 l a .... w = gk(l+k a+ ;~), 4 (5.14a) k a or ½ I 2 2 I axx w = (gk) (I+ 2 k a+ 8 k2a). (5.14b) (Notice that this is the same as the dispersion relation (5.4), when, in (5.4), 2 2 2 2 2 8 a/8T is replaced by c a a/ax . g Remark: It is noted that (5. 13) is now not simply obtained from a~aa = 0, because &..is not only a function of a but also from its derivatives. - 37 - Because the arguments of &-are functions and not functionals one obtains from ~a ffg,(w,k,a,a ,a ,a ,a ,a )dxdt = 0, u t X tt XX tX the condition a&- a a~ a a~ a2 a~ a2 ag, a2 a~ - - -(-)- -(-)+ -(--)+ "t"x(~a)+ 2(--;.;--a ) = (5. 15) cla clt cla clx cla 2 cla a a a -a a o. t X d t t t tx X XX In the derivation of (5.13) from (cS/cSa) JJS.-dxdt = 0 with &.-given by (5.10) all terms with derivatives of wand k to x and/or t have been neglected. As usual, the system of equations is completed by addition of the consistency relation (5. I 6) As Eq. (5.12) and the first of Eqs. (5.3) are equivalent within the order of 2 approximation for which they are derived (terms like a clw /clT were neglected), 0 the resulting sets of equations (5.3), (5.4) and (5.12), (5.14b), (5.16) are equivalent. 2 2 2 2 Without the condition a k << a k a more complicated expression for~ can be X X derived by using expressions (5.8) and (5.9) for ~(x,t) and ~(x,z,t); this results in equations for the propagation of deep-water waves which have the same range of validity as the more complicated equations (5.49) for waves on water of finite depth. The important point of the 1975 paper of Yuen and Lake is that it is shown that the method of the average Lagrangian is also applica ble for E ~ o, once one consistently takes all terms into account. Whitham (1974, p. 526) gave a general form for the average Lagrangian when - -1 higher-order dispersion is to be included. Writing L = (pg) &, expression (5.11) can be written as (5. I 7) - 38 - w2 I 2 where G(w,k) gk - I and G = - k . 2 2 2 Note that G(w,k) = 0 is just the linear dispersion relation and c a is the 2 non-linear correction to it obtained by Whitham's theory. Upon variation of - L to a one obtains, using (5.15), 3 2Ga + 4G a -{G a + 2G ka + Gkka } = 0. 2 WW tt W Xt XX This expression does not result in the dispersion relation (5.14b). It turns out to be necessary to have -2Gwkatax in (5.17), resulting in 3 I Ga+ 2G a + -{-G a + 2G ka - Gkkaxx} 0. (5.18a) 2 2 WW tt . W xt Note that this expression was also given by Davey (1972). We conclude there fore that the general expression for the average Lagrangian in which higher order dispersion is accounted for is: (5. 18b) Furthermore, it is noted that Hui and Hamilton (1979) also used an expression like (5.18b) instead of (5.17) although they referred to page 526 of Whitham (1974). We remark that in the example of Whitham both Gwk and Gkk were zero, as is 2 the case when G(w,k) is written as w - gk and the dimension of ~ is taken differently; in that case (5.14a) is obtained immediately, without the necessity of using the order relation 8a/8t + c, 8a/8x = O. Yuen and Lake 2 (1975) also considered G(w,k) = w - gk in thei; Appendix. Remark: Writing G(w,k) = w - 0.(k) in (5.18a) there results 2 3 I 8 0. {w - 0.(k)}a + 2G a 2 + 2 8k2 .axx = O, and the dispersion relation can therefore be written as W-oo:... n(k) - 2G a----•--2 I (820.·.) axx 2 2 ak2 a ' 0 - 39 - where ( ) denotes that the expression between brackets has to be eval- 0 2 uated at the carrier wave number. The term-2G a is now written as 2 2 2 (aw/aa ) a (see Davey (1972)) so that 0 a XX w = rt(k) + a Defining now (5.18c) the dispersion relation is written as a 2 XX w = rt(k) +Va - A (5.18d) 1 1 a These coefficients V and A are the coefficients in the NLS equation 1 1 ( 5. I) : a = IAI • See also Karpman (1975, Section 27) or Kadomtsev and Karpman (1971), where an heuristic derivation of the NLS equation is given. By extending the method of derivation one order higher, Roskes (1977) derived the following fourth-order envelope equation: 2 2 3 · where \ = -w/(8k ), v = wk /2, a = w/(16k ), 8 -wk/8, s = ka. 1 1 1 1 I The quantities are non-dimensional and w = k 1 is the linear dispersion rela- . tion; a moving coordinate frame moving with the group velocity w/(2k) has 3 3 . been adopted. The term a a A/ax represents the effect of fourth-order 1 2 2 dispersion and JAl aA/8x + A aA* /ax represents the higher-order effect o1 y1 - 40 - of non-linearity. It is striking that even to this order no terms with derivatives of k and w to x and tare present; these terms could be neglected in the derivation of the NLS equation (5.1) because of the "constant" carrier wave. See also the very general multiple-scale technique as described by Catignol (1977, 1972). 2 • • ( ) • The E non-linear terms in Eq. 5.19a act as symmetry-breaking terms, as observed in the experiments of Feir (1967) on stability of Stokes' wave trains. Some solutions of the NLS equation are given in Appendix C. The simplest solution of (5.1) is the Stokes' wave train in which A is a function of T only: 2 A= A exp(-ivlA T) A constant. 0 0 0 Much attention has been paid to the stability of Stokes' waves in literature; instability is found to occur when AV < 0. Because A < 0 when no surface 1 1 1 tension is present, instability occurs when v 1 > O; for waves on water of finite depth hone finds v = 0 for kh ~ 1.363. Near V = 0 higher-order 1 1 terms have to be included; Johnson (1977) derived for this case 2 cl A cl A I , 2 4 . I , 2 clA cl I , 2 i clT - ala,2 - a2 A A+ a31AI A+ i(a4 A ar: - a5A ar:( A )) + (5.19b) 2 2 4 with ell/;/ cl = A , the a. are real numbers and s = E (x-c t) and T = E t s I I i g 2 (and thus T =ET, s = E,). In case of instability soliton-like wave envelopes are produced, which are, however, flatter than true soliton solutions of the NLS equation (see, e.g., Eq. (C6)) and are skewed, which phenomenon was also predicted by Roskes (1977), who obtained terms similar to those with coefficient~ a 4,a5 here (see Eq. (5.19a)). Only one-dimensional wave equations have been given in this Section. Atten tion is drawn to the fact that a two-dimensional non-linear Schrodinger equation for deep water waves is obtained as the limit of the Davey and Stewartson equations (see next Section); for kh + 00 one finds (see Eq. (5.29)) - 41 - with n = cx and µ > O, and the waves are propagating in the x direction. 2 1 1 A short discussion of this equation is given in Appendix D. Special solutions are given by Hui and Hamilton (1979), who also applied this equation to ship waves. 5.3 Propagation in two horizontal spatial dimensions; constant depth In this Section the propagation of wave packets in a fluid domain which is unbounded in (x ,x ) is considered; the still water depth his taken constant. 1 2 Davey and Stewartson (1974) derived a coupled set of equations for the complex amplitude A of the envelope and a variable Q which is related to the wave induced mean flow. At= 0 the free surface elevation is given as gl;(x,O)• = icwa(ox ,ox )exp{ikx} + CC. (5. 20) 1 2 1 This represents thus a progressive wave in the x -direction with a slowly 1 . 2 -I varying amplitude; a is measured in ms i.e., a is a measure for the velocity potential q>(*,z,t). The governing equations are the Laplace equation for q> in -h < z < l;, the kinematic conditions at z ~-hand z = l; and the dynamic condition at z = l;. A solution of these equations is supposed to be of the form 00 00 • l;(x,t) = E l; Em q>(*,z,t) = E cp Em m m m=-oo ' m=-oo (5.21) l; = ,. , cp• = , E = exp{i(kx -wt)}. -m m -m cpm 1 Subsequently, l; (*,t) and cp (~,z,t) are written as m m - 42 - 00 i'.; = I Eni'.; (s,n;r) m nm n=m 00 n = I E 2 (E = o) • = s(x 1 -c g t) , n = EX 2 , T =Et c is the linear group velocity g (5.23a) based on the linear dispersion relation I w = {gko}2 a= tanh kh. (5.23b) The zeroth-order, zeroth-harmonic terms i'.; and 2 2 . acpnm 2 2 = km Using the bottom condition a/az = 0 at z =-hit is seen that cosh k(z+h) cosh 2k(z+h) = A(s,n,T). = F(s,n,T). cosh k(z+h) 3A (z+h)sinh k(z+h)-hocosh k(z+h) = D(s,n,T) - i Subsequently the expressions for i'.; and are substituted into the kinematic and dynamic boundary conditions. The coefficients of snEm are equated to zero - 43 - for n = 1,2,3 and m = 0,1,2. The quantities n = 1,2, m = 0,1,2 can then snm , be calculated. For n = 3 the component a~ /3z of the vertical velocity gives 3 0 a contribution in the kinematic condition; eliminating a differential s20 equation for~ is found: 1 0 (5.24) where c = w/k is the phase velocity. Also an equation for ~ follows; this 30 3 one is ~ot used further, the significance of the E terms is that O(E) quan tities can be calculated fully. 3 1 The coefficients of E E in the kinematic and dynamic free surface conditions ield two equations form and r at z = 0. Elimination of [ (or m ) Y - r31 "31 '31 ' '31' from these two equations shows that these equations are only compatible when the following condition is fulfilled: 2 • 3A { 2 2 a A 2lW - - C gh(l-o )(l-kho)}-2 + dT g at.: (5.25) 1 4 2 2 4 2 2 2 d~lO = 2 k {9o- - 12 + 130 - 2o }!Al A+ k {2cp + cg(l-o )}A~ It is noted that Eqs. (5.24) and (5.25) describe the evolution of the pro gressive wave to first order ins;~ is the first-order mean current and 1 0 A is the amplitude of ~II' evaluated at z = 0. Equation (5.24) is an equation of Poisson-type and Eq. (5.25) is a NLS equation, coupled to the mean flow. Introduce now a quantity Q(~,n,-r) by (5.26a) It is noted that the mean surface elevation is then given by s20 2 2 = k Q _ k{o + 2kh(;-o )} (5.26b) g gh - c- g Eqs. (5.24) and (5.25) can be written as - 44 - 2 2 clA cl A cl A l -+ A -- + 1-1-= \) IA I 2A + \) QA 1 2 clT lc)~2 i an2 (5.27) 2 2 a Q a2 K a2IAl2 (gh-c ) - + gh _g_ = 2 g a~2 an i an2 where < w' (k) C g > A = _!_ w" (k) = 0 = 0 1 2 1-11 2k - 2k 4 2 20 2 2 = --k [ 9-100 2 +90 4 - 4c c (l-0) + gh( l-0 /}] \) 1 2 2 {4c! + p g 4w0 gh - C g (5.28) k4 2 > = ~ {2c + c (l-0 )} = 0 \) 2 2wc p g g K = ghc }2cp 1 gi gh - Equations (5.27) are sometimes referred to as the Davey and Stewartson equa tions (D&S) Eqs. It is noted that, in absence of capillarity, one always 2 has for waves of finite length gh - c > O, and thus Q satisfies an equation g of Poisson-type (i.e., an elliptic equation). Note that Equations (5.27) are derived under the condition that E = ka, cS =A/A= 6k/k and 6k /6k are all I 2 1 of the same order of magnitude, where k = (k~+k;) 2 • The deep water and shallow water limits of Eqs. (5.27) are considered now and subsequently the inclusion of surface tension is considered. Deep water limit In the limit for deep water one has kh + 00 and 0 + I; keeping now Ekh << I (and thus, in fact, cSkh << I or the length of the wave group is much larger + than the depth), Eq. (5.24) reduces to grad ~ = 0 and Eq. (5.25) reduces to 1 0 - 45 - Notice that (5.29) is in fact uniformly valid for kh • 00 (the condi~ion Ekh << 1 need not be imposed) because the troublesome terms cancel in the next approximation, see Longuet-Higgins (1976). When A is independent of n, this equation reduces further to the NLS equation. Notice that here A is the amplitude of the velocity potential evaluated at z = 0 and therefore the coefficients differ from those in Eq. (5.6), where A is the amplitude of the free surface elevation. Shallow water limit Davey and Stewartson gave the following equations as the shallow water limit (kh + O, c + c) of Eqs. (5.24) and (5.25): g p (5.30) a2cp i 0 +--- an2 The validity of these equations was checked by Freeman and Davey (1975) who started from the original equations of motion and introduced the appropriate long wave scaling. They showed that the double limit E • 0, kh • 0 in the 2 asymptotic expansions for 4i and I;; was uniform when E/ (kh) « I. Because E 2 measures the wave slope, i.e., E '\J ka, the condition E/(kh) << 1 can be a 1 2 written ash. kh « I. The Stokes parameter is defined by (a/h)/(kh) ; 2 therefore, the condition E/(kh). << 1 does not necessarily imply that also the Stokes parameter is taken much smaller than one; note that Djordjevic 2 and Redekopp (1978) defined E/(kh) to be the Stokes parameter, while E me.<1s11red the wave slope. 2 When E/(kh) is taken to be 0(1), Freeman and Dav~y obtained the two-dimen sional generalization of the Korteweg-de Vries equation as given by Kadomtsev and Petviashvillii. When A and (5.31) Inclusion of surface tension Djordjevic and Redekopp (1977) extended the analysis of Davey and Stewartson (1974) in that they took into account the effect of surface tension. The linear dispersion relation then becomes ~ l ~ w {gko(I+T)} 2 o = tanh kh T = (5.32) and T is the surface tension of the fluid. The group velocity is then 2 c = c }o + kh(1-o ) + T l c = w/k. (5.33) g P} 20 I + T ~ p -I -3 -2 For water with T = 0.074 Nm and p = 1000 kgm , and with g 9.81 ms one 2 has T = 7.54 ~ 10-i k . Djordjevic and Redekopp used the same method to derive the evolution equations as was used by Davey and Stewartson. It is found that the second harmonic term 2 2 becomes singular when = o /(3-o ); in this case one has second harmonic s22 T resonance, that is, when the wave numbers of the first and second harmonic wave component fulfil this condition, then the phase velocities are equal. Assuming that the wave number is not too close to that for which 2 2 T = o /(3-o ), the third-order ~erms can be considered. Instead of Eq. (5.24) now the following equation is obtained for the leading-order mean flow ~ : 10 (5.34) Introducing a quantity Q by Q (5.35) - 47 - the set of equations (5.27) is recovered, where the constants are now given by C (Jj I (k) ___£ > A = _!_ w"(k) = - 2k O, 1 2 2k 4 ~ 2 2 ~ 2 2 2~ = .!:_ (1-0 )(9-0) + T(3-0 )(7-0) + 802 _ 30 T + \\ 4w 02 - T0-02) I + T 2 2c g 2 ~ (gh-c ·) ( I +T) g . 4 K i 2 ~ l > = (1-0 )(l+T)i = o, \\ (Jj l:: + 2 2 ~ 2c + c (l-0 )(l+T) p g ghc . Kl = 2 ~ g (gh-c )(l+T) g ~ Relations (5.36) reduce of course to relations (5.28) when T + 0. Notice that in Djordjevic and Redekopp (1977, Eqs. (2.17)) a misprint is present; the third term between square brackets in the formula for v 1 in (5.36) should have the numerical factor one and not four, see also Eqs. (5.28) and Eqs. (5.42) which are due to Djordjevic and Redekopp (1978). Djordjevic and Redekopp (1977) now mention the following properties which are different from the case where surface tension is neglected. - The influence of surface tension is to increase the group velocity; it becomes then possible that c > /gh. g By expanding 0 for small values of kh and taking T to be of order (kh), a 2 first assessment gives, neglecting terms of O{(kh) } that for c to be ~ 2 -4 g greater than /gh there should be 9T > 4kh and thus k . I O > 5. 89 kh. For kh = .I one obtains k > 76.8, or A= 2n/k < 8.2 cm and h .13 cm. We thus conclude that, in our case, we have usually c < /gh and the equation for g Q will remain elliptic. . . I ,2 . . = vghr-;- d - The coefficient v 1 of the term A A is singular when c an when 2 2 g T = 0 /(3-0 ); for c = /gh one has long wave/short wave resonance in which g the group velocity of the short (capillary) waves equals the phase velocity 2 of the long (gravity) waves. Near c = gh another scaling of the independent g - 48 - variables has to be used and the asymptotic expansion of~ and, is also . 2/3 4/3 213 different:~=€ ~ + €~ + € ~ + ... , and~= s (x -c t) n 1 2 1 g ' 2/3 4/3 ° , c x 2 T = € t. In this case the following set of equations are derived where then dependence is dropped: = BA -a (5.37) 1 C 2 ~ where B(~,T) = cS d 0. 1 This set of equations implies that the evolution of the long wave is forced by the self-interaction of the short wave and that the short wave is modu lated by the long wave. Such a·phenomenon is important in wind wave growth studies. An application of the D&S equations One of the important points is that, on water of finite depth, the modulation of a wave gives rise to a variation of the mean water level and a variation of the wave-induced mean current; these variations are long waves and are - !2 propagated with characteristic velocities close to+ (gh) • Besides the for- mation of a non-linear wave group, these long waves are also generated. Larsen (1978a) considered the effect of these long wave components on the pressure in the fluid and correlated the beat of surface waves to the observed pressure fluctuations at a depth of 4000 m. Larsen (1978b) investigated the depth-dependence of coherent wave packets for kh > 1,363 and derived an ex pression for the pressure fluctuation at the bottom. Therefore, the passage of a wave group results in pressure fluctuations with a period that is much longer than the period of the carrier wave and these fluctuations can be of importance for the onset of sand transport, for waves with carrier wave lengths which themselves are not long compared to the depth. 5.4 Propagation in one horizontal spatial dimension; uneven bottom In the same way as a dispersive long wave equation such as the Korteweg-de Vries (KdV) equation for water of constant depth can be extended to a KdV like equation for the case of varying depth, h = h(x), an inhomogeneous NLS equation can be derived for the propagation of wave packets on an uneven bottom. In both cases the reflection has to be neglected because both the - 49 - NLS and the KdV equations describe waves propagating in one direction only. Djordjevic and Redekopp (1978) gave a derivation of an inhomogeneous NLS equation in a way which is similar to that in which the Davey and Stewartson equations (5.27) are derived. The depth is slowly varying and is supposed to 2 2 be a function of E x, that is, h = h(E x), where Eis again a wave slope 2 parameter. In fact, because o =Eis taken, we have h = h(o x), with o = A/A; because o is the modulation parameter and the modulation of the carrier wave gives rise to a wave group, A may be seen as a measure for the horizontal 2 extent of the wave group. The condition h = h(o x) is equivalent to saying that the depth only changes little over an horizontal length A. Because the group velocity is now a function of hand therefore a function 2 of Ex, the following multiple scales are introduced now rfx d 2 T = E{_/ C (~) - t} € x. (5.38) g Notice that the role of T and~ is reversed compared to Section 5.2 and 5.3. It is supposed that w is constant, i.e., no temporal variation of the medium is considered. It is supposed that the group velocity c , the phase velocity g c and the wave number k can be locally defined as function of the local p depth h(~) and therefore the variation of c , c , k is with~' cg(~), cp(~), g p k(~). The free surface s(x,t) is expanded as 00 E n=l with E = exp{i[Jx k(~)dx - wt]}, sn,-m = s~m' A similar expansion for~ is used. It is seen that this expansion is similar to the one used earlier in Section 5.3, the difference being an adoption to the non-µniform depth which neces sitates a coordinate frame moving with a non-uniform velocity so as to remain near the centre of the wave group. Proceeding in the usual way, one finds 3 0 3 1 from the E E terms an equation for ~ 10 (~,T) and from the E E terms an equa tion for the amplitude A(~,T) of the solution for ~11 (~,z,T) emerges (~ 11 has the same sort of solution as given below Eq. (5.23)). Introducing the quantity Q(~,T) by - so - 2 2 a,1, k c 't' 1 0 gQ = -- + -~g-2 (5.39) dT gh-c g the equation for c/l becomes simply 1 0 aq - with the solution Q Q (s), (5.40) 3T - o, 0 and the equation for A becomes 2 \) f Al A + \) Q A, (5.41) 1 2 0 where the coefficients are given by 2 ( I -0 ) ( I -kh0) d(kh) 0(l-kh0) k' µl = 2 k 0 + kh( l-0 ) as 2 gh 2 a w ;\ = --- { I - -(l-kh0)(l-0 )} - 1 2wc 2 3 2 g C 2c ak g g (5.42) 2 2 20 C 2 C k4 g 2 \) l = [9-100 2+90 4- + 4 ic1-0 ) + 2 2 C :~(l-a2J2(] )<:) a 4w0 c gh-c g g b g C C 2 _.E. {2 _.E.+ 1-0 } ~ o. C C g g 2 2 Notice that (l-0 ) occurs in the last term between curly brackets in the 2 expression for v and not (l-0) as is given by Djordjevic and Redekopp 1 ( 19 7 8, Eq. ( 2. I 7)) . It is noted that Equation (5.41) can be written in a simpler form upon appli cation of the transformation (5.43) - 51 - the resulting equation for B then reads -iµ B. (5.44) 1 It is thus clear that the term v Q A only gives a phase shift. The difference 2 0 between the NLS equation (5.15) and the inhomogeneous NLS equation (5.44) is A the term -iµ 1 B in (5.44); another difference is that 1 and V1 in (5.44) are functions oft;;. Equation (5.44) now describes the evolution of wave packets propagating over an uneven bottom, under the condition that reflection can be neglected and, 2 consequently, the depth varies very slowly (h = h(E x)). Only for constant depth one has µ = 0 and A ,v are constants. 1 1 1 It is furthermore noted that Eq. (5.44) can be reduced to the homogeneous NLS equation with t;;-dependent coefficients by introducing the transformation B(t;;,T) = a(t;;)D(t;;,T). By substitution it is found that choosing (da/dt;;)/a = -µ one obtains 1 (5.44a) with (5.44b) 2 In the shallow water limit kh + 0 one obtains, when E/(kh) << I (and thus, (a/h)/(kh) « I), the following_ expressions for the coefficients µ , A , V 1 1 1 and v : 2 I k' I h' w l = A = hz µl -zk= 4h 1 2g3/2 (5.45) _2__w_h-7/2 l_ ~ h-3/2 = V = Vl 4 3/2 2 2 - g g2 where a prime denotes differentiation to the argument. For AV > 0 (which is always the case in shallow water) the governing equa l 1 tion for the envelope-hole solution has to reduce to the inhomogeneous KdV - 52 - equation with variable coefficients. That Eq. (5.44) reduces to this gener alized KdV equation can be seen as follows. The real and imaginary parts of Bare separated by writing B = R(~,T) exp{i!T 0(~,T)dT; Rand 0 are then ~ ~ ~2 ~ ~2 expanded in a small parameter o, R = Ro+oRl+o R2+ ..• , 0 = 081+0 02+ ... where o is a measure of the slope of the modulation of a wave train about a uniform finite amplitude state. Note that the discussion is restricted to any small amplitude long perturbation of a wave which is modulationally stable (A 1V 1 > 0). The following further coordinate stretching is introduced: T = 8½{f~ ~ - T} X = 63/2~. C(X) Substitution of the expansions for Rand 0 yields expression for R (X) and 0 C(X) and a relation between 0 (X,T) and R (X,T). The secularity condition 1 1 for R2 and 02 yields the generalized KdV equation, which can be written, with R = h?/S H(X,T) as 1 9w w 3 oH o h-5/8H oH + 0 hi I /4 o H = 0 312 312 3 ax 2g ar 12r g oT 0 -1 /4 where R = r h . 0 0 5.5 Propagation in two horizontal spatial dimension; uneven bottom In this Section we at last deal with the problem of surface wave propagation in two horizontal spatial dimension in water of depth h(x1,x2) under the condition that the wave modulation parameter o is of the same order of mag nitude as the non-linearity parameter E (= ka). Chu and Mei (1970) derived evolution equations for the wave parameter in this case. They considered modulated Stokes waves over a slowly varying bottom. With o = s, the slow scales X = E~ and T = Et are introduced and the expan sions for ~(1,z,t) and s(t,t) are taken as - 53 - co n n -+ 1 -+ co n n -+ I -+ z;; (1, z, t) = I € I (X,T)exp{- 1m X(X,T)} (5.46) snm s n=l m=-n t-n,-m and the depth is slowly varying, h = h(X). The wave number vector k and the frequency ware, as usual, given by k. = ax/ax. and w = -aX/aT. Note that the -+ J-+ J -+ variation of k is allowed to be 0(1) on X scale. X, k and ware expanded as 00 00 2j = " 2jx = = E: 7) X t... E: 2 j k w w .• (5. 4 J=O 2J The expansions are substituted into the governing equations for irrotational wave motion. For each pair of indices (n,m) a set of equations is obtained. The evolution equations for the wave parameters now consist of the solvability conditions which are needed to ensure bounded solutions at higher order. A detailed account of the method can be found in Chu (1970). Here, we only quote the results. Writing the solutions of cp and l;; as 1 1 1 1 = -i 1L A cosh k(z+h) = A cosh kh = a, (5.48a) W 11 11 2 0 (and thus a is the amplutide of the first-order, first-harmonic wave), and introducing the following quantities 2 E = ga b = r B. = a,h /ax · (5.48b) 2 c, 2 0 J 'I' 1 0 , for the wave energy, the mean surface elevation and the mean current respec tively, the following set of evolution equations is obtained. - 54 - c1b a E "'T + -;:,--X (h[3. + - k . ) = 0 o o . J W OJ J 0 (5.49) as. a I i 2 _i + - {gb + - k ~ E} = 0 0 = tanh k h aT ax. 2 o 0 0 i where 0 = tanh k hand the terms w represent the effects of Stokes amplitude 2 . . S o D U • dispersion w, mean depth change, w, mean current, w, spatial modulation, X 2 T 2 2 w , and temporal modulation, w • These terms are given by 2 2 2 (k a) s 0 = w D w2 0 16 0 u w k (5. 50a) 2 0 j f3 j 2 D _ 1 kb l-0 w2 - 2 -0- and 2 T 1 a a w2 = -2 -2(-) (= _1_(_1_ a2~ /aT2) I ) 2w ~ 11 z=O a dT Wo 0 11 + + k ~ }I ' + (5.5Gb) acos ~ k h [vh.{-i v~ 11 o 21 z=-h 0 +Jo V.(k ~ )}cash k (z+h)dz] -h 0 21 0 Upon comparison with Whitham's results, it is seen that the terms (5.50a) - 55 - are the only ones included in the equation for conservation of wave crest in Whitham's case, see Eq. (4.15). The modulation contributions w; and w; are included because o and E are taken of the same order of magnitude. The ex- X pression for w can be worked out by substituting the (known) expressions for 2 ~ and ~ ; for the case h = h(X ) this is carried out in Appendix B of Chu 11 21 1 and Mei (1970). We only remark here that there occur first and second deriva tives to X of h, k and A in that expression. l o 11 Once the solution of the set of equations (5.49) is known, all functions~ , nm snm' n=l,2, m=0,1,2 can be calculated and the complete second-order solution of slowly varying Stokes waves is known then. It is noted that ~ and s 21 21 are a consequence of the modulation o only, they are linear in a. The resulting solution can be written as l 3 (5.51b) It is thus seen that the wave front is not strictly vertical because 0 depends on z. Considering the evolution equations (5.49) it is seen that the principal difference with Whitham's equat~ons, Eqs. (4.13) and (4.14), is the inclusion of the higher order dispersion terms w! and w; in (5.49). It is noted that the first three equations of Eqs. (5.49) coincide exactly with the corre sponding equations given in Whitham (1967a), but the second of Eqs. (4.14) 2 for the evolution of S contains½ s in the flux, whereas this term is absent in the corresponding equation of (5.49); the reasdn is that, in Chapter 4, only the variation of S was supposed to be a small quantity and S could be 0(1) (for example a pre-existing current) whereas in this Section Sis the wave induced current and thus S itself is already a small quantity. With = 3~ /3X. an O(E) quantity, it follows from the second of Eqs. (5.49) S. 10 J J 2 2 2 3 that as.fax. is O(E ), because b = O(E) and thus (a/aX.)(S) is O(E ). J J J - 56 - Hoogstraten and Van der Heide (1972) derived by a slightly different method for the case of an horizontal bottom a set of equations which is the same as the set to which (5.49) reduces when h = constant is substituted. When, in Eqs. (5.49), the variation of the wave number and the frequency on -+ • X,T scales is restricted to be O(E), then the bottom slope has to be also of -+ • 2 O(E) on X scale, that is, of O(E) on natural scales (wave length scale). -+ For such small variations of k and w, the Davey and Stewartson equations (5.27) should result when the bottom is horizontal and the waves are nearly uni-directional (i.e., ok /k << ok /k). In fact, the set of equations (5.49) 2 1 includes all other model equations given in this Chapter, except the ones in which effects of surface tension are accounted for. When solving Eqs. (5.49) numerically, special attention is required for those cases in which a becomes zero or closely to zero because of the term 2 2 (1/2a)o (a/w )/oT in w; and similar terms which occur in w~. This is pre 0 cisely the reason why the NLS equation for the complex wave amplitude is preferred to the set of equations for the real amplitude and the phase, see, e.g., Eqs. (BIO). - 57 - 6 A discussion of the significance of the two methods Essentially two different methods of derivation of the evolution equations for the wave parameters are considered in the previous Chapters. Both methods rest on multiple scale expansion, in the first case (Chapter 3) applied to the variational principle and in the second case applied to the governing equations. First Whitham's method of using an averaged Lagrangian density was considered, resulting in an hyperbolic system for the wave parameters. Because of the hyperbolicity and non-linearity of the system, there will occur ultimately crossing of characteristics and the equations become invalid. This can be prevented by taking higher-order dispersion effects into account; it is then necessary to take o and E of the same order of magnitude. By taking account of these higher order modulation terms, the average Lagrangian approach is able to yield the same evolution equations as are obtained from a direct multiple scale expansion with E = o, The results of Whitham's theory may be obtained from the NLS equation (5.1) on making the assumption that the phase variations, although small, are much larger than the amplitude variations; putting A(s,T) = R(s,T) exp(i0(s,T)) 2 2 2 this means R(d0/ds) >> a R/as (see also Eq. (BIS)). For the deep-water limit of the D&S equations, Eq. (5.29), Whitham's results are recovered by 2 2 2 2 neglecting the terms with a R/as and a R/an ; these terms are, however, responsible for the possibility of wave groups traveling in a direction dif ferent from that of the carrier wave, see, e.g., Longuet-Higgins (1976) or Hui and Hamilton (1979). A uniform solution of the NLS equation (5.1) may be obtained on assuming A 2 to be independent of s: A(T) = A exp(-iv A T), A constant. This solution is 0 l o O stable to relatively small long wave disturbances only for A V > O, a condi- 1 1 tion which leads to kh < 1.363. The reason why the NLS equation gives the same stability criterion as obtained from Whitham's theory, in which the 2 2 2 simplifying assumption R(a0/as) » a R/as is made, is that all disturbances of sufficiently long wave lengths are unstable for A V < 0 and these include 1 1 those for which this assumption is valid. Whereas it followed from Whitham's approach that the instabilities were growing in time without bound (and Whitham's approach is equivalent to the sideband-disturbances approach of Benjamin and Feir (1967)), it was found by Chu and Mei (1971) that the instability of a uniform wave develops into - 58 - stronger modulations that settle down into groups of waves of finite form. These groups of waves are nowadays found to be described by so-called envelope solitons of the form (C6). Lake et al (1977) and Yuen and Lake (1978) considered the evolution of non-linear wave trains as described by the NLS equation for deep water waves and found the time periodic return of the un stable wave train to its initial form; this behaviour was confirmed by experiments. When generating a wave group in a wave flume by a slightly modulated oscillating wave board, it is found that this wave group changes form and returns to its original shape after some distance, unless the initially generated wave envelope is precisely of the form (C6), in which case this form remains constant in shape over very long distances. This re current behaviour can be compared with the recurrent behaviour of fairly long waves in wave flumes, This recurr.ent behaviour of wave groups could be expected by analogy with fairly long waves with Stokes number 0(1), because it was seen earlier that E ~ o is analogous to taking the Stokes number order 2 2 one for fairly long waves (i.e., a/h ~ k h ). Most of the attention has been directed to bottom geometries which are hori zontal. The average Lagrangian technique can be applied to those cases where • • • • h = h(ox), U = U(ox,ot), where o is the non-uniformity parameter of the medium. (Note that in Chapters 3 and 4 µ was used instead of o.) When wave groups are considered then the variation of k is O(os) and the variation of h has to be of the same order of magnitude. In the one-dimensional case a perturbed NLS equation, with coefficients which are a function of~ can be derived, see Eq. (5.44). When the perturbation term can be considered to be small, perturba tion techniques can be used in which the solution of the unperturbed NLS equation, which can be obtained by the inverse spectral transform, acts as the "zeroth-order" solution, see, e.g., Kaup and Newell (1978), Newell (1978a), or the review article of Newell (1978b). The idea behind it is closely akin to the cnoidal wave refraction which is treated in next Chapter; note, how ever, that Eq. (5.44) is a one-dimensional wave equation. We consider it to be possible to derive a set of D&S-like equations, in which ~ is the coordinate along a wave ray and n the coordinate perpendicular to the ray. This seems only feasible for so-called quasi-two dimensional situa tions, in analogy with the inhomogeneous KdV equations. Because the propaga tion of a-wave group is then the issue, the wave rays are to be calculated by taking the group velocity c of the carrier wave; c is just the "phase- g g velocity" of the wave envelope. The concept of basic group velocity, defined - 59 - by Hayes (1973) as 8~8A8k., A wave action, (see Chapter 3) can be of use J then. Note that, according to Djordjevic and Redekopp (1978), the inhomoge- neous NLS equation (5.44) can be reduced to an inhomogeneous Korteweg-de 2 Vries equation for kh + O, with (ka)/(kh) << 1 (see page 52). The most general approach to solve the problem of non-linear wave propagation over an uneven bottom which is mentioned in the previous chapters is to solve Eqs. (5.49) with (5.50) numerically, perhaps after the inclusion of the pos sibility of pre-existing slowly varying currents U(t,t). In order to be able to interpret the results, various aspects of the non-linear Schrodinger equa tion descriptions have to be further investigated, especially for uneven bottom geometries. - 60 - 7 Fairly long waves 7.1 Introduction Fairly long waves are waves with h/A rates such that the Stokes number 2 (a/h)(h/\) is of order one, or, otherwise stated, long waves in which the vertical acceleration has to be taken into account. It is known that these waves can be described with reasonable accuracy by Boussinesq-like differen tial equations, which can be obtained from the governing equations of irrotational wave motion by elimination of the vertical coordinate. As an example of a set of Boussinesq-like equations one has the following one, in which u(i,t) is the horizontal velocity averaged over the depth: clli (->, ) • at+ u.V u + gV~ = ( 7. I) ~~ + V.{(h+s)t} = O. A number of other sets of Boussinesq-like equations can be derived, in which the velocity variable ~(l,t) has a different meaning (e.g., the velocity at the bottom etc.), Usually these sets of equations are given only in one spatial coordinate x for a bottom geometry given by h(x); in two spatial coordinates x ,x the corresponding equations are readily obtained by letting 1 2 u be a vector and by replacing 8/clx by the grad and div operations. Boussinesq-like equations as Eqs. (7.1) constitute the basis for investigating the propagation of fairly long water waves over a non-horizontal bottom geometry given by z = -h(x ,x ), The quantities which are to be calculated 1 2 are s(1,t) and t(1,t); once these are found, the horizontal velocity t(1,z,t), the vertical velocity w(l,z,t) and the pressure p(1,z,t) in the fluid can be obtained because these quantities can be expressed in terms of ••u(x,t) and s(x,t)• and their derivatives; these expressions are found during the derivation of Eqs, (7.1). As an example the expression for u(x,z,t) for the case of one horizontal spatial coordinate x is given below. Other ex pressions can be found in Dingemans (1973, Chapter 5), also for~= (x,O). u(x,t) is written as u. 2 - 8 u(x,z,t) = ~(x,t) -(z+ __!_h) (hu) (7. 2) 2 clx2 - 61 - It is known that Boussinesq-like equations such as Eqs. (7.1) describe weakly dispersive, weakly non-linear shallow water waves; the bottom slopes !Vhi are restricted to values less than h/A, A being the wave length of the wave under consideration. For a discussion of the properties of these equa tions the reader is referred to Vis and Dingemans (1978, Chapter 4) and especially to Sections 4.3 and 4.5.4 therein and to the literature quoted in that Chapter. Formally the Boussinesq-like equations are derived under the condition that 2 both h/A and a/hare small quantities, with (a/h)/(h/A) = 0(1). Terms of 2 2 O{(a/h) ,a/A,(h/A) } are neglected in Eqs. (7.1). For applications to non breaking periodic waves especially the condition h/A << I is the limiting one. The smallest value of A/h, or equivalently, of T/g/h, for which Bous sinesq-like equations give a reasonable description of the physics involveu depends on which criterium is used. Generally it can be stated that Boussinesq like equations can yield a useful description for T/g/h ~ 8; see also Appendix E. A solution of the problem of propagation of fairly long waves over an uneven bottom can be obtained by means of a direct numerical solution of the Bous sinesq-like equations (7.1). The result is t(x.,y.,t ), ,(x.,y.,t) in a l J m i J m large number of mesh points (x.,y.,t ). Another method, which resembles more l J m the methods treated in Chapter 5 of this Report, is to derive evolution equations for the parameters of the wave so that Eqs. (7.1) are satisfied in an approximate way. The purpose then is to derive ray equations and an equa tion for the amplitude along a ray, which equations are, of course, usually coupled. In Section 7.2 a method of Ostrovskii and Pelinovskii (1975) will be dis cussed. Their method is based on a set of Boussinesq-like equations from which, using a multiple-scale technique, ray equations are derived which are the same as those following from linear non-dispersive long wave equations; along these rays an equation of Korteweg-de Vries type results for the free surface elevation. A seemingly much different method of Shen and Keller (1973) is commented upon in Section 7.3. It turns out that this method is in effect closely analogous to that of Ostrovskii and Pelinovskii. The result of Shen and Keller's approach is also a ray which follows from linear theory and an equation of - 62 - KdV-type is obtained for the free surface elevation along that ray. The important result of both methods is that the ray and amplitude equations are uncoupled. Because it is known that, for the case of an horizontal bottom, the fairly long wave equations, simplified to admit only waves into one direction, admit permanent wave solutions of cnoidal wave type, a simple procedure might be to take a cnoidal wave as a plane wave approximation in 1,t space and to suppose that the wave remains of cnoidal wave type when progressing. The role of the cnoidal wave is then the same as that of the sinusoidal wave in linear re fraction problems. This course is followed by Skovgaard and Petersen (1977) and is discussed in Section 7.4. It has to be remarked that Ostrovskii and Pelinovskii also considered the case of refraction of cnoidal waves. The difference between the method of Ostrovskii and Pelinovskii and that of Skovgaard and Petersen consists of different calculations of rays. The latter authors simply define rays by the usual geometric consideration of wave front propagation, with as result that for a bottom configuration h = h(x ), the 1 refraction law of Snel remains valid; the wave celerity c is taken as the expression which follows from the cnoidal wave solution, and c thus depends on the wave height. The direct numerical solution of Eqs. (7.1) is discussed in Section 7.5 and some comments on variable coefficients KdV equations are collected in Section 7.6. A short discussion follows in Section 7.7. 7.2 Multiple-scale expansion of Boussinesq-like equations Ostrovskii and Pelinovskii (1975) considered a set of Boussinesq-like equa tions which is different from the set (7.1). On inspection it is seen that they use as horizontal velocity variable t(t,t) the velocity evaluated at z = O; the two dimensional analogue of Eqs, (5.7) of Dingemans (1973) is easily seen to be • ~~ + ci.v)i + gVs = o (7.3) ~~ + V.{(h+s)ti} It is noted that the term in curly brackets in the right-hand member of the second of Eqs. (7.3) can be written as - 63 - (7.4) • When h(~) is taken to be slowly varying in~, i.e., h = h(ox), the second and the third term in (7.4) can be neglected with respect to the first term. 2 Taking o to be of equal order of magnitude asµ= (h/A) , o ~ µ, it is seen that neglecting these terms is in accordance with the approximation$ already 2 implicit in Eqs. (7.3) (terms of O(µ) were dropped). Ostrovskii and Pelinovskii based their investigations on the following set of Boussinesq like equations, valid for small bottom slopes, clti • • 3t + (u.V)u + gVs = 0 (7.5) cls { -+ I 3 2+} I3t + V. (h+s)u + 3 h Vu = 0 ,. h = h(oi), o = µ. Terms of o(s,µ,o) are neglected here. Ostrovskii and Pelinovskii considered quasi-plane waves; that is, •u and s are supposed to be functions of • s = T(k) - t and X where the variation with respect to sis faster than that with respect to i. In effect, it is easily seen that a multiple scale technique is applied; with • • X = ox the assumption of quasi-plane waves can be written more explicitly as • • sei,t) s{s(X,t), X} tci,t) = o{scx,t), x} (7.6) • I -+ • with s(X,t) 6° T(X)- t X = oi, o << I. It is noted that~ and tare supposed to be no function of a slow time scale because no temporal variations of the medium are considered here. Furthermore, note the similarity of the coordinate transformation here and that used in Section 5.4, Eqs. (5.38). • Equations (7.5) can be written in the new coordinates sand X by noting that cl/clt and cl/cJx. transform as L - 64 - eh 3 -----+ (7.7) 3X. dS ]_ Because it is much easier to work with dimensionless equations, the following non-dimensional quantities (denoted by~) are introduced I x h = h/h ~ = r,/a ct = t/{E(gh ) 2}, 0 0 I t = { (gh ) 2 / ;\}t E = a/h and a and h being a (7 .8) 0 0 0 representative wave amplitude and water depth respectively;µ = (h /;\/. 0 Equations (7.5) then become in non-dimensional form (the tilde is dropped for convenience): ~~ + E(O.V)ti + vr, 0 (7.9) cH;; { -+ 1 3 2-+} at + I/. (h+El:_;)u + 3 µh V u o. -+ These equations become with the new coordinates s, X, dropping terms of o(E,µ,o), au. ~ ~ ~ dU. ~ i + ~ ~ + _o_'T _ __i + _o_l:;_ = O as" ax. as EUJ, ax. as 0 ax. ]_ J ]_ ~ ~ au. ~ ~ ~ ol:_; + h _o'T_ __J o'T · o ( ) ~ o ( (7. I O) 8s ax. as + E ax. 8s l:_;uj + u ax. huj) + J J J 3 1 3 ~ 2 31: a uj + - µh (VT) - -- = 0, 3 ax. ~ 3 J as ~ ~ 2 where VT = aT/ax. and (VT) = (aT/aX.)(a1:/ax.). ]_ ]_ ]_ A solution of (7.10) is found by expanding I:_; and 11 as (7. 11) - 65 - With E ~ µ ~ o one obtains from the zeroth-order equations the following equation for the eikonal T: I = 2 = h ' C or, in variables with dimension, 2 • 2 c = gh(X). (7. 12) C In other words, in first approximation the rays are the same as those obtained for the linear non-dispersive long wave equations. For the solution of the equations for s(i) and t(l) to be bounded, the usual orthogonality condition has to be fulfilled, yielding the solvability condition which can be written in terms of s(o) along the ray already found in zeroth-order. This condition, according to Ostrovskii and Pelinovskii, can be written in terms of variables with dimension as (with n written for s(o)) 2 a 3 a h a n 1 d 2 c °'~ + n n + - -- + -- n -(hf'i ) = o ( 7. 13) ON 2h 3s 6g as3 4hf'i2 d9v ' where 9v is the distance reckoned along the ray and L'i is the cross section of the ray tube. Whereas the wave ray follows from (7.12), the wave profile along that ray is governed by Eq. (7.13). It is noted that when his constant, Eq. (7.13) reduces to the KdV equation. Ostrovskii and Pelinovskii subsequently considered the case of straight isobaths, parallel to the coastline; then h = h(x ). Denoting the angle be 1 tween the wave vector and the isobath at some depth h by TT - a, the eikonal 0 2 T(5t) is found as 2 !2 x 1-(h/h )sin at· 1l 0 sina = . h dx1 + x 2 + canst. (7. I 4) 10 . g (gh )2 0 Equation (7.13) can now be written as - 66 - o, (7.15) with 1 h 2 2 6 = (I - h sin a) , ( 7. I 6) 0 Ostrovskii and Pelinovskii took Eq. (7.15) as the basis for their investiga tions of the solutions. At first they considered the case in which Tlg/h » I; that is, the effects of frequency dispersion can be neglected and the set of equations (7.5) or (7.3) reduces .to the classical shallow water equations. The term with the third derivative in Eq. (7.15) can then be neglected; the resulting equation can be solved by the method of characteristics. More relevant in the present context, Ostrovskii and Pelinovskii also dis cussed the case that the amplitude and frequency dispersion are of the same order; the bottom slopes were assumed to be so small that the term with dh/dx l in Eq. (7.15) can be considered to be a perturbing term. Because Eq. (7.15) was derived already under the condition of small bottom slopes and the bottom I 2 slope dh/dx (where hand X have dimension) had to be 0(µ 8), which can be 1 1 '\., µ '\., estimated, using s cS ' as O(a/'A), the slope of the wave, this assumption about small slopes means now that 1 « a/ 'A (see also Appendix G). ldh/dx1 It is well known that the KdV equation yields as stationary travelling waves, solutions of cnoidal wave type. For the case of non-horizontal bottom geometry the solution of (7.15) is taken to be a cnoidal wave with slowly varying parameters, similarly as in the. linear refraction theory where a sine wave with slowly varying parameters is considered. Due to the slowness of the change of the wave parameters, the following conservation equation for the energy flux along the ray tube can be obtained from (7.15): l 2 6 (gh) 2 < n > = constant, (7.17) where the average<·> is taken over the period of the wave. Ostrovskii and Pelinovskii give the following equations from which the variation in wave height of the cnoidal wave can be obtained. - 67 - constant 2 4w H = - - m K(m) (7.18) 2 3'1T g 2 1-m _ E y (m) = K4 ( ) [ 4-2m E (m) (m)] m 3 K(m) 3 2 K (m) where K(m) and E(m) are the complete elliptic integrals, m the elliptic parameter and w the frequency of the wave. The limiting values of the cnoidal wave are the linear (sine) wave (m • O) and the solitary wave (m • I).For 2 linear waves one has Y ~ m and Eqs. (7.18) yield Green's Law: l l Hh 4 6 2 constant. (7. 19) 3 In the solitary wave approximation one has Y ~ K and one obtains constant. (7.20) 7.3 Method of Shen and Keller Shen and Keller (1973) use a ray method which can be viewed as a non-linear extension of the ray method used for linear problems. The mathematical pro cedure is given in a theoretical paper of Choquet-Bruhat (1969). The result is that the rays found are the same as those which are found from linear theory and along each ray the wave amplitude is given by a Korteweg-de Vries type of equation with variable coefficients. In this respect the results are similar to those obtained by Ostrovskii and Pelinovskii (1975). The method of Shen and Keller is based upon a "new" stretching of the van_ ables which differs from the Friedrichs-Keller stretching technique (for a discussion of the latter method see, e.g. Section 4.4 of Vis and Dingemans (1978)) in the stretching of the vertical velocity component. It will shortly be seen that Shen and Keller in fact used the same scaling technique as has to be used in the derivation of Boussinesq-like equations in which it is subsequently supposed that the medium is slowly varying as was done in the previous Section. The difficulty in understanding the implications of Shen and Keller's method is due to their introduction of a horizontal length scale L' without stating whether it is the wave length scale or the scale of the - 68 - inhomogeneity of the medium. On close reading it becomes clear that L' is the inhomogeneity scale; this becomes clear by their introduction later of a fast phase variable. In order to substantiate these remarks, we follow their method of non dimensionalization. Shen and Keller considerer a reference frame which is rotating• about the z'-axis with the constant angular velocity~• -+ I . Furthermore 1 the density p' is a function of 1 , z' and t'. With L' h' and 6' being the scales for horizontal length, water depth and ' 0 density the following non-dimensional quantities are introduced (dimensional variables are denoted in this Section by a prime): (x,y) = (x' ,y')/L' (z,s,h) = (z' ,s' ,h')/h~ t = (U'/L')t' ; (u,v) ( u I 'VI) /U I (7.21) w = w'/{(h'/L')I/JU'} p = p'/6' ; p = p'/(g'h~6') 0 l • I 1 2 1 (L /h')(g'/h') ~ where U' (g'h')2. 0 0 ' 0 In the Friedrichs-Keller type of expansion and in the derivation of the Boussinesq-like equations the scale for the horizontal length is taken to 1 be the wave length A • The scaling of the vertical velocity component is done in these two methods as follows: w' WFK = (7. 22a) erA~ (g'h~)2, and w' WB = ( 7. 22b) C)A~ (g'h~)2, These expressions are to be compared with the present non-dimensionalization of w': w' WSK = (7.22c) h'\1/3 1 ( L~) (g'h~)2 - 69 - Shen and Keller use (7.21) to non-dimensionalize the governing equations which consist of the equation of continuity, the three equations of motion, the kinematic free surface and bottom condition, the dynamic free surface condition and the adiabatic condition. The result is a set of non-dimensional equations in which the small parameter 1 3 \J = (h'/L // 0 appears. Upon comparison with the non-dimensional equations resulting from the Friedrichs-Keller approach, it is seen that in the last equations the 2 small parameterµ= (h'/A') appears at the same places as \Jin the equations 0 of Shen and Keller. That this has to be the case can be made clear by con- sidering the forms of the non-dimensional continuity equations; for ease of exposition p' is taken to be constant. awFK au av) n FK µ(- + + = V (7. 23a) ax ay ~ au awSK SK \J(- + av) + = 0 (7.23b) ax ay ~ au av awB B + + = 0, (7.23c) ax ay az where for w the various expressions of (7.22) have to be taken. 2 It follows from (7.22) that wFK = µwB and wSK = (h~/A') /J wB when L' A1 is chosen. We now identify L' with the inhomogeneity scale and the rate of wave length A1 to L' is denoted by: o = (A'/L') << I. When it is assumed that o andµ are of the same order of magnitude, the small l 1/3 quantity \J 2 = (h'/L') is seen to be of the same order of magnitude as I 0 µ2 = h'/>-:': 0 - 70 - I hL~\1/3 = (h~)l/3 A' I/J = 1 h' /A I µ2. ( -; L' . (L') 0 The result is that, in effect, the non-dimensionalization of w' of Shen and Keller is the same as done in the derivations of the Boussinesq-like approach. The non-dimensional independent variables x and t of Shen and Keller are then + + X I A' - X' +, I seen to be slow scales, because x = x'/L' =I"'. L' - o I"', and x /A was the non-dimensional x used in Section 7.2. Shen and Keller, after obtaining their non-dimensional equations, suppose that s, t, w, p and p also depend upon the fast phase variable~=~ S(i,t), so that s = ~(~,1,t) and the 1 and t derivatives in their equations have to be replaced by a -+at In order to keep the notation in this Report as uniform as possible, the dimensionless 1 and t of Shen and Keller are subsequently written as X and + + + + + T, where X and T and slow scales and x = x'/A', X = ox, T = ot. Because V ~ µ ~ o, (7.23b) can be written upon introducing the phase variable and . + . reverting to the X,T notation as d dU. +~+,(' __J=o az u ax. · J It is seen that Eq. (7,23c) reduces to the same equation after introduction I + + of the dependence of son (8 S(X,T), X,T), ~ = S/o. We conclude therefore that the method of Shen and'Keller and the one adopted by Ostrovskii and Pelinovskii are equivalent in underlying assumptions. A short sketch of the procedure followed by Shen and Keller is given in Appen dix F. - 71 - 7.4 Cnoidal wave refraction Introduction Skovgaard and Petersen (1977) presented a method to calculate the depth refraction of cnoidal waves which is in effect a direct extension of the linear refraction method. The usual geometric optics assumptions are made, i.e., reflection is ignored and the wave energy flux is taken to be constant between adjacent wave rays. Locally, the water surface elevation is supposed to be given by the corre sponding cnoidal wave solution in case of a horizontal bed; as a consequence, the bottom slope has to be very small, in effect IVhl << H/A. Such a condi tion was also found in Section 7.2 from the condition that the term with dh/dx in Eq. (7.15) could be regarded as a perturbing term, The same condition is found in Appendix G from a direct investigation of Eqs. (7.1). It is the assumption that an initial cnoidal wave profile remains of cnoidal form when progressing into a region with a non-horizontal bottom which requires the bottom slope to be so very small; neglect of reflection is permitted already for larger bottom slopes. The free surface elevation, measured from the mean water level, is given locally by I E t 9, i'._; ( s) s = (7.24) = H ~ - mK 2K(m) (r - I), K(m) and E(m) being the complete elliptic integrals of the first and second kind respectively, m is the elliptic parameter, 9, is the coordinate measured along the ray and T is the wave period. The cnoidal wave solution chosen by Skovgaard and Petersen is in effect the one corresponding to Boussinesq's original equation (see, e.g., Dingemans (1974, p. 16)). The parameters can be found from , 1 E A -(l+m-3-) m 1-m m 1 K 1 (7.25) A= cT and his the mean water depth, - 72 - Equations (7.24) and (7.25) now substitute the sinusoidal wave solution and dispersion relation as are used in linear refraction calculations. As in linear calculations the ray equations and an amplitude equation are needed now. Ray equations Skovgaard and Petersen define the wave rays by the same differential equations as in linear theory; however, the phase velicity c is now the phase velocity of the cnoidal wave, that is, c is to be evaluated from (7.25) and the rays follow from = cos8 sine ' (7.26) dC cos8 .-"'-), ox 2 where 8 is the angle between the x-axis and the wave ray, positive counter clockwise. It is readily understood that, for finite-amplitude waves, the location of the rays also depends on the wave amplitude, but whether the result is equa tions (7.26) together with c from (7.25), remains to be investigated further. We do note, however, that the fact that the determination of the rays is coupled here to the determination of the wave amplitude is consistent with the results found in Chapters 3-5. Determination of the ~ave amplitude The wave height is determined from the condition of constant mean energy flux between adjacent wave rays. One obtains to leading order the following -+ expressions for the mean energy flux Ef (see Appendix H): (7.27a) and, with (7.24), - 73 - 1 2 B = - -{(2 E -m )(1-2 !) + (!K) }. 0. 27b) K 1 K 3mz Quasi -two-dimensionai situation Skovgaard and Petersen used (7.27) as the expression for the mean energy flux. They only solved the problem of refraction of cnoidal waves for a quasi two dimensional situation, that is, the bottom contours are parallel and phase independent parameters do not vary with distance along-shore. The x 1 coordinate is taken to be orthogonal to the bottom contours. In this situation the wave ray equations (7.26) can be replaced by Snel's law: / 7 ')Q\ C =constant= f (say). \ I • L.UJ sine 1 The conservation of the mean energy flux in a wave tube can then be put in the form Efcos8 = constant r (say)' (7.29) 2 because, in the geometric optics approach, the energy flux is perpendicular to the wave front. • • In this respect we note that it is indeed well-known that Ef is perpendicular to the wave front, but that this is also true for non-linear waves, albeit of permanent form, remains to be proven rigorousiy; in fact the assumption was already made in (7.27a). The four unknown parameters c, H, 8 and m (note that the wave period T is fixed) can now be obtained from the four equations (7.25), (7.28) and (7.29). Elimination of c and 8 results in two non-linear algebraic equations in Hand m; it is stressed that this is only the case for the quasi two-dimensional case. These two equations can be written as 2 c2E2Cr2-c2) = r2r2 E- = pgBH 1 1 2 with (7.30) 2 Hc 2 = _!_§_ T-2mK2 c· = gh ( I +AH/h) , h3 3 - 74 - and have to be solved iteratively for each particular depth. The constants f and f are obtained from boundary conditions at some depth-contour 1 2 h = h . Because the free surface elevation is cnoidal in the whole considered st region, it would be natural to prescribe a cnoidal wave at the boundary line h = h which there has a direction of propagation 8 However, Skovgaard st st and Petersen chose to give the parameters at that location in terms of linear wave theory parameters. The connection between linear and cnoidal theory is obtained from demanding that the energy flux between adjacent rays is con stant. The constants r and r then become 1 2 1 2 r C /sin 8 r = H e 1 st st 2 8 pg stcg,st cos st' where c and c are obtained from (5.23). Substitution of these expres- st g,st 2 sions into Eqs. (7.30) and scaling H,h with A = (g/2n)T and c,c with 0 g c = A /T yields then two equations for the unknowns m and the dimensionless 0 0 wave height H, see Skovgaard and Petersen (1977, Eqs. (19) and (20)). It is noted that for shoaling, i.e., 8 = O, the first of Eqs. (7.30) st 2 2 becomes c = f , which yields the cnoidal shoaling formula on which the E 2 tables of Skovgaard et al (1974) are based, Skovgaard and Petersen compared the results of some computations of quasi two-dimensional cnoidal wave refraction with those obtained from linear theory and also with a method of Dean (1974) which was based on the stream function. Some of their conclusions are: I) The cnoidal wave ray is refracted less than the sinusoidal one; that is, using subscripts c and s to denote cnoidal and sinusoidal wave respectively, one has 8 > 8 for the same values h/A , H /A and 8 . C S O O O 0 (A subscript zero denotes deep-water values.) For 8 = 60° and H /A = 0.03 and 0.0005 one has le -8 l/8 = 25% and 31% 0 0 0 C S S respectively. 2) For fixed values of 8 , h/A and H /A ~ 0.05, the cnoidal wave height H . 0 0 0 0 ~ C before breaking-inception is greater than H . For 8 = 60° and H /A = S O O 0 0.05 and 0.0005 one has IH -H I/H ~ 3% and 110% respectively. C S S - 75 - 3) It is not possible to draw a general conclusion from a comparison of the angles 8 as obtained from the cnoidal theory and from Dean's theory. 4) Dean's stream function theory usually leads to greater wave heights than are obtained from the present first-order cnoidal wave theory. At the breaker line of Dean's stream function wave ray one has for 8 = 60° and 0 H />, = 0.08 and 0.002 IHe -HDI /RD = 23% and 46% respectively. 0 0 5) For H /">-. ; 0.01 one has RD; 1.12 He and H ; 1 .25 H. For H /A ; 0.001 0 0 C S O 0 RD can be up to 50% higher than H, which again can be up to 100% higher C than H . s It is remarked that the conclusion that the cnoidal wave is refracted less than the sinusoidal one is in fact a direct consequence of the fact that the cnoidal wave is a finite-amplitude wave and that the ray equations are coupled to the amplitude equation. It is a general property of non-linear waves that the rays do not converge as fast as is the case for linear waves (see, e.g., Hayes (1973, pp. 214-215)). The non-linearity can thus be seen as an opposing effect concerning the change of distance between two adjacent rays due to depth variation. Consider now subsequently a region of converging wave rays and one of diverging wave rays. Converging wave rays The effect of the non-linearity on the calculation of the wave rays would lead to lower wave heights for the non-linear case compared to the linear case. Shoaling of non-linear waves leads to higher wave heights than shoaling of linear waves. (A region in space in which both convergence of wave rays and shoaling occurs is, e.g., the front side of a cylindrical shoal.) It is no simple matter to estimate which of the two opposing effects have the greater effect on the resulting wave height. The wave heights as calculated from the method of Ostrovskii and Pelinovskii (1975) are expected to be larger than the wave heights as resulting from the cnoidal wave calculations of Skovgaard and Petersen, because the wave rays in the method of Ostrovskii and Pelinovskii follow from the linear long wave theory. Diverging wave rays The effect of non-linearity is a slower divergence of wave rays as obtained - 76 - from linear theory and non-linearity would yield higher wave heights in such a region. The effect of non-linearity diminishes soon in such regions because the wave height diminishes because of the divergence of the rays. The calculations of Ostrovskii and Pelinovskii would lead to smaller wave heights compared to those of Skovgaard and Petersen. An estimation of which method would yield higher wave heights in regions in which both convergence and divergence of wave rays occurs can not be given beforehand. Using only the calculations of several examples on which the conclusions of Skovgaard and Petersen are based, it is not possible to decide if cnoidal wave refraction gives better results than linear wave refraction. In theory one might expect better results by using a finite-amplitude wave theory, if only because the finiteness of Lhe waves is taken into account, but which is needed is some experimental verification. Such a verification does not exist, to this author's knowledge, for the case of refraction. Some indication might be obtained from experiments in which wave shoaling is considered. It is not the purpose of this Report to investigate shoaling thoroughly, but some indica tions can be readily obtained from the paper by Walker (1976). Walker (1976) has carried out measurements in order to ascertain the influence of wave height and wave breaking on wave refraction over a shoal. From aerial photographs, taken near Hawaii, it was seen that breaking waves had a diver ging pattern over the centerline of the shoal where linear theory predicted a strong convergence. The "observed" wave orthogonals tended to be considerably less affected by the bathymetry than were the theoretical orthogonals (based on linear theory). The effects of diffraction, wave induced currents, reflec tion, and energy dissipation other than breaker decay were assumed to be negligible, so that refraction was the principal effect to be studied. Before doing experiments with a shoal, Walker carried out a number of experi ments concerning wave shoaling over a constant sloping bottom. In order to minimize the effect of the current system which is generated by small varia tions in the waves which are generated, only the first five to ten waves were used. As a result of these experiments it was found that the wave shoaling as measured in the model increased much more rapidly than the one as computed from linear theory, especially just before breaking. Walker compared the ex perimental values found for shoaling with the theoretical ones based on shoaling of cnoidal waves and the ones which result from Dean's (1974) theory. A difficulty in his comparisons was that, in the experiments, an equivalent - 77 - deep water wave height H was calculated by applying linear shoaling formulae 0 to the wave heights obtained from the wave gauges located in the deeper part of the model, whereas for the theories H was the true deep water wave height. o< The general conclusion is that, for h/A = 0.03 (or T/g/h = 14.47), cnoidal 0 wave shoaling gives results which are closer to the experimental results than are the results from Dean's theory (see Figure 7 of Walker (1976), which is reproduced below). Furthermore, Walker found from the experiments that the wave celerity was predicted rather accurately by the simple expression = gT C = Zn tanh kh. From the results of Walker we draw the conclusion that, for long waves, cnoidal wave shoaling.gives better results than shoaling as resulting from Dean's theory. Whether this is also the case when Chaplin's (1978) reformu lation of Dean's stream function theory is used is not known to us. A result of Chaplin was that he found a smaller wave celerity than Dean, especially for higher wave s~eepness. 2.6 .-....,,-'""T'...,."TTTT"T""TT'1""---r--,--.,.-.,.....--,-...... ,_,__ 2.4 0.002 ISOLINES OF I I H /L \ 0 0 \ 2.0 \• \ \ \ 0.01 0 1.6 :t ', &. \1 :t ,:~ \\ ' 1.2 ',~' \, ~' .... s ---- MEASURED WAVE SHOAL I NG ...... _ --....,~~~..... ~-~--..f 0.8 -:::. - - - - CNOIOAL WAVE SHOALING (AFTER SVENDSEN AND KJAER, 1971) 0.4 ------STREAM FUNCTION THEORY (AFTER DEAN, 1974) NOTE:BREAKER INDEX B=H/d:(1:!) (H0 )(!:..2.) 0.0 ...... __..__..__,,_.....,_,,_...__...._1,,_ _ H_,;.....,__,;.....__.,__,,__._.i....1,0 Lo d ...... ____ 0.001 0.01 0.1 d/L0 7.5 A direct numerical solution of the Boussinesq-like equations In Sections 7.2-7.4 the free surface elevation had to be a quasi-stationary wave, a consequence being that the bottom slopes IVhl had to be very small. - 78 - When the bottom slopes are larger than about a/A, then the quasi-stationary wave approach cannot be followed anymore. The Boussinesq-like equations as Eqs. (7.1) describe also non-stationary wave solutions. In the derivation of Eqs. (7.1) it was assumed that !Vhl = O(h/A). Recently, Abbott et al (1978) published a paper in which some numerical ex periments are described, based on Eqs. (7.1). In defining difference schemes they freely use the order relations such as at/at+ gVs = 0 and as/at+ V.(ht) = 0. Indeed, the resulting equations are asymptotically equi valent upon using these order relations, but the dynamical behaviour of the resulting Boussinesq-like equations has also to be taken into account, especially when solving these equations numerically. By the process of dis cretizing, short waves with wave lengths of order of the mesh-size are gener ated; the behaviour of these waves should not lead to disastrous results in the final computed results. A very useful method to ensure good dynamical behaviour of the differential equations describing the wave motion is a Hamiltonian approach, see, e.g. Broer et al (1976). The idea is that an approximate Hamiltonian is constructed for the case of long waves; higher order terms may be added in such a way that the resulting Hamiltonian also ensures good dynamical behaviour for short waves; the specific set of Boussi nesq-like equations then follows from the variational principle. Once the approximate Hamiltonian is chosen, no further approximations are made. Abbott et al did not mention these questions of good dynamic behaviour; this is not to say that they had trouble in their results, because it is especially the very short waves which cause trouble. Only results for rather long waves are given in the paper; because the number of points per wave length chosen by Abbott et al is rather low, the shortest waves generated numerically (with wave lengths of order of the mesh size) are not so very short. The number of point per wave length, N, which is (according to Abbott et al) X needed for most applications is very low. For a Courant number of about one, N = 6 would suffice, For practical situations a time step of 1 second is X recommended. For A/h = 9 Abbott et al give two graphs of the experimental phase amplification versus N for H/h = 0.01 and H/h = 0,1. The phase ampli x fication for N = 6 and Courant number= 1 as read from these figures is .94 X .92 respectively. Nevertheless, in our opinion only six points per wave length is too low for practical use. Furthermore it is noted that these tests are 2 done for rather small Stokes numbers (a/h)/(h/A) which are (with a= H/2) .4 and 4 respectively; secondary waves in a wave signal become only visible for Stokes numbers above about 15 (see, e.g., Dingemans (1976)). - 79 - During the last years a number of papers have been published, all concerning finite difference methods for KdV or BBM equations. (Note that the BBM equation is also known under the name regularized long wave (RLW) equation and is proposed by Benjamin, Bona and Mahony (1972) as a better model equation for fairly long water waves compared to the KdV equation; the equation is given in (7.31) .) A short review is given by Eilbeck (1978). Numerical studies of the KdV and the BBM equation are often concerned with the interaction of solitary waves (solitons). Whereas the effect of interaction of two solitons obeying the KdV equation is only a phase-shift (the solitons themselves reappear further unchanged in shape, which is in effect used as definition of a soliton), the BBM equation has no true soliton behaviour: when two large solitary waves (with JO as a measure for the amplitudes) collide in the BBM equation, a very small osciallatory -3 tail (~ 10 ) appears. Finite In problems of long wave propagation over a shelf h = h(x), Madsen and Mei (1969) used an integration method along the characteristics of a set of Boussinesq-like equations. For choosing the grid size they recommend the rule of thumb that 6x = h/4 in the shallower part. This choice can be shown to result in a number of 50-150 points per wave period, depending on the geo metry and the wave parameters, and a typical number is 70. This constrasts very much with 6 points per wave period as recommended by Abbott et al (1978). Eilbeck and McGuire (1975) devised a three-level scheme of second-order accuracy for numerical solution of the BBM equation. In variables with dimen sion this equation reads (7.31) Eilbeck and McGuire considered the non-dimensional BBM equation 1.n the form - 80 - (7.32) and took for the mesh size, 6x = 6t = 0.2. Because (7.32) follows from (7.31) with the scaling n = ¾s/h, x = x/(h/6), t = tilgh/(h/6), 6x = 0.2 means that < 6x/h = .49, or, 6x h/2. In one wave length there are thus 2\/h mesh points, which is also considerably more than the value taken by Abbott et al. Furthermore the paper by Fornberg and Whitham (1978) is mentioned in which a very accurate numerical scheme is developed for solving non-linear one-dimen sional wave equations which is based on a pseudospectral treatment of the space dependence together with a leap-frog scheme in time. The numerical solution of the KdV equation is treated and also the development of non-linear wave train instabilities is accurately solved numerically. Typically, 128 mesh points in a wave period are chosen. Linear stability of u + u + u = 0 3 2 t X XXX is obtained for 6t/ (6x) < 3/ (21r ) where 6x = 1T /N and the wave length is scaled to 2 ; this stability condition is obtained for the maximum wave num- ber, k = N. For further details and a discussion of the non-linear stabil- max ity of the numerical scheme the reader is referred to Fernberg and Whitham (1978). 7.6 Some comments on the generalized KdV equation and its solutions The form of the generalized KdV equation As is known,the constant-coefficient KdV equation describes the propagation of uni-directional fairly long waves of nearly permanent form in case of an horizontal bottom; that is, the free surface elevation s(x,t) is given as X 2 s(x-ct,T) ors(- -t,X), where c = gh X = Ex T = ET E = a/h << 1 and 0' ' ' 0 2 2 C E ~ µ = h /\ . With s(x-ct,T) the KdV equation reads 0 (7. 33) For the case of an uneven bottom, h = h(x), the following variable-coefficient KdV equation can be derived (Ostrovskii and Pelinovskii (1970), Kakutani (1971), Johnson (1973)): - 81 - 2 3 ~ + 2- (eh) -1 s ~ + .!_ ~ U + _1 dh s = 0 .(7. 34) clx 2 cls 6 3 ~ 3 4h dx ' C oS where x dx' 2 (7.35) s = f c(x') - t c (x) = gh(x). Equation (7.34) is derived under the condition that "A dh a/h = O(s). (7. 36) h dx Writing h = h(ox), o << 1, this is equivalent too~ E: ~µ,and the depth is thus slowly varying. One reason for the depth to be slowly varying is the fact that the KdV equation describes uni-directional waves and the reflected wave is therefore neglected. For uni-directional waves in channels of slowly varying depth hand breadth b, with ("A/b)(db/dx) = O(s), Shuto (1974) derived the following generalized KdV equation: (7. 37) This equation is seen to be equivalent to Eq. (7.13) upon substitution of c = /gh and b = 6. Writing the generalized KdV equation (7.37) as 3 8 ell'.;+ f (x)i;; ~s + f (x) s3 + f (x)i;; = 0, (7 .38) dX 1 oS 2 dS 3 it is seen that the inhomogeneous term f s may be eliminated by writing 3 s(s,x) = f (x)Z(s,x) with f (x) = exp{-Jx f (x')dx'}, (7.39) 4 4 3 resulting in fs = flf4, (7. 40) - 82 - where the dimension of Z is now different from that of s, Applying this l transformation to Eq. (7.34), one obtains Z = h 4 s and the resulting KdV equation with variable coefficients becomes 3 az 3 -! -7/4 az 1 -3/2 12 a z + - g h Z - + - a h -- = 0. (7.41) ax 2 as 6 b 3 as It is noted that it follows from (7.36) that the bottom slope dh/dx is of the same order of magnitude as the wave slope a/A under the condition that the 2 2 Stokes number (a/h)/(h /A ) is 0(1); under this condition also dh/dx = 3 3 O(h /A ). It is well-known that the KdV equation (7.33) has as permanent wave solutions the solitary wave and cnoidal wave solution; the latter is a periodic wave. For an uneven bottom of the form of a shelf (h h = constant for x < x and 0 0 h = h = constant for x > x > x ) the problem often is to determine the 1 1 0 evolution of an initial permanent wave (in the region h = h) upon progressing 0 onto the shelf. It is always required that the permanent wave remains of similar shape to leading order; perturbations to the solitary or cnoidal wave have to be of higher order of magnitude. In other words, the phase function is slowly varying. This has as a consequence that the inhomogeneous term (f 3 s in (7.38)) has to be small so that it constitutes a small perturbation to the homogeneous KdV equation, and thus, from (7.34), dh/dx << a/A (see also Appendix G).This is most easily seen by writing the generalized KdV equation (7.34) in non-dimensional form, with n = s/a, h = h/h, x = x/A, 0 s = (c /A)s, c = c/c , c = vgh; dropping the tildes and using n = n(s,X), 0 0 0 0 X = ox, the result is 3 an 3 -1 an 1 h2 -3 a n o dh o clX + 2 s(ch) n 8s + 6 µ c as3 + 4h dX n = O, (7 .42) 2 or, because c (X) = h(X), o. (7.43) Note again that in the derivation of this equation it was supposed that h = h(X) and E: 'v µ 'v o. The inhomogeneous term is small compared to the - 83 - other terms when h-ldh/dX << 1; this can be achieved by writing h = h(oX) with a<< I. The relation between 0 and Eis not known yet; this has to follow from an investigation of the solution. The non-dimensional equation (7.43) is written as 3 3n + a (ax)n ~n + 6 (0x)~ = -y(0X)n, (7.44) 3X 1 oS 1 dS3 which can also be put in the following form by using a transformation equiva lent to (7.39), namely n = F(X)q with F = exp{-?- y(X')dX'}, a3 + S (ax)--1 = o (7,45) 1 3 3s Note that a.,f3 = 0(1) and y = 0(0). l 1 Often again another form of the generalized KdV equation is seen in the literature, especially in papers dealing with the inverse spectral transform; this form is written here as 3 3 3 3 P + 6p _E.+ ___e.= -r(or)p. (7.46) ar as as3 This (non-dimensional) equation can be obtained from (7.43) by substitution of n(s,X) = f(X)p{s,r(X)}; (7 .47) one obtains ~ 3Efh-3/2(dr)-l ~+J..l:'..h½(dr)-l{J..df+_l dh} =O 3r + 28 \dx P 3s 6 o \dx f dX 4h dX P • The functions r(X) and f(X) then follow by choosing the coefficients of 3 3 p3p/3s and of 3 p/3s equal to 6 and to 1, resulting in r(X) = _!__ }:'.. JX h(X')dX' and f (X) (7.48a) 6 o - 84 - 3 The coefficient of p then becomes f(X) 'l:2 i h- /z dh/dX· considering h to 2 µ ' be a function of r, one obtains = 2_ h-1 dh f(r) 4 dr ' (7.48b) For r to be of O(o), there is taken h = h(crr); this means that in (7.48) should be taken h(oX) in both expressions. Several forms of the generalized KdV equation (also called inhomogeneous or variable-coefficient KdV Eq.) are given above, both in dimensional as well as in non-dimensional form. In non-dimensional form, Eqs. as (7.44), (7.45) and (7.46) are usually considered; the latter two are usually given in a notation where u stands for p or q, t for X or rand x for s; t and x are then simply denoted as time and distance and energy of the motion is given 2 as Ju dx. When using solutions as appear in mathematically oriented papers, care should be taken to obtain a physically correct interpretation of those solutions. Some solution techniques are mentioned below for the case of solitary waves and for cnoidal waves; 0 is a small parameter and thus the bottom is mildly sloping. S0Uta1~y waves Ko and Kuehl (1978) solved Eq. (7.45) by considering an asymptotic approxima tion of the form (o) (1) 2 (2) q = q (0,oX) + oq (8,CTX) + 0 q (0,oX) + •.•• , (7.49) where 0 is a phase variable which reduces to kX-ws for a ,S constant. For 2 1 2 ~ q(o) the KdV soliton q(o) = asech ~' ~ = b(0+0) is obtained as a solution where a and b can be expressed in terms of a , '08/os, 08/o(crX) and is 2 ,S 1 0 an arbitrary function of ox. a, band 0 are completely determined by con sidering the first-order equations (in 0) and 0 follows from consideration of the second-order equations. The inclusion of the phase-shift 8 is new compared to the method followed by Johnson ( 1973); the contribution of 0 causes the soliton to lose energy when progressing in a non-uniform medium. The solution q = q 2 corre t to O(o) and qd is the distortion of the soliton; q = (l-~)asech {a. 7 8 (l-~) 2t} with O < ~ << in the region h = h(x). The soliton energy, defined as q~ ds (see remark above concerning energy), is decreasing, for both J:00 decreasing and increasing depths h(x). The total energy and the mass of q = q +qd are not calculated by Ko and Kuehl. 8 A different method consists of investigating the perturbation of the inverse spectral method for the KdV equation, see, e.g. Karpman and Maslov (1977a,b) and Kaup and Newell (1978). (For an account of inverse spectral solution techniques for the constant-coefficient KdV equation the reader is referred to the reviews by Scott et al (1973) and Miura (1976).) It is shown by Kaup and Newell that a soliton solution of (7.46) can not fulfil both conservation of mass and energy. [r-1ore precise,· soliton solutions could not fulfil both exact conservation laws of (7. 46): 0 00 00 J p ds = -r J p ds ar -oo -oo (7.50) 0 2 ar J p ds The subsequent motion of the soliton 2 2 - p (s,r) = 2v sech {v(s-s)}, (7.51) 8 was found to be governed by dv 2 ds · 2 1 = r.v = 4v + r/v. (7.52a) dr 3 dr 3 These equations can be integrated to yield uniform solutions over scales r.r = 0(1) and the result is (7.52b) 3 s + {(h/h ) /Z -I}, 2v 0 0 where the index o indicates values taken in the region r < r where h = h = 0 0 constant (note that we have dimensionless quantities). The soliton solution - 86 - (7.51) with (7.52b) does not fulfil the first conservation law of (7.50). The radiation part of the solution, also called the oscillatory tail, (see Appendix C for the case of the NLS equation) has to be investigated also. Kaup and Newell (1978) gave an expression for p due to the continuous C spectrum (their equation (9.10)) which is approximately given as r p (s,r) = for r < r < r = 0 elsewhere. (7.53) C 3vtanh 2 {v(s-s)}- 0 With p = p + pc both conservation laws (7.50) are fulfilled. The effect of 8 p is that a KdV soliton moving into deeper water will leave a depression of C the water level behind it, which extends back to where the radiation solution starts, at about r ~ r . As the soliton moves foreward, the length of this 0 depression increases, If the soliton propagates through a region of decreasing depth, the solution pin the region r > r (for which h h = constant) will 1 1 consist of the original soliton adjusted for the new depth plus a number of secondary solitons produced by the arrival at s = s of the elevation and of 1 some left-over radiation. (The number of secondary solitons will be propor- - 2 l2 tional to {-(s-s) /(3v)} .) 0 Karpman and Maslov (1977b) consider a perturbed KdV equation in the form (7.54) where R is an operator working on p. With R[p] = yp, the generalized KdV equation (7.46) is recovered when substituting -p for p and r =cry.Presumably, Karpman and Maslov only consider the example y = constant. In the adiabatic approximation (in which the distortion of the soliton and the tail formation 2 2 are neglected)they have the soliton solution p s = -2v sech z, z = v(r). {s-~(r)} and v and s evolve according to dv a co [ 2 = - v J_co R p (z)Jsech z dz dr 4 8 (7. 55) 2 0 1 2 ds = 4v - fco R[p (z)J (z+ - sinh 2z)sech z dz. dr 2v2 -co 8 2 which becomes, with oR[pJ = yp , 8 - 87 - v(r) = v exp(2yr/3) 0 As a first perturbation a solution of the perturbed KdV equation is sought 1.n the form 2 2 p(s,r) = -2v [sech z + w(z,r)J. where z = v(r){s-~(r)} and v and s evolve according to the previously given equations. For w is obtained the solution (0Rp = yp) w(z, r) (7.56) + tanh z - I + 2z(tanh z - I) + z 2 tanh z , 2 Ji cosh z 3 2 where 0(y) = I, y > 0 and 0(y) = O, y < 0 and p(r) is given by dp/dt = -8v p r 3 and thus, 1/p(r) = 8 J v (r')dr' with r. an arbitrary integration constant. 1 ~ co r 1 2 co Eq It is noted that J w(z,r)sech z dz = 0 and J w(z r)dz = - - = -co -co ' 4p 5 2 -(16pv )-l .0 Jco R[p (z)]tanh z dz; this expression has to be small 1.n -co 8 magnitude. Writing w = w + w + w one obtains 1 2 3 w 1 (z,r) 08(-0q) tanh z roo 2 5 2 }_ R[p 8 (z')]tanh z'dz' 8v p cosh z 00 and an expression for w (z,r) which 1.s of O(o) anc;l whose asymptotic behaviour 3 for lzj • 00 is given by 00 0sign z 2 -2jzl / [ dz' w (z,r) • - z e R p (z') J . 3 5 8 2 8v • - 00 cosh z' w (z,r) characterizes the tail at the solution, the length of which is in 1 creasing for increasing r (decreasing p). The contribution w (z,r) describes 2 - 88 - the distortion of the soliton and is localized approximately in the same region as the soliton; the amplitude of this distortion is relatively large, O(o,p) The contribution w3 is also a distortion of the soliton, but not localized to the region of the soliton. It is well-known that the KdV, the NLS and some other equations have an in finite number of conserved quantities of the form I {p, p• } = /" q [P, p*] ds n=l ,2,3, ... (7.57) n - 00 n where the q [P,P~ are polynomials of the functions p and p* and their spatial n derivatives. With o # 0 in Eq. (7.54) and a similar equation for the NLS equation, these quantities I are no longer conserved (for the NLS equation, n see (C12) and (Cl3) or (CIS)). Karpman and Maslov give the following expres sion for the change of I with r: n dl n 00 j or or n=l,2,3, ... , (7.58) dr = o op(:) R[u(s)] + n 100 op*(s) where ol {p,p*}/op(s) is the variational derivative of the functional n I{p,p*} with respect top at the points for fixed p*. Equations (7.58) are called the conservation laws, modified for the case of perturbations, or, briefly, the modified conservation laws. A clear discussion about slowly varying solutions of the generalized KdV equation (7.37) is given by Miles (1979). The following quantities I and J are invariants of Eq. (7.37) (note that we have now variables with dimension): l oo 00 2 = 2 f l; = I (be) -oo ds J be ! -oo r; ds. (7.59) Mass Hand momentum mare defined by l 3 l M = pl(bc) 2 m = pl(bc ) 2 (7.60) and thus, except for special combinations band h, Mand mare conserved 00 only when r; ds = 0. Because the energy is defined by E pg J, energy J-oo = - 89 - 00 i;; is conserved. Because a solitary wave does not fulfil f -oo ds = 0 (note we use the adiabatic approximation, see above), mass and momentum are not con- served for a slowly varying solitary wave. The change in mass must be com pensated by a reflected wave. For cnoidal waves the integration in (7.59) is over one wave period; the I 2 invariants are then written as I= (bc) 2 < s > and J =be< s >. Slowly varying cnoidal wave solutions are obtained by imposing the conditions J is 2 2 constant and< N > = O, where s = a(x) N(0,x) and N = en (2K0/m) - < en >; see further Miles (1979). In order to conserve mass for the solitary wave, the solution is written as s = sl + s+ + s_, where sl is the primary wave given as the solitary wave with varying parameters (as resulting from the adiabatic approximation) and s+ and 1;;_ are right and left moving secondary waves for which the length scale is the length of the channel variation; s± are given by ~ = _1 ± - X (be) 2 f± (s±), s = +t + J dx/c. As amplitude a of s just behind the ± X • • 0 3/2 I 5/2 + + 9/2 primary wave, Miles finds a+ = -4~3- a- 2b (d/dx)(log(bh )), where a is the amplitude of the primary wave. The solutions for s+ and s can be compared to the occurrence of the forma tion of an elevation or depression as found by Kaup and Newell (1978). Apparently, Miles found not a distortion of the solitary wave as found, e.g., by Karpman and Maslov (1977b). Cnoidal waves Svendsen and Buhr Hansen (1978) consider the question of evolution of periodic fairly long waves over a slowly varying bottom and consider the generalized KdV equation in the form (7 .6 I) 2 -I where c (x) = gh(x). Introducing in (7.61) the coordinates s = cS T(X)-t, 1 X = ox (see (7.6)) and using -r(X) = JX c- (X)dX (see (7.35)), one obtains, 3 3 3 3 3 with a/at+ -a/as, a/ax + c-l a/as+ cSa/ax, a /ax + c- a /as + O(cS), equation (7.34) within the same order of approximation (s ~ µ ~ cS), that is, terms of O(scS,µcS) are neglected. Introducing non-dimensional quantities as n = s/a, x = x/A, h = h/h, = (c /A)t, c = c/c , c =~(see also (7.8)), 0 t 0 0 0 0 - 90 - Equation (7.61) can be written in non-dimensional form (dropping the tildes for ease of notation) as 3 an an 3 c an 1 2 a n 1 de at+ C ax + 2 Ehn ax + 6 µh C ax3 + 2 dx n = o. (7.62) Because h = h(X), X = ox and thus also c = c(X), one has an+ c = 0(o) at ~ax and !(dc/dx)n 0(o); compare with Eq. (7.42). Supposing the depth variation be given by h = h(oX), o « 1, the term !(dc/dx)n becomes of relative order 0(o); Svendsen and Buhr Hansen now use a multiple scale expansion for n(0,X): kn = n (o) ~ (7.63) I 1• ' ~ = k (o) + ok ~ J where k = ae/ax and only time-periodic waves are considered, i.e., w = -ae/at = constant and thus k does not depend upon t. For n(o) a slowly varying cnoidal wave solution is obtained and all parameters can be established to this order. From the equation for n(i), it follows, according to Svendsen and Buhr Hansen, that k(l) = O(E) and that therefore some terms with k(l) can be omitted from the differential equation for n(l). The solution of n( 1 )(0,X) can then easily be obtained; three arbitrary functions of X are still present in this solu 1 tion. Two of them are determined by requiring that both n(l) and an( )/30 are bounded and continuous at the crest and the through of n(o); it turns out . ( 1) . . that these functions are zero then and that also k = 0. The third arbitrary function is seen to be zero also by requiring that n(l) + 0 for vanishing b ottom s 1 ape. Requiring· · tath n.(i) is· perio. d'ic in. 0 imp. 1·ies t h e constancy o f energy flux for n(o), which condition was needed in order to determine the parameters of the slowly varying cnoidal wave solution for n(o). The solution for n(l) yields the distortion of the cnoidal wave. We note that the solution in case of cnoidal waves as given by Svendsen and Buhr Hansen (1978) is, qualitatively, quite different from th~ one given by Ko and Kuehl (1978) for the case of solitary waves. In the latter case it was found necessary to include a phase-shift in the soliton solution, which phase-shift 2 was determined from the 0(o ) equation; this resulted in an O(o) change of the soliton part of the solution while the other terms of the first order solution gave the distortion of the soliton. That this is not the case for the solution of Svendsen and Buhr Hansen for cnoidal waves is due to imposing - 91 - their conditions at the crest and trough of the zeroth order solution which lead to k(l) = 0. Furthermore, the distortion of the solitary wave solution had a length scale of the order of the length scale of the inhomogeneity, see Miles (1979), whereas the distortion of the cnoidal wave was periodic on wave length scale. Whether this different behaviour between cnoidal and solitary waves is only due to the fact that mass is not conserved for a slowly varying solitary wave, is not known to us. A comparison of the solution of Svendsen and Buhr Hansen with measurements in a wave flume gave fair agreement both for wave heights as well as wave shapes, even for waves close to breaking; the measured values were corrected for energy losses in the flume. For both measured and calculated wave profiles, s = 0 was defined at their respective mean water levels and not as still water level so that possible differences in mean water level can not be observed and thus a check whether the calculated distortion is truely periodic can not be made. As is stated perviously, the depth is varying with aox = aX, h = h(aX), and a<< 1, o ~ E ~ µ << 1. The order of magnitude of a compared to, e.g., Eis given by Svendsen and Buhr Hansen as E < (a/h).(a/A) << jdh/dxl << a/A, or, The restriction of order of magnitude is correct, but for different reasons than are put forward by Svendsen and Buhr Hansen. They argue that a complete asymptotic expansion of n is of the type 1 (ij) n I Ej I a n µ = 0 = E, j=o i=o - 92 - and that for sn(oi) to be negligible compared to on(io) one should have E << 0 (both n(oi) and n(io) are 0(1) in the present scaling). An asymptotic expansion of n with respect to the non-linearity parameter E in Eq. (7.48) or Eq. (7.42) has no physical relevance, because the KdV equation is derived from a set of Boussinesq-like equations (see, e.g., (7.9)) by neglect of terms of o(s,µ,8) and the Boussinesq-like equations themselves are also 2 accurate only up to O(s,µ), terms of O(Eµ,µ ) are neglected. Higher-order solutions with respect to E, where as zeroth-order solution a permanent wave solution is taken, should be obtained from a set of higher-order Boussinesq like equations, such as, e.g., given in Dingemans (1973, Appendix D) or, better still, from the original governing equations (see (2.12)-(2.15)). The 2 requirement 0 >> E, and thus oo >> s , is a consequence of neglect of higher order terms in the KdV and Boussinesq-like equations. The derivations of Svendsen and Buhr Hansen are difficult to follow, which is due, in our opinion, to their inconvenient scaling of the variables and their use of only h/A as small parameter; due to the exclusive use of h/\, 2 2 instead of, at least, a/hand h /\ , they falsely state that h + 0 has to X lead to vanishing wave amplitude and vanishing h/A (below their equation (3)), whereas only ,S + 0; note in this connection that h + 0 leads to n(l) + O X and n(o) + constant depth cnoidal wave. Only when requiring that 0(8) terms in (7.42) remain of the same order of magnitude as the 0(£) and O(µ) terms then 8 + 0 yields both E + O, µ + 0 with s/µ = 0(1) for consistency. But for E + 0 the ordinary linear wave theory is expected to be sufficient. The solution given by Svendsen and Buhr Hansen seems to be very useful, but the consistency of their treatment of the n(i) solution and the effect of their choice of requirements to determine the arbitrary functions of X in ( 1) . . . . . tenh solution needs some attention, in our opinion. 7.7 Discussion A few papers concerning the behaviour of fairly long waves over an uneven bottom h = h(x ,x ) are reviewed in this Chapter. 'The common basis is in all 1 2 cases the set of Boussinesq-like equations, although this fact is disguised in the approach of Shen and Keller (1973) (see Section 7.3). In effect there are two methods of approach: a direct numerical integration of the Boussinesq like equations, of Section 7.5, and a geometric optics approach employed on the Boussinesq-like equations, cf. Sect.ions 7 .2-7 .4. Asis usual in the geometric optics approach, reflected waves are ignored and - 93 - the wave is locally given by the permanent wave solution found for that particular depth if the bottom were horizontal. As a consequence, the varia tion of the water depth has to be smaller than in the case for which the Boussinesq-like equations are derived; instead of IVhl ~ h/A it is now necessary that IVhl << a/A (see Appendix G).Three different small parameters are present, viz., the modulation parameter o == A/L where Lis the inhomo geneity scale, the non-linearity parameter€= a/hand the dispersion para 2 meterµ== (h/A) ; the investigations are carried out under the assumption that E ~ µ ~ o. Both Ostrovskii and Pelinovskii (1975) and Shen and Keller (1973) found, by using a multiple scale expansion in which the free surface sand the horizontal velocity u are considered to be functions of (T(x ,x ),ox ,ox ), that the wave rays are found from linear non-dispersive 1 2 1 2 long wave theory (the phase velocity is c = /gh(cSx) ) and the wave amplitude is found from a Korteweg-de Vries-like equation. The fact that the amplitude equation and the wave ray equation are not coupled, or, otherwise stated, the fact that the wave direction is independent of the amplitude, is alto gether different from what might be expected from, e.g., the Lagrangian approach. Ostrovskii (1976) discusses several examples for which the linear ray approximations for a finite-amplitude distortion in the presence of smooth inhomogeneity of the medium. This needs further consideration. The effect of non-linearity on the wave rays was just the coupling, which had the effect that the formation of caustics was counter-acted. An heuristic method of cnoidal wave refraction is proposed by Skovgaard and Petersen (1977). Their method is also a geometric one. They consider a cnoidal wave with slowly varying parameters. The rays are supposed to be given by the usual linear wave ray equations in which the phase velocity c is replaced by the phase velocity resulting from the cnoidal wave formulae. By taking the energy flux between two adjacent rays constant, a set of non linear algebraic relations can be obtained and has to be solved iteratively. The method is only practical when the isobaths are straight as well as parallel. In this case the difference with the results of Ostrovskii and Pelinovskii is, in effect, only due to the different refraction law. Whereas Ostrovskii and Pelinovskii obtain Snel 's law with c == /gh, Skovgaard and Petersen take Snel's law as being valid with a non-linear expression for c, resulting in a wave direction which is also a function of the local wave height. In our opinion it is possible to check this result by applying Whitham's approach for the case of cnoidal waves; it is recommended to do this in a subsequent study. - 94 - Only one numerical approach for solving the Boussinesq-like equations in two horizontal spatial dimensions by a finite-difference method is mentioned in Section 7.5, the method of Abbott et al (1978). Unfortunately, the descrip tion of the numerical treatment of the pertinent differential equations is very perfunctory so that it does not becomes clear from the paper what state ments as "time steps of about I second can be used in many practical situa tions, giving distance steps of the order of 10 m" are really worth, especially because the range of depths in these practical situations does not become clear. For fairly long wave propagation in one horizontal spatial dimension, espe cially with h = constant, a number of difference schemes are investigated and are adequately described in the literature. Some of these papers are mentioned in Section 7.5. It would be feasible to numerically solve Boussinesq-like equations by using these papers as a basis to devise a well-suited difference scheme. It is recommended, when solving numerically, to use a set of Boussi nesq-like equations which is derivable from an Hamiltonian density which is positive definite and in which the operators remain bounded also for very short waves. In Section 7.6 the generalized KdV equation for one-dimensional wave propaga tion over an uneven bottom is discussed and some analytical approaches are mentioned; distortion terms of the solitary wave and the cnoidal wave are discussed. The question whether the different behaviour of the distortion solutions for the solitary wave and for the cnoidal wave is only due to the impossibility for mass conservation for slowly varying solitary waves,need still to be resolved. - 95 - 8 Long waves Long waves are waves in which the effect of frequency dispersion is neglected, i.e., terms of O(h/A) are neglected. The governing equations then become clu. au. l l ell; -- + u. +g--=O clt J clx. dX. J l (8. 1) ~ + cl {(h+l;)u.} = 0 clt dX. J J I 2 and 11 = ll(x,t). The linear group and phase velocity are both equal to (gh) • Equations (8.1) are the so-called classical shallow water equations. The amplitude of the wave is not supposed to be small. It is known that, for a horizontal bottom, Eqs. (8.1) can be completely solved for plane waves in terms of Riemann-invariants. Often equations (8.1), together with terms describing energy dissipation, e.g. due to bottom friction, are solved numerically; this is done in tidal flow calculations for large parts of the Southern North Sea, Our interest here is directed not to tides but to wind generated waves and the dimension of the regions of interest is much smaller. For the classical shallow water equations to be a reasonable description of the physical situa tion, one should have at least about Tlg/h > 20, T being the wave period; this means that these equations can in our case only be used quite close to the coast. (For T = 6 s one should thus have h < .9 m and for h = 10 m one should have T > 20 s.) The bottom is again taken to be slowly varying in 1, i.e., the water depth is changing only a little within a wave length: h = h(ol), o =A/A<< 1 where A is the inhomogeneity scale. Varley, Venkataraman and Cumberbatch (1971) (denoted by VVC) presented a method to study the behaviour of large amplitude long waves in a beach zone. They considered slowly modulated simple waves. (A simple wave is a wave which propagates into water of rest and it has therefore a pulse-like shape. Important papers concerning simple wave approximations are Broer and Schuur mans (1970, 1971) .) Only plane waves were considered. Cumberbatch and Wen (1973) gave some results of this approach and they also considered non-plane - 96 - waves. It can be seen that, in effect, the basic approximation is again a multiple scale expansion in which sand tare supposed to be functions of a phase variables• (•x,t ) and the slow variable• •X = ux:~• -I • • ij s (i' t) s(o s(X,ot),X) (8.2) -+- -I • • ~ t(i, t) = u(o s(X,ot),X). By noting that 3/3t • (3s/3t)3/3s, 3/3x. • (3s/3X.)3/3s + o3/3X. it is possible i i i to write Eqs. (8.1) as 3u. dU. 3s i 3s i (~ + u. + ----3s - -o ~ ou. 3t J 3X.)3s 3X. 3s 3X. J 3X. J i i J (8.3) d dU. 3 (~ + u. ~)~+ (h+s)_s___ J = -o ax. {(s+h)uj}. 3t J ax. 3s 3X. 3s J J J • • For a simple wave (where s = s(s), u = u(s), h = constant) the terms in the right-hand side of (8.3) are zero. Cumberbatch and Wen now state that when an approximation scheme is adopted based on the right-hand side being small, the condition of the first approxi mation remaining bounded yields the two compatibility conditions 3s 3s + u. A!Vsl (8.4) at J 3X. J and dU, __i + 3s 3 2 An. (u. = O, (8.5) i J 3X. 3X.)+ 3x. 2 where = -v's/lv'sl, A = h+s and v' - (3/3X, 3/3X). The independent equa- ri l 2, tions of the first iteration are (8.6) Equations (8.4)-(8.6) are now the governing equations for the first approxi mation. The ray equations associated with (8.4) are - 97 - dX. J_ = u. + An., (8. 7) dt J_ J_ and dn. J_ oA = (n.n.-8 . . )(~x (8.8) dt J_ J 1-J O • J By putting Ji = (cose' sine)' (8.9) equations (8.8) for the bending of the rays can be written as (8. 10) in which a a -sin8 -;:;--x + cos8 -- (8. 11) o l ax2 A local one-dimensional approximation is made by Cumberbatch and Wen; it is supposed that the particle velocity is parallel to ti. Note that such an approximation was also made in Section 7.4 in the quasi-two dimensional situa tion for cnoidal wave refraction. For a beach with straight contours, the ray equations (8.7) and (8.10) become dx 1 d8 de = c cos8 , = c sin8 , = sine (8. 12) dt dt dx1 from which Snel's law of refraction follows: c/sine = constant. Note that c is here the characteristic velocity of the non-linear set of equations (8.1); c thus depends on the amplitude of the wave. A conclusion of Cumberbatch and Wen is therefore that oblique waves are reduced in ampli tude by ray curvature. - 98 - Several results for the amplitude of the long waves upon refraction can be obtained from algebraic relations in terms of the Riemann invariants. For these results we refer to Cumberbatch and Wen (1973) and to Varley et al (1971). The important point for us is that again Snel's law of refraction is used in which the velocity is a function of the amplitude. Because the ray equations (8.7) and (8.8) were taken by Cumberbatch and Wen from an earlier paper of Varley and Cumberbatch (1965), an examination of this paper is necessary in our opinion in order to settle the question of the validity of extending the linear ray equations by taking a non-linear expression for c. Furthermore, an investigation as carried out by Varley et al (1971) and by Cumberbatch et al (1973) can be useful for grasping the effect of non-linear long wave refraction. Note that Ostrovskii and Pelinovskii (1975) gave also a discussion of long wave refraction, based on the long wave limit of the inhomogeneous KdV equation. - 99 - 9 Some comments on non-linear irregular wave propagation It was seen in the previous Report, Dingemans (1978), that the evolution of irregular surface waves could be described by an evolution equation for the spectral action. density. N(+ k,x,t): + a a~ a ~ _a_) + + ++ (at+ dk. dx. - dx. dk. N(k,x,t) = S(k,x,t) i i i i where N = E/w, E the spectral density and w the relative frequency, and + + r. . + r + w = ~ (k,x,t) is the dispersion relation; S(k,x,t) is the source term in which, amongst others, the effects of wave interactions are collected. Here,attention is directed to the change in action density due to non-linear processes. Studies of evolution of non-linear surface water waves have, in the random point of vue, mostly been restricted to the study of wave-wave energy transfer within a broad spectrum due to non-linear coupling in a nearly random ocean (see, e.g., the review by Hasselmann (1968)). In terms + of the spectral action density N(k), the net rate of increase of action . + . density at wave number k 1 due to resonance between quartets for which + + = + + w = w + w (9. 1) k.3 k4 2 3 It is given by 3N l -F = (9.2) +++:.r +++ • 8 (w +w -w -w ) 8 (k +k -k -k) dk dk dk , l 2 3 It l 2 34 2 3 It where o(·) is the delta function and w(k) the linear frequency dispersion + + l2 relation, w(k.) = w. = {glk, ltanh(lk, lh)} , and G ,is a complicated function i i i i of its arguments. Note that it follows from (9.2) that 3N /at= 0 when all l N. are equal; furthermore, it follows that, with N 2 = N3 =Nit= N, Nl / N, i 2 the term in square brackets becomes N (N-N) and N +Nast evolves. In l 1 other words, Eq. (9.2) predicts that the net result of all interactions is to redistribute the energy of the spectrum more uniformly over all wave numbers. For a clear discussion of the behaviour of Eq. (9.2) the reader is referred to Webb (1978a) and also Webb (1978b). Insight can be gained by - 100 - writing the terms in square brackets in (9.2) as [N N (N -N ) + N N (N -N )]. 1 3 4 2 2 4 3 1 Because Eq. (9.2) describes the transfer between the pairs of wave numbers -+ -+ -+ -+ k ,k and k ,k , the term N N (N -N ) can be considered as describing a dif- 1 2 3 4 2 4 3 1 + • • • • fusive transfer between k and k ; this diffusive transfer now "pumps" the 1 3 -+ -+ transfer between k and k in order that conservation of energy and momentum 2 4 is retained. The entropy of the sea wave spectrum is given by H = K f logN(k) dk and the total rate of entropy production dH/dt; 0. By considering the entropy production separately for the diffusive transfer and the pumped transfer, > (dH/dt)d and (dH/dt)p respectively, Webb showed that (dH/dt)d = O, whereas (dH/ dt) is of equal sign as (1/N -1/N )(1/N -l/N ). this gives an explana- p 1 3 2 4 tion for the growth of the spectral peak and its shifting to lower frequen cies during the wave generation stage. Equation (9.2) has been derived under the condition that the wave field is homogeneous. Furthermore it has been assumed that the distribution of the wave field is initially normal (Gaussian) and that it remains Gaussian; that is, the fourth order cumulant is taken to be zero. In weak turbulence theory this condition is known as the Millionshchikov zero-fourth-cumulant hypothesis (see, e.g., Monin and Yaglom (1975), p. 241 and Section 19.3). It is known that this closure hypothesis leads ultimately to negative energy densities. In wave interaction problems the hypothesis of the zero-fourth-cumulant is often called the random-phase approximation (see, e.g., Davidson (1972, p. 136) or Tsytovich ( 1970, p. 51)). Hasselmann (1962) used the random-phase approximation together with an up dating scheme and Benney and Saffman (1966) showed by using a multiple scale expansion that,for a simplified model equation, a closure assumption is not -2 needed on time scales of O(c ), where Eis the root-mean-square wave slope -4 parameter. Because the time scale of the energy flux is O(c ), this result is not enough, but Newell (1968) showed that, for water waves, even at this order no closure hypothesis is needed. A general discussion of weak coupling theory is given in Olbers (1979) and Hasselmann (1977) discusses coupling of systems with widely different intrinsic time scales. We note that for water waves exchange of energy takes place due to resonance between four wave components, as is expressed by Eq. (9.2); this description is obtained as a lowest order correction to the linear description. At higher order also resonances between more components takes place, but the lowest order spectral transport equations are already quite involved, Because the energy density spectrum is the Fourier transform - 101 - of the covariance, and thus of second order moments, of the free surface elevation, which is regarded as a stochastic variable, higher order moments also occur in the evolution equations for the covariances. Therefore there is need for some closure hypothesis. The hypothesis which is taken is that the random phase approximation is valid, i.e. the phases of the wave compo nents are uncorrelated, or, otherwise stated, the Gaussian distribution which is taken as initial condition is assumed to remain Gaussian for increasing time. In other words, this means that all higher-order cumulants are taken to be zero at that order of approximation of the wave field. In order to test this hypothesis at least bi-spectra, and possibly also tri-spectra have to be calculated along with the usual spectrum; an effect of non-linearity is namely that the initially uncorrelated phases become correlated to some extent, or, otherwise stated, bounded wave components are generated with the result that higher order moments become non-zero, as expressed for example by the skewness and kurtosis of the wave field. Due to the non-linearity the probability density function of the surface elevation becomes a Gram-Charlier distribution function of which the Gauss distribution is the first term when expanding into a series of the non-linearity parameter. We note that there is evidence that the deviation of normality is more pronounced for the velocity field and the pressure field of water waves than it is for the free surface elevation (restricting one-self to gravity waves); see, e.g., Imasato and Ichikawa (1977), where the influence in the water is responsible for this effect. These results are all valid for very weakly non-linear waves, close to the resonance curve. Non-resonant waves yield, as a result of interacting, already at lower order bounded waves, or, otherwise stated, they give a coherent phase shift. A rather complete investigation of the first non-zero correction to the lowest order gravity wave dispersion relation (for deep water) is given by Weber and Barrick (1977) by extending Stokes' method to random waves, see also the companion paper Barrick and Weber (1977). Recently Holloway (1978) proposed another kind of tumulant-discard hypothesis with which it would be possible to tackle also strong-interaction problems. The basic idea is as follows. An n-th order moment of random quantities a , ••• an can be reduced to a sum of products of lower order moments, plus a 1 irreducible term, the n-th order cumulant: - 102 - n-1 < a ... a > = I: < a ... a. > < a. ... a > + < a ... a (9. 3) 1 n 1 J J+ 1 n 1 n j=l where<·> is the ensemble mean. (For a discussion of the definition of cumu lants both for random variables and random fields, see Manin and Yaglom (1971, pp. 223 ff.).) Instead of taking the n-th order cumulant < a 1 ••• an >C O, Holloway substitutes for the n-th order cumulant a term linear in the (n-l)th cumulant with a forefactor which is an unknown function of lower cumulants. The problem is then to get a prescription for this forefactor. Depending on the equations under consideration, a conjecture is made to indentify this forefactor. It turns out that the manipulations concerning cumulants closely resemble the method of renormalizations used in quantum mechanics. We do not know yet how to apply this method to the problem of non-linear evolution of random water waves, but the method is mentioned because it might lead to a significant step forwards to strong wave interaction theories. A very much different method of describing non-linear random waves, is by supposing that the basic equations are given by the Davey and Stewartson equations, Eqs. (5.27). Longuet-Higgins (1976) considered the transfer of energy near the spectral peak; he considered a wave train of slowly modulated amplitude and phase and used the deep-water limit of the Davey and Stewartson equations. Writing s = Re EA exp{i(lx+iy-wt)}, (9.4) • -2 where k = o:,;) lS the carrier wave number and w = gk, k = 1t I, the D&S 0 0 equations (5.27) become in the deep water limit in units of length and time • - such that g = 1 ' k = ( 1 , 0) , w = 1 ' 2i c3A (9.5) dT where~= o(x-c t), n oy, T = oEt, o = E. g Longuet-Higgins now presented the envelope function A(~,n,T) in the form A= I: a (T)exp{i(A ~+µ n-w T)}, (9.6) n n n n n - 103 - where w = (9. 7) n The magnitudes of the relatively slowly varying amplitudes a (T) are small n and the phases of a are supposed to be uncorrelated to first order; in the n limit the a become density distributed ink-space in such a way that, when n -+ summed over an element dk, -+ -+ E 1 a a• '\; N(k)dk. 2 n n n -+ . N(k) is the local action density. The constant Min (9.7) is the total action density. The An' µn and wn can be related to the physical wave number and frequency. The result of the investigations of Longuet-Higgins was that for transfer of energy in the peak of the spectrum (i.e., k , k and k in (9.2) nearly equal) 2 3 4 the otherwise very complicated function G becomes simply (in dimensional 6 quantities) 4Tik . Numerical calculations of Fox (1976) based upom Longuet- o Higgin's (1976) model showed that most of the energy transfer occurs among groups of almost identical wave-numbers. For narrow spectra therefore Eq. (9.2) can be replaced by, for deep water waves, near the spectral peak at i,ave number k 0 3N 1 6 = 4Tik JJJ[(N +N )N N - (N +N )N N ] 3t O 1 2 3 4 3 4 1 2 Furthermore, it was shown that energy from an isolated peak in the spectrum tends to spread outwards along two characteristic lines in wave number space having angles+ a with the direction of the wave field, with a= arctan(l//2) = 35°. The important point for us is that Longuet-Higgins joined the two view points of regular slowly varying non-linear wave propagation and interaction of irregular (random) waves together. It is noted in passing that the random phase approximation remains necessary. - 104 - Alber (1978) derived an evolution equation for the wave spectrum by using the full set of the Davey and Stewartson equations and used this evolution equa tion to study the effect of randomness on the stability of two-dimensional wave trains (that is, the effect of randomness on the Benjamin-Feir type of side-band instability was studied). Alber assumed that the D&S equations (5.27) remained valid when the complex amplitude A(s,n,T) is a random func tion of (s,n). Introducing the two point correlation function p(x,r,T) = < A(x+! t,T) A + with X = (s,n), and<•> denotes the ensemble average, Alber derived an evolu- tion equation for p from the D&S equations. (A short account of his derivation is given in Appendix J.) The variations of p occur over length scales of - -L 2 order E A and time scales of order E- \ (as is the case for A); the 0 0 spectral width must be of order Ek, k = 2TT/A being the carrier wave num- o O 0 ber. By taking the Fourier transform of the evolution equation for p tor, an evolution equation for the wave spectrum is found (cf, Eq. (JI!)). Alber found in his transport equation for a narrow-banded spectrum non-linear re fraction terms which are similar to those found by Willebrand (1975) and by Watson and West (1975). These terms are of second order in the wave spectrum and vanish only when the spectrum is homogeneous in (s,n) or when the wave system is stable to small spatial perturbations (cf, Eq. (Jl3)). Specializing the transport equation for the spectrum for deep water waves, Alber studied the homogeneous solution (Jl4) (the equivalent of the Stokes' wave train, Eq. (C2)) for its stability to small-amplitude, long wave length perturba tions in the form of oblique spatial waves, The principal result is that the instability diminishes and vanishes as the spectral breadth increases. In other words, the decorrelation of the phases of the wave envelope leads to stabilizing of the wave train. Lake and Yuen (1978) even proposed to describe non-linear wind-wave systems by coherent bound wave systems, The idea is that the individual components in the wind-wave spectrum do not propagate as free waves and do not obey the usual dispersion relation. Much of their argumentation for taking the non linearity more important than the randomness rests on results of experiments of Ramamonjiarisoa andCoantic (1976) carried out in a wind-wave flume, which amounted to the fact that, measured in _the dominant direction of wave propa gation, the phase velocities of wave components with frequencies f abov.e the - 105 - peak frequency f in the spectrum were the same for increasing frequency. p Otherwise stated, the coherence of wave components with f > f is nearly p constant for increasing f, an observation already made by Yefimov et al (1972) as a result of measurements at sea. In this connection it is noted that Longuet-Higgins (1977) gave an explanation for this behaviour which amounts to a propagation with the direction± 0 of the dominant wave direc tion of components with equal f, such that c/c = cos8. 0 Recently,Yuen and Lake (1979) gave a description of the theoretical model which amounts to a mathematical description of their (1978) observations (hypothesis) that a developing wind-wave system under steady wind condition can be well characterized by a single non-linear wave train with a carrier frequency equal to the dominant frequency (peak frequency) of the wind-wave system. The model consists of the· non-linear Schrodinger equation for deep water, Eq. (5.6), used together with a simple form drag model of Deardorff to describe the input of wind energy into the dominant wave. The NLS equation has now slowly varying time-dependent coefficients. For the wave spectrum an evolution equation similar to (J13) is derived, now without dependence on the transverse direction n. No quantitative results are given in this paper. It is stressed that the model proposed is a purely one-dimensional model. Mollo-Christensen and Ramamonjiarisoa (1978) proposed a wind-wave model in which wave groups (solitons) occur spatially random in space; these wave groups fulfil the NLS equation. Dorman and Mollo-Christensen (1977) gave arguments for the suggestion that wind waves may be tended to be generated in groups. Extending the instability theory of Miles of the early stages of wave generation due to wind to slowly varying wave trains, it is found that the growth rate is the largest where the initial amplitude is the largest and that therefore modulational maxima will be amplified the most. Because of the Benjamin-Feir type of instability of waves, the wave group remains as a group. Another cause for the generation of wave groups might be the gustiness of the wind field, but the instability of finite amplitude Stokes waves is likely to be the most important cause for the formation of wave groups. Above a few papers are mentioned in which the wave envelope description of the one-dimensional NLS equation or the two-dimensional D&S-equations is used to describe random wave fields in which group behaviour (solitons) is to be thought to be of more importance than is found from the usual descrip- - 106 - tion of random wave fields as an infinite sum of sine waves. A difficulty is that in the energy density spectral approach the "groupiness" of a time series cannot be identified very good other than by the spectral breadth. However, it has been shown recently that realizations with the same spectrum can have a completely different behaviour with respect to wave groups (see, e.g., Burcharth (1979)). Coastal structures respond differently to wave patterns having a different group structure, albeit the same energy density spectrum. It is thus evident, at least in problems where the stability of coastal structures is concerned, that the usual (linear) spectral approach has to be reconsidered. In this sense an approach as the one of Mollo-Christensen and Ramamonjiarisoa (1978) might be fruitful, especially in wave-flume problems. It has to be noted that the D&S equations yield solutions of wave groups that are propagating under angles+ a of the direction of propagation of the carrier wave. This is easiest seen from the deep water limit of the D&S equations, Eq. (5.29). Note that this property of different directions of the group and phase velocity vectors is clearly a non-linear effect, because the bottom is taken to be horizontal and no pre-existing currents are taken into account in the derivation of the D&S equations. A rather complete account of the wave group solutions of Eq. (5.29) in various directions a is given recently by Hui and Hamilton (1979). Because the spectral transport equation as given by Alber (1978) is derived directly from the D&S equations, the property of energy transport in a direction different from the direction of the wave field should also be included in that transport equation. So far only the case of a horizontal bottom is discussed, and the wave field is homogeneous (statistically). The spectral transport equations of Wille brand (1975) and of Watson and West (1975) give corrections, which are of second order in the spectral density, to the transport equation as derived by Hasselmann; these corrections are due to the inhomogeneity of the wave field. Watson and West (1975) take into account a pre-existing current U(1,t), to account for the inhomogeneity of the wave field. Willebrand (1975) uses Whitham's method to derive the spectral transport equation for irregular waves propagating in water with a non-constant depth h(i). Both models are non-linear and in both cases the group velocity vector need not be collinear with the phase velocity vector, even in the homogeneous case (i.e., h = constant, U = O). Terms which represent- higher order dispersion are not present here, but the models of Willebrand (1975) and of Watson and West - 107 - (1975) have the advantage over the model of Alber (1978) that they are not restricted to narrow spectra and that non-uniformity of the medium is taken into account. A spectral transport model which includes all advantages of the models of Alber (1978), based on the D&S equations, and of Willebrand (1975), based on Whitham's method, is then a spectral transport model based on Chu and Mei's (1970) method. Presumably equations (5.49) could be used to extend them to random wave propagation in a similar way as was done by Alber (1978) taking the D&S equations (5.27). For the interpretation of such a intricate spectral transport model, simpler spectral transport models such as the model based on the deep water limit (5.29) of the D&S equations, Alber's (1978) spectral model or Willebrand's (1975) model or the one of Watson and West (1975) could be useful. Because these so-called 11 simpler11 models are themselves already quite complicated, a study of the diffusion of wave action density along lines as given by McComas and Bretherton (1977) for resonant interaction of internal waves or West (1978) might be useful. Lastly, we mention the finding that all one-dimensional solitons are unstable with respect to long transverse perturbations; that is, the wave group solu tions of the D&S equations are unstable, (cf. Ablowitz and Segur (1979)). Such an instability cannot be observed in wave flumes, because the width of the flume is too small for oblique long wave perturbations to be able to exist. In this respect it is noted that the wind-wave model as proposed by Yen and Lake (1979) is based almost exclusively on the interpretation of wave flume experiments. Another paper worth mentioning is the recent paper by Kawahara and Jeffrey (1979) in which two forms of Boussinesq's equation are taken to study the dynamic behaviour of an ensemble of short waves interacting with a long wave. Several asyn1ptotic kinetic equations for a wave system composed of a single long wave and an ensemble of short waves with a corltinuous spectrum are derived. Only wave propagation in one spatial dimension is considered. One of the two model equations allows for four wave resonance, and the other one allows for three wave resonance. The evolution equations result from non-secularity requirements resulting from a multiple-scale expansion technique (to be precise, the derivative-expansion technique, see Nayfeh (1973, Section 6.2), is used); no closure hypotheses are needed. Among the evolution equations - 108 - discussed is one which reduces upon neglection of the long wave to a gener alized NLS equation as obtained by Hasegawa (1975) by invoking the random phase approximation. The reason why the paper of Kawahara and Jeffrey (1979) is mentioned, is threefold; firstly some simple spectral transport equations are discussed lucidly; secondly, it appears to be not necessary to invoke some statistical closure hypothesis; thirdly, although the technique is only applied to a simple model equation, the method seems to be applicable to other governing equations. Note in this respect that multiple scale techniques are often illustrated by using simple equations such as Boussinesq equations (cf., i.e., Catignol (1977)). - 109 - 10 Conclusions and recommendations Some methods are described in the previous Chapters with which problems of non-linear wave propagation may be described. Apart from a numerical method to solve Boussinesq-like equations for fairly long waves, the basic assump tion is in all cases that the wave field is slowly varying. In the case of fairly long waves this leads to a generalized KdV equation from which the refraction of cnoidal waves may be computed. Most attention has been paid to the propagation of modulated Stokes' waves. One of the principal effects of non-linearity of the wave field in water of finite depth is that the waves and the underlying mean flow field are coupled; an energy exchange exists between these two fields. The wave energy is then not a conserved quantity; the quantity that is conserved is wave action which has the dimension energy x time. Wave action is usually regarded as being an adiabatic invariant, that is, wave action is only defined asymptotically. Recently, Andrews and McIntyre succeeded to give a definition of wave action which is exact to all orders of aµproximation; a formulation in Lagrangian variables was required. The solution of slowly varying non-linear wave systems rests always in some sense on a multiple scale expansion technique. Because we are concerned with wave motions in which a phase function can be identified, the multiple scale technique used here is in fact a non-linear extension of geometric optics and the WKBJ technique (for the latter techniques one is referred to Jones (1964), Felsen and Marcuvitz (1973), Froman and Froman (1965) and Olver (1974)). Two different ways to apply the multiple scale technique are described. One in which the multiple scale technique is applied directly to the variational principle from which the governing equations may be derived. This results in the average Lagrangian technique, see Chapter 3. The way in which Whitham applies this average Lagrangian technique is valid only for the modulation parameter o to be much smaller than the non-linearity parameter E (= ka); this can also be stated as foflows: the effect of group velocity dispersion is much smaller than the effect of non-linearity. When these effects are taken to be of the same order of magnitude, higher order derivatives of the amplitude to x and t occur; these terms can be viewed as diffraction-like terms and do extend the validity of the resulting evolution equations for the wave parameters over longer scales. It is seen that, with E ~ o, the average Lagrangian technique leads again to the same results as a multiple scale expansion, see Section 5.2. - 110 - In Chapter 5 attention is given to multiple scale expansion techniques applied directly tc the governing equations; sand o are taken of equal order of magnitude. Much attention has been given to the so-called nearly mono chromatic nearly one-dimensional waves in which case the Davey and Stewartson equations are obtained; for purely one-dimensional waves these equations reduce to the one-dimensional non-linear Schrodinger equation. These equa tions are discussed also in Appendices C and D. Furthermore, the equations of Chu and Mei (1970) are given in Section 5.5; the variation of k is here O(o) instead of O(os) as in the D&S equations (measured on wave length scale). In Chapter 7 a few methods are described for the description of slowly varying fairly long waves in case of mild bottom slopes. It is then found to be necessary that jVhj << a/A, as is also the case for wave groups on water of varying depth in the NLS description, in order that terms in which deriva- tives of h occur are small compared to the other terms so that a cnoidal wave on water of constant depth remains cnoidal in form in the region with sloping bottom. The generalized KdV equation is obtained from a set of Boussinesq like equations by applying the multiple scale expansion technique, see Sec tions 7.2 and 7.6. One paper concerning a direct numerical solution of a set of Boussinesq-like equations for two spatial dimensions is mentioned in Section 7.4, the paper by Abbott et al (1978). Whereas a direct numerical solution of some set of Boussinesq-like equations could yield useful results, we do not recommend to use the specific method of solution of Abbott et al (1978) without further investigations. Especially the applicability for interesting cases (rather short fairly long waves) has to be investigated because the examples shown in the paper do not substantiate their claims of applicability. A direct numerical solution of Boussinesq-like equations can be compared to some extent to the numerical solution of tidal flow equations (very long waves). Whereas in the first instance the region of interest is many wave lengths long,in the second instance the dimension of the region of interest is much smaller than the wave length. Long wave equations can be of use very close to tne coast; a few comments are made therefore on long non-dispersive wave equations in Chapter 8. In Chapter 9 a short discussion is given on non-linear irregular wave propa gation problems. In essence three different aspects are discussed. At first Hasselmann's weak interaction theory is mentioned and a few implications of the change of wave action due to resonance are discussed. Secondly, the aspect of a closure hypothesis is briefly discussed. Holloway (1978) sketched - 111 - a method of closure which would also be valid to strong wave interactions and not only to weak interactions; this method requires more study in order to determine the implications for random wave propagation problems clearly. Thirdly, the use of evolution equations derived for slowly varying regular waves for a description of irregular waves is discussed. Both the one-dimen sional NLS equation and the two-dimensional D&S equations for nearly one dimensional wave trains are used as model equation. These descriptions are only valid for very narrow spectra because both are derived for nearly monochromatic wave trains. In this Report the consequences of various non-linear models are only dis cussed to a limited extent. This is due to the large number of often mathe matically very involved papers on· special cases of non-linear wave propaga tion problems, often being of importance to some aspects of non-linear water wave propagation problems. In order to decide which method should be worked out, possibly numerically, for TOW purposes, it has to be kept in mind that the principal aim of TOW Coastal Research Programme is to obtain good estimates for the sand transport in coastal areas. In our view the investigation of non-linear regular water wave propagation problems has to be carried out in such a way that results of regular wave propagation problems lead to a considerably better understanding of random wave propagation problems, because it is random wave propagation which is really of importance for coastal studies. Nearly all wave propagation models are based on small amplitude and are valid for slowly varying waves. In order to be able to make considerably progress towards essential non-linear wave propagation problems, the wave action concept of Andrews and McIntyre (1978b) should be studied closely and, in order to achieve a similar result in random wave problems, the closure hypotheses have to be studied; possibly a step forward can be made by using concepts as proposed by Holloway (1978). A simpler extension of linear wave propagation problems is provided by a numerical solution of the set of evolution equations of Chu and Mei (1970), Eqs. (5.49). This set of equations gives the most general model for studying wave behaviour on water of slowly varying depth in the weakly non-linear approximation. Special results can be obtained from simpler model equations in which further approximations have been made, such as the Davey and - 112 - Stewartson equations (5.27), which equations are only valid for horizontal bottom geometries. We think it to be possible to derive a D&S-like set of equations, with coefficients depending on the spatial coordinates, in which inhomogeneous terms occur, just as is the case in the one-dimensional uneven bottom extension of the NLS equation, see Eq. (5.44). For very small bottom slopes the inhomogeneous terms can be regarded as perturbation terms. An investigation of the behaviour of special solutions of the ordinary D&S equations due to the inhomogeneity is useful, in our view, for interpretation purposes of Chu and Mei's equations. Recommendations Our recommendations for further research on regular and irregular non-linear wave propagation problems now comprise the following parts, numbered in order of increasing complexity. I. Adiabatic approximation for the behaviour of permanent waves or soliton like solutions due to slow non-uniformities of the medium. 1 .I. Cnoidal wave refraction for quasi-two-dimensional situations as proposed by Skovgaard and Petersen (1977) seems to be a useful method for engineering problems. A comparison between the formulae derived by Miles (1979) for a slowly varying cnoidal wave and those on which Skovgaards and Petersen results are based has to be carried out. The average Lagrangian approach for cnoidal waves might yield a check on the assumption that en~rgy is transported in a direction perpendicular to the wave front also for non-linear waves. 1 .2. An inhomogeneous version of the D&S equations has to be derived and the consequences for small bottom slopes have to be studied; these inhomogeneous equations might be obtained'from Chu and Mei's equations by taking the variation of k and w to be small on X,T scales. As mentioned in Chapter 6, an inhomogeneous NLS equation along rays could result from such an investigation. Problems in solving such an equation are of the same kind as the ones for solving the inhomoge neous KdV equation. Refraction .of non-linear wave groups can be con sidered then. - 113 - 1 .3. An extension of the method of Peregrine and Thomas (1976) for the refraction of essentially non-linear waves due to currents might be possible for non-linear waves in water of varying depth. This approach is based on the use of an average Lagrangian which is defined by using the integral relations for energy, etc. as given by Cokelet (1977). It is not yet clear to us whether it is useful to consider such a method without accounting for the mean motion variations as occur always when non-linear waves propagate in water of finite depth. (In passing, we note that such a phenomenon also occurred in solving the KdV equation for variable depth; a depression or elevation was found when considering the next approximation.) Another problem, at least conceptually, is the symmetric wave forms of Cokelet and thus distortion of the wave shape seems not to be possible; this has to be investigated further. 2. Higher-order approximation (in bottom slope parameter) of initially perma nent waves and soliton-like solutions due to slow non-uniformities of the medium. 2.1. A check of and interpretation of the results of the method of Svendsen and Buhr Hansen (1978) is necessary for the reasons mentioned in Section 7.6. Especially the distortion of the cnoidal wave profile and the absence of a variation of the mean motion needs attention. 2.2. An investigation of the distortion of non-linear wave groups as obtained from the NLS equation has to be carried out. Some solutions are available in the literature (e.g., Karpman and Maslov (1977b), Kaup and Newell (1978) and many other papers). Again, in the first instance, it is an interpretation of the available results which is needed. 3. Numerical solutions. 3.1. A numerical solution of a set of Boussinesq-like equations. Care should be taken to use a set of equations which has good dynamic behaviour. One set with varying depth is given by Broer et al (1976), however, only for small bottom slopes CIVhl << h/A). It seems possi ble to derive such a set of equations also with relative ease, for slopes IVhl = O(h/A). - 114 - 3.2. A numerical solution of Chu and Mei's equations. Because of possible instability of the waves, much attention has to be given to the finite-difference method to be chosen. Note that the applicability of these equations is larger, for TOW purposes, than that of Boussi nesq-like equations. 4. The wave action concept of Andrews and McIntyre has to be studied closer. Note that it becomes possible now also to consider refraction due to • currents which are not uniform in the vertical coordinate, U(i,z,t). Another extension can be made to waves which may be rotational. The most important aspect of the wave action concept of Andrews and McIntyre is that it does not rest on small amplitude approximation or on slowly varying waves; fast varying waves as occur in diffraction of waves due to coastal constructions may also be treated. 5. For random wave propagation problems, the methods of Willebrand (1975) or of Watson and West (1975) seem the most appropriate ones to start with, especially because these models account for the effect of the statistical non-uniformity of the wave field in regions with sloping bottom. 6. It has to be investigated in which way the wave action concepts of Andrews and McIntyre can be used to extend the presently available spectral transport models based on weak interaction of waves (Hasselmann's models). When it can be used, the random phase approximation has to be reconsidered, e.g., by using concepts as proposed by Holloway (1978). 7. For problems of harbour oscillations the "groupiness" of the free surface elevation may be important, because it has been shown that non-linear wave groups excite long wave components; the so-called mean free surface ele vation and induced mean current are seen to have a long-wave character and can only be regarded as being a (slowly varying) mean on wave length scale of the individual waves. - 115 - APPENDIX A - The scaling in the NLS approximation In order to see why the coordinates T and~ in the NLS equation (5.1) are 2 T = s t, ~ = s(x-c t), we follow simple arguments of Asano (1974). Consider g two waves with (w,k) and (w' ,k') as pairs' of frequency and wave number. When the differences n = w' - wand K = k' - k are small, the resulting wave has a long-wave evelope with wave number Kand frequency n; the dispersion rela tion is + • • . ' (Al) where c(o) = 8w/8k is the group velocity of the carrier wave. The linear g phase velocity of the wave envelope, V = n/K, is then, + ••. (A2) On the other hand, considering the resulting wave with (n,K) to be a non linear long wave (i.e., without frequency dispersion), its velocity may be described by the characteristic velocity dx/dt, which can be written, ex panded in powers of s, as 2 ( 2) dx = c(o) + sc(l) + E C + ••• (A3) dt g g g The coupling between the modulation and the non-linearity is strongest when both are of the same order, and thus K = O(s); because we already had K = O(o), this means o ~ s. Note, moreover, that this is similar to the case of stationary long waves where the linear dispersive long wave velocity ,-;- I 2 c ~ Y gh { 1 - (kh) + ••• } and the non-linear non-dispersive long wave 3 ' 2 velocity is c ~ /gh {! + ½a/h + ••. }; for permanency (kh) ~ a/h was obtained. Because K = O(o) measured with respect to the carrier wave number k, the scale on which the wave group has to be considered is A= A/o, and thus X = ox where x is scaled with A= 2TI/k. In order to remain near the centre of the wave group, a moving coordinate system has to be taken; the coordinate along the characteristic curve is then - 116 - ~ = o(x-c(o)t). Because one has x = o-l~ it follows that dx/dt = -I g (o) o dt;/dt + c . From (A3) one has dx/dt + cC 0 ). In order that these g g expressions are the same, one has I d~ _ ( ) oE.: dt - C g l • Introducing the slow time T = oE.:t, one obtaines d~/dT = C(l) (~,T), g This produces ' T = OE.:t. (A4) - 117 - APPENDIX B - Derivation of the NLS equation from Eqs. (5.3) and (5.4). In this Appendix the NLS equation is derived from Eqs. (5.3) and (5.4): 2 2 a (a ) + ..L (c ~) 0 clT w ax g w 0 0 (BI) elk + clw = aT ax 0- · and (B2) I where w = (gk) 2 and c = dw /dk. 0 g 0 Let the variation of k on (X,T) scale be O(E): k = k + EK(X,T). (B3) This means that a nearly monochromatic wave train is considered and the wave envelope (with carrier wave number k) is varying O(I) on scale A/ (Eo), or A/E, where A= 2n/k. It follows from Taylor series expansions that then de 2 __g K2 w = w + Ee K + E + ... 0 g 2 dk (B4) de _£ C = C + E + g g ... dk- ' I 2 I - ½ where w = w (k) = (gk) and C = dw/dk = 2(g/k) " 0 g The first of Eqs. (B 1) can be written as 2 clc cl a 2 a a a 2 + C + a _£ = 0. (B5) (clT + cg ax)a w (3T g ax)wo ax 0 Substitution of (B4) yields then - 118 - a - 3 2 deg a 2 a2 - a - a 2 ("T + C '\)a + E - -;:;-- (Ka ) - E - C (-;:;-- + cg "X)K + O(E ) o. o g oX dk oX W g oT o (B6) Substitution of (B4) into the second of Eqs. (BI) yields It follows thus from (B7) that 3K/3T + c 3K/3X = O(E) and the third term of 2 g (B6) is thus O(E) and can thus be neglected, The result of the ansatz (B3) is that equations (BI) reduce for nearly uniform wave trains to Eq. (B7) and (BS) It is noted that Eqs. (B7) and (BS) correspond to Eqs, (2.3a,b) of Chu and 2 2 Mei (1971) except for the term (3 a/3T )/(2wa) in (B7) which was replaced by - 2 2 - them by -c (3 a/3X )/(2wa) in Eq. (2.3b) of Chu and Mei (1971). Using the g order relation which follows from (BS), it is possible to write _2 a2a c -- + O(E), g ax2 and Eq. (B7) can be written to the same order of approximation as _2 2 2 2 2 (__§__ + -c __§__) K + _!_ E __§__ ldeg K + w-k- a + _cg, ~i + O(E)2 = 0. (B9) "T g "X 2 "X 2 0 0 0 dk wa ax 2 2 We conclude therefore that the minus-sign in front of a a/ax in Eq. (2.3b) of Chu and Mei (1971) is a printing error. Introduction of the moving coordinate frame - 119 - X - c T T = ET, g then yields 3a2 3 de 2 a:r- +~(~Ka)= 0 dk (BIO) -2 + cg 32a~ = - 2 0. wa 3~ Introducing de W = -2 ~ K > 0, (B 11) dk one obtains 2 3a 3 I 2 F + ~ (- 2 Wa) = 0 (B 12) 2 3W + ~ ~ _ J.. W2 I - 2 + gk - 3 _I_ 3 a i = O 3T 3~) 4 + 4 gka 16a ~2~ ' 3 1 312 where w = (gk)½, - (g/k)½, d~ /dk = - J.. g½k- have been substituted. ~ g - 2 g 4 Eqs. (Bl2) are the dimensional form of Eqs. (2.Sa,b) of Chu and Mei (1971). Davey (1972) now introduced 3 3a2 3 3,1, 2 F +~cat a) = o (Bl4) and (BIS) Equation (BIS) is integrated with respect to~- The constant of integration - 120 - (a function of T) may be set equal to zero because~ is undefined to within an additive arbitrary constant of T. Equation (BIS) can thus be written as 2 - 2 a~ ..!_(~) 2 gka g a a = (Bl6) dT + 2 d<; 8 -3 -2 o. 32k a cl.; Equations (Bl4) and (Bl6) can be combined into a single, complex equation by means of the transformation I A(.;,T) = a(t;:, T) exp{U ~ (.;, T)}, (B 17) 1 I ½--3/2 - l ~ g k , or, with w = (gk) 2, Al = I The result is: (BI 8) with - 2 I __ ) -w/(Sk) and v = - wk 1 2 (Bl8a) Equation (BIS) is the NLS equation (5.1) and the coefficients A1 and v 1 are, for the case of deep water given by (Bl8a). Remark. The precise form of the transformation (Bl7) is actually found by trying A = a exp(if~), where f do.es not depend on .; and T and has the dimension 2 m- s so that f~ is dimensionless. Supposing the resulting equation in A to be of the form (BIS), with un known coefficients A and V , a exp(if$) is substituted into (BIS) 1 1 A= and the real and imaginary part are both set equal to zero. These equa tions have to be reworked so as to obtain equations similar to (B14) and (Bl6); multiplication of (Bl6) with -fa and division of (Bl4) by 2a results in equations of similar form. From the condition that coefficients of similar terms are equal, three independent conditions for the three 2 unknowns f, A and v result. Choosing for f the negative root of f = 1 1 results in A < 0, v > 0; the ch~ice f > 0 would result in A > 0 and 1 1 1 v1 < 0 with the same values for IA 1 1, lv 1 1. Notice that these two choices of sign for f result in equations for A or A* respectively. - 121 - Equation (BIS) can be written as 0. (B 19) The complex conjugated equation becomes 2 "A*0 "0 A* - 2 1. + A - + A A* = 0 , (B20) I I - I\) 1 I I I F 1 at:2 or, from (B 18) , A normalized form of the NLS equation is often used in the literature. It is practical to write this equation in dimensionless quantities. Therefore the dimensionless T and E: are defined by T and A is made dimensionless with a characteristic amplitude a: B = A* /a. One obtains from (B21) 1 2 1 3B 2 2 2 - --a B +\/a 2 - I I 2 = k (gk) A1 (gk) B B 0. ai ai2 i Introducing the dimensionless c~efficients (B22) this equation reads 1.n dimensionless form (B23) The normalized form of the dimensionless NLS equation is usually written as - 122 - clq a2q 2 1. -+ 0 = o, (B24) clt -2 + 1lql q dX where ~ Z:1 z: ½ X = = = (B25) t -Ai1 E, 01 sign(_/\.) q l_l I B. ' 1 1 Note that /1. 1 < 0 because A1 < 0 and that x and tare here scaled dimensionless 2 variables; t = -k A1T. For the case that A is the amplitude of the velocity potential, the dimension 2 -1 l -2 of A isms and for a may be chosen (gk) 2 k . It is seen that 01 -1 corresponds with A1\\ > 0 and thus with kh < 1. 363; 0 1 = +1 corresponds with A1V1 < O, kh > 1.363. Soliton behaviour is obtained for 0 = +1. 1 The inhomogeneous NLS equation Consider the inhomogeneous NLS equation (5.44) with coefficients given by (5.42): -iµ B. (B26) 1 2 -1 Notice that B has the dimension of velocity potential, ms . This equation may be written in dimensionless form by introducing the following dimensionless variables and coefficients: _l I ~ 2 q = a (g/k) zB E, = kt, T = (gk) T -1 I (B27) 2 -2 -1 = = a k gV = k µ. Al gAl Ll 1 fl Dropping the tildes, Eq. (B26) becomes in dimensionless form 2 2 1 ~~+A (E,)~ - z: (t;,)lql q = -ir (E,)q. (B28) OL, 1 dT 2 1 1 This equation can be transformed into an inhomogeneous NLS equation in which - 123 - the coefficients in the homogeneous part are constants; the transformation is similar to the one used in the case of the generalized KdV equation, see (7.47). Write (B29) with which Eq. (B28) becomes . ap dr - 1 a 2p 2 dr)-1 2 _ • ( dr)-1 1 df) i at;: + A1 ( dt;:) a·r2 - E 1 If I ~dt;: IP I P - -i dt;: ~r 1 + I dt;: P. (BJO) For envelope soliton solutions to be possible, one has E > 0 (and thus 1 A E < 0 because A < O). In this· case the conditions for rand fare taken 1 1 1 as and (B31) resulting in dr dt; = -Al (t;:) > 0 The function f(t;) may be taken real because if f(t;) is taken complex, then the coefficient of pin the right-hand member of (B30) also has a real part; notice that such a real part was cancelled from (5.41) by invoking the transformation (5.43) so as to obtain (5.44) or (B26). With the choice f(t;) > 0 one obtains (B32) Denoting the right-hand memher of (B30) by -iy (t;:)p, one obtains by substitu- 1 tion of (B32) and, subsequently, supposing y (and thus also A , E and f ) 1 1 1 1 to be a function of r, (B33) with - 124 - 'Y 1 (r) (B34) Equation (B33) is usually taken as the perturbed NLS equation for which solutions are sought by perturbing soliton solutions of the homogeneous NLS equation; see Karpman and Maslov ( 1977b) and Kaup and Newell ( 1978). Similar procedures are used for the perturbed KdV equation, see Section 7.6. Notice that different signs in (B33) may be obtained by considering p* instead 2 2 2 2 E < of p. If 1 O, the signs of a p/3T and IPJ p are equal and lfl = Al/El > o. - 125 - APPENDIX C- Some solutions of the non-linear Schrodinger equation Consider the NLS equation 1 ~A+ A cl2A2 - V IA/2A = o. (C 1) oT lclt; 1 Note that \ = I w" (k) and thus \ < 0 for water waves without surface ten sion. The coefficient V , as given by (5.28), is negative for kh < 1.363 and 1 positive above this value. (The actual change of sign of v 1 is obtained for a value of kh just below kh = 1.3628.) The simplest solution of (Cl) is ·the Stokes' wave train which is independent oft;. The solution is then 2 A(T) = A exp(-iv A ,). (C2) 0 1 0 These solutions are found to be unstable for long wave perturbations (side band perturbations) when kh > I .363, and thus when A V < 0. For A V > 0 1 1 1 1 stable solutions of (Cl) can be found (see also Appendix K). A stationary wave packet solution of equation (Cl) can be obatined by putting A(t;,T) = b(X) exp{i(rt;-sT)} , (C3) where X = t;-vT is a moving coordinate with respect to the frame t;,T and v, r and s are constants. It is noted that there is thus = o{x-(c +sv)t}, where X g x and tare the physical independent coordinates. It becomes clear by substi- tuting (C3) into (Cl) that it is necessary to chooser= v/(2A) in order for 1 2 b to be real. Putting s = v /(4A -S), the following ordinary differential 1 equation for b(X) is obtained: 3 A b" - Sb+ v b = O, (C4) 1 l where a prime denotes differentiation to the argument. ~ Taking b,~' • 0 for X +; 00 , this differential equation has the following solution, provided that 8A > 0 and V A < O, 1 1 1 - 126 - l I 2 2 b = (£) sech{(~) (~-vT)}. (CS) -v I\ 1 1 l 2 Introducing the amplitude b = (-2S/v ) , the solution for A becomes 0 1 2 A(~,T) = b sech{(- V~ ) ½b (~-vT)}.expri{ ~ ~ -(v - 0 2 0 2 4 /\1 l' /\1 "Al The solution (C6) is a one-soliton solution of the NLS equation (Cl); there are two free parameters, the amplitude b and the velocity v. In general, 0 for an initial condition A(~,0) with A+ 0 fast enough for~++ 00 , the solution of the NLS equation (Cl) consists of a train of solitons (C6) with parameters a ,v and an oscillatory tail. This tail contains a relatively n n small amount of energy. Such an oscillatory tail is given in the literature (see, e.g., Ablowitz and Segur (1979)) in the following form. Using instead of (Cl) the normalized and dimensionless version (B24) of the NLS equation, (C7) an exact solution of (C7) is given by q (CB) where A and ~ are real constants. 2 2 In terms of A(~,T) the solution (CB) becomes A(~,T) (C9) Hasimoto and Ono (1972) found the following special solutions of (Cl) by writing A(~,T) = b(K~) exp(iw2 T): - 127 - 2w 2 < 0 b = b dn b - "1A ~Im m = 2- --2 \'-\ 0 l 0 ( 2 y ( 1 \\ bo (CIO) 1 A v > 0 b = m = - ( I+--2w, 1 1 b0 sn)b 0 (~~ m) slm( 2r 1 \!lbo When the elliptic parameter m = I, these solutions reduce to the envelope soliton and to the phase-jump soliton respectively; the latter one is a special case of the envelope-hole soliton, A v < 0 A(~ ,r) 1 1 (C I I) A(~,T) Note that the first expression of (Cl!) is obtained from (C6) by choosing V = 0, The NLS equation (Cl) has an infinite set of conservation laws. The conserved quantities of the scaled NLS equation (C7) can be given as (cf. Zakharov and Shabat (1972), who were the first to solve the NLS equation by the inverse spectral transform) CX) (2i)nC = !_ fn(x)dx n= I , 2, ••• (C12) n 00 with f f = d (~) + f (Cl3) n+l q dx q 1: fjfk j+k=n 1 The first four conserved quantities are - 128 - 01 2 2iC = /'' lq(x,t) 1 dx 1 2 -co 0 2 1 rco (2i) c = J (q*q_ -qq*)dx 2 -4 -co X X (Cl4) 0 0 3 1 co 1 (2i) c = J 0 4 01 1 (2i) c = Jco (qq* + 3 q q* I q I 2) dx . 4 2 -co XXX 2 X Usually real conserved quantities are used with other numerical factors. With 0 = I the quantities In as given by Watanabe et al (1979) are 1 co I = J_co (q* q -q/1' q) dx 2 2i X X (CIS) 2 4 I = Jco (lq l -lql )dx 3 -co X I = Jco {(q*q -q* q)+ 3lql2(q*q -q*q)}dx. 4 2i -co XXX XXX X X The conserved quantities C or I are calculated from the initial condition n n q(x,O). Besides these conserved quantities, one has also the following one for Eq. (C7) (with 0 = I), as ~eported by Watanabe et al (1979): 1 I = Jco {xlql2 - 2t.(q*q -qq*)}dx. (Cl6) 0 -co 1. X X Differentiating I (t), one obtains 0 d Jco xlql2dx = (Cl 7) dt -co and because 1 is time independen~ the left-hand side 1.s also time independent. 2 It is now possible to define from (Cl7) the velocity of the centre of gravity of q(x,t); defining the centre of gravity xE by - 129 - (Cl8) it follows that (Cl9) and dxE/dt, being the velocity of the centre of gravity is thus a conserved quantity. An envelope soliton thus propagates with a constant velocity, equal to the velocity of its centre of ''mass'' (in fact the centre of gravity of 2 energy because A in (Cl) is a measure for the energy in the wave train). Note the similarity with the non~linear group velocity discussed in Chapter 3. Note also the correspondence between (C19) and (D12). The conserved quantities can be used to construct purely soliton solutions of the NLS equation; radiation is then ignored. For a one soliton the first two conserved quantities I ,I are required and for n-soliton solutions the l 2 first 2n conserved quantities are needed. A simple physical interpretation of the conserved quantities is provided by Ablowitz and Segur (1979). Because the model equation is obtained by invoking 3 perturbation series as (5.21) and O(E ) equations are used, the mass, momentum and energy are also calculated to this order of approximation, The mass M of oo n a wave is then expressed as M = E E c ; with M constant all c have to be n= 1 n n constant too. Ablowitz and Segur obtain then expressions of the general form: 2 M = Ea I + E a I + 0(~3) 1 1 2 2 2 3 m = E b I + O(E ) X 2 2 (C20) 2 3 = Ee I + E C I + O(E ) Ek 1 l 2 2 3 3 E = Ec 1 + E c 1 + O(E). p 1 1 2 2 where the. coefficients (a.,b.,c.) can be calculated from the conservation i i i laws of mass M, momentum m and energy E. Thus, not I ,I ,I represent the conservation of mass, momentum and energy -- l 2 3 of the water wave, but expressions (C20) do. - 130 - Inhomogeneous NLS equations are obtained when the depth is a function of x, . 1 h = h(x ), or when dissipation terms are considered. Sometimes it is possible 1 to transform such equations back to the standard NLS equation. We mention here a form of the NLS equation with linearly x-dependent coefficients as given by Calogero and Degasperis (1978) which has this property and which equation can be solved by the inverse spectral method. Consider the following equation (given in the notation of Calogero and· Degasperis) 2 o o a 2 i -1 + (y +iµ +µ x)q + i(y +µ x) -1 + (y +µ x)(____g_ + 2lql q) + at o 1 o 1 1 ax 2 2 ax 2 (C21) 00 1 2 1 + 2µ (~ - q ! lq(x ,t) i dx ) = O, 2 OX X 2 2 2 whereµ. ,y. are real constants. Because the sign of o q/ox and jqi q is i i equal, soliton solutions are possible. When in (C21) µ = O, then (C21) can be reduced to the standard NLS form by 2 the transformation = a(t)exp[Hn(t) + x t t 2 n (t) = - ft dt'{y (t')qi (t') + y (t') f3(t) = - ft dt' a(t'){y (t') + 2y (t') = With (C22) and (C23) Eq. (C21) with µ 2 0 transforms into - 131 - 2 i a~+a~ y2(t)a 2(t) [aayi + 2lcil 2]ci = o q = q(y,t) (C24) A time-dependent rescaling oft leads then to the standard form. The most difficult part of the solution of the NLS equation (and others which are solvable by the inverse spectral transform) is the radiation part of the solution. Especially with waves over a shelf radiation plays an important part because of the mass conservation. A one-soliton in an homogeneous medium traveling into an inhomogeneous medium experiences some reflection and this yields radiation solutions, which, how ever, are often neglected. This results in the impossibility of obeying both mass and momentum conservation. The problem here is quite similar to the problem of solitary waves over a shelf. Although the energy of the wave which is reflected is only a small part of the energy of the incoming wave, it is the impossibility to conserve mass which is the more important aspect. See, e.g., the clear discussion in the recent paper of Miles, Miles (1979). For a discussion of radiation solutions for homogeneous NLS equations, one is referred to Segur and Ablowitz (1976) and Segur (1976). - 132 - APPENDIX D - Some properties of the Davey and Stewartson equations The D&S equations are usually written as (cf. Eqs. (5.27)). (DI) 2 2 2 2 2 3 Q 3 Q 3 IAl ( gh-c )-- + gh - = K - z , 2 2 1 g 3~ 3n 3n where the coefficients A ,µ ,v ,~ ,K are given by (5.28), or, when surface 1 1 1 2 1 tension is included, by (5.36), and Q is given by (5.26a) or (5.35) respec tively. It is noted that A < O, µ > 0 and v > O; when no surface tension 1 1 2 is present, then K > O. 1 For interpretation purposes it is often practical not to introduce the quan tity Q, but to keep the component ~ of the velocity potential~. Writing 10 simply~ for~ and introducing the non-dimensional quantities (with tilde) 1 0 by l 2 ~ = k~ n = kn T = (gk) T (D2) 2 _1 2 _1 zA = 2 A = k (gk) ~ = ~ 1 0 k (gk) ~ 1 0 the equations for A and~ becomes, dropping the tildes for ease of notation, (D3) where the (dimensionless) coefficients are given by - 133 - 2 2 2 w = gko(l+T) , w = gk, o = tanh kh, T = k T/g 0 (.Q, ,m) w 2 2 ~ 2 2 2 X = : {(1-o )(9-o) + T(2-o )(7-o) + 802 _ Z(l-o2)2(l+T) _ 3o_,T}, 4 2 2 o - T(3-o ) l+T (D4) 2 kc gh - C + _g (I-o2)(l+T) > 0 a= ---=g 2w gh kc w 2 > = ( 1-o ) + _2_} = f3 w kh { (/ 0 0 l+T = V X - X1 f3/a. Note that Vis the dimensionless equivalent of V in (DJ). Notice furthermore 1 that the coefficients are evaluated at m = 0 and that here a clearer distinc- tion is made between lkl and the direction of propagation of the waves. When T = O, equations (D3) follow directly from Eqs. (5.25) and (5.24) upon non dimensionalization with (D2). Eqs. (D3) with (D4) are given in this form by Ablowitz and Segur ( 1979). The figure below, rn which the changes of sign of the parameters are given as a function of T and kh is taken from Ablowitz and Segur (Jg79); a similar picture for the coefficients of (DI) (see Eqs. (5.36)) is given also by Djordjevic and ~edekopp (1977). 4 X>0, X<0, v>0 1•<0 A>O 3 F D ~ 2 B 1·>0 0·25 0·5 l·O 1·25 1·5 - 134 - An important property of the D&S equations (DI) and (D3) is that they admit solitons traveling at almost any acute angle relative to the group velocity vector of the wave packet. Limiting cases of (D3) are found by setting either a/an or a/as= O. With a/an= 0 the second of Eqs. (D3) can be integrated and one obtains (DS) and (D6) Equation (DS) is again the one-dimensional NLS equation; the induced mean current ~scan be obtained from (D6) when A is solved from (DS). Note that a~/as = 0 for !Al = 0 is implied in (D6). When AV> 0 no soliton solutions exist, only decaying solutions of the form (C9); this is called radiation. AV> 0 occurs in regions A, Band E of the parameter space given in the figure above. Soliton solutions exist for A.V < 0 and occur in region C, D and F; such solitons are called envelope solitons and are of the form of Eq. (C6). Wave flume experiments showed that (C6) gives, when taking the amplitude of the soliton the same as the peak amplitude as measured, a wave envelope which is very close to the measured envelope; see, e.g., a figure in Ablowitz and Segur (1979) taken from an unpublished paper of Hammack, or Yuen and Lake (1978) or Lake at al (1977). Taking now a/as 0 in Eqs. (D3), the system reduces to 2 aA a A 2 i - + µ - ~ XIA! A. (D7) aT an2 For µX < 0 this equation has also soliton-like soiutions; Eq. (D7) is similar in structure to Eq. (DS). However, whereas the wave crests move ins direc tion (the D&S equation were derived under this condition), the modulations travel inn direction, i.e., along the crests of the waves. In optics A nn represents diffraction of light and equation (D7) provides a non-linear correction to Fraunhofer diffraction. Soliton solutions of (D7) are also called wave guide solitons. - 135 - In our opinion an equation as (D7) can be used to predict the variations of the amplitude of a wave along its crest. Such a variation is quite common in experiments in wide wave basins where it is difficult to obtain a proper incoming wave field in, e.g., studies of wave penetration in harbours. 2 In the shallow water limit kh + 0 under the constraint ka << (kh) , Eqs. (DI) reduce to a set of equations similar to Eqs. (5.30) where the surface tension T was neglected. This set of equations can be written in the following form after rescaling of the variables. (D8) . I " " 2 where I: = sign( -T), T = T/ (gh ) . 2 3 The difference with Eqs. (5.30) is the inclusion of surface tension. Equations (D8) can be solved by the inverse spectral transform (IST) method; the equa tions (D3) for arbitrary depth are not completely solvable by the IST method. An exact decaying solution of the D&S equations (D3) is reported by Ablowitz and Segur (1979). Instead of (D3) the following scaled version was considered (scaled in a similar way as (C7) is a scaled version of (Cl)) 'A + a A + A i t 1 XX YY (D9) a,+. + ,+. 'l'xx 'l'yy An exact solution is A2 2 + a + B(t) + 2 t (DIO) -1 where A and 8 are arbitrary constants. Note that A decreases like t 2 - 136 - _1 whereas the decrease in the one-dimensional case is given by t 2 (see (C9)). It is now anticipated that the part of the solution that decays in time can be described in terms of a slowly varying modulation of (DIO). The following conserved quantities of Eqs. (D3) are given by Ablowitz and Segur (1979): I = J f (A aA* - A* aA)dc:-d 2 a~ a~ s n (D 11) I = JJ (A aA* - A* aA) d C d 3 an an s n 11 [{\J aAI 2 + µJ aA 2} _ a~ an 1 2 and it also follows from (D3) that (DI 2) 1 is interpreted as the mass of the wave to leading order in£ and the 1 integral (Dl2) is then interpreted as the moment of inertia. I now gives a 4 condition for focussing; when 1 < 0 no global solution of Eqs. (D3) exist 4 and for 1 • O+ focussing is seen to occur because mass 1 remains conserved. 4 1 Regions A and Dare clearly of most interest for us. Because in these regions \ < 0 andµ> O, the integral of (Dl2) is not of definite sign and no con clusions can be obtained from the sign of 1 • 4 In Region F focussing occurs, in region Eno focussing occurs in the deep 2 water limit, and in regions Band Cone has a< O, or gh-c < 0 (the equation g for Qin (DI) is hyperbolic), and integrals involving~ are generally un- bounded. The deep water limit of Eqs. (DI) is, in absence of surface tension, given by (5.29): (DI 3) - 137 - When taking for A twice the value as is taken previously, so that the free surface elevation is given by g(; = Re [iE A exp{ i (9,x-wt) }] , instead of by g(; = iEAexp{i(R-x-wt)} + CC, as is taken in Section 5,3 (see Eq. (5.20)), and introducing dimensionless • quantities such that w = I, k = (9-,m) = (1,0) and g = I, then Eq. (Dl3) becomes (Dl4) Longuet-Higgins (1976) and Hui and Hamilton (1979) studied properties of this equation. Longuet-Higgins (1976) gave the following conserved quantities for 2 2 l2 - solutions of (Dl4) which approach zero fast enough for(~ +n ) • + 00 • Multi- plying (Dl4) by A* and subtracting the complex conjugated equation, one obtains 2i ~(AA*)eh (DIS) Longuet-Higgins considered irregular waves. Supposing A to be statistically uniform he found, by integrating over the whole ~,n plane, the following conserved quantities: M =½ I I = aA A* - A aA* > 2 2i < a~ a~ (Dl6) I I = aA A* - A aA* > 3 2i < an an I < aA aA* _ aA aA* > I 2 2E 2 + - < AA*> -13 a~ a~ an an 2 - 138 - where the conservation of M, I and I , E correspond to the conservation of 2 3 wave action, momentum, energy respectively;<·> denotes the averaged value with respect to (~,n). Hui and Hamilton (1979) gave several special solutions of Eq. (Dl4) in the ~,n plane. Denoting the angle between the direction of the carrier wave and the direction in which solutions are sought by ljJ, the ~,n plane is split into regions according to the sign of the quantity y defined by 2 2 y = cos l)J - 2sin l)J. (DI 7) 2 2 y = 0 corresponds to tan l)J = 2 , y > 0 yields tan l)J <½and y < 0 yields 2 > _!_ tan ljJ 2 • Some results of their investigation are as follows: - For y > 0 and k and w constant solutions for the group envelope of the elliptic form dn and en always exist; i.e., infinite groups of permanent waves exist whose envelope varies periodically in space and time, Their common limit is the sech profile. - The only waves of constant frequency/wave number type having their group propagating at constant velocity along the characteristic directions 2 ljJ =+\)Jc (i.e., tan l)J =½and y = O) are the constant amplitude plane pro gressive waves, _1 - When y = 0 and characteristic coordinates are introduced as r = ~ + 2 2 n, _1 s = ~ - 2 2 n, and defining the real functions Rand 0 by A= R exp(i0), the most general solution of the deep water equation (Dl4) is given by = R(s-vT) (DI 8) I 2 = - - TR (s-vT) - 2vr + F(s-vT) ~: 2 with F and R arbitrary functions. The group shape (i.e., the envelope) 2 propagates ins-direction at constant speed EV 0note that T =Et and ~ = E(x -cgt), n = Ex ). The wave number and frequency are not constant in 1 2 this case, One has 9, = I + 0 and tLns X 9, = I - ETRR' - 2Ev +Eh'. (Dl9a) The x -component of the wave slope is given by 1 - 139 - a~/3x E(ER 1 +iRt) exp{i(x-t+0)}. (D19b) 1 It is concluded from (D18) and (D19a,b) that, while the group propagates at constant velocity without changing its shape, the individual waves of which the group is composed tend to become more steepened in time in regions where R' < 0 and more flattened in regions where R' > O. It follows from (D19a,b) that the wave slope increases• as-ERR2 I T; in• terms 4 of the physical time (measured in wave periods) this become E RR't. This growth is thus a small quantity on time scales for which (D14) is valid. In our opinion therefore this steepening behaviour can only be used as an indi cation of the behaviour of the 8olution (D18). A pertinent evaluation of steepening of waves has to be obtained from a higher-order NLS-like equation, for example from the two-dimensional analogue of Roskes 1 (i977) fourth-order wave envelope equation (see Eq. (5.19a)). - 140 - APPENDIX E - On the applicability of the Boussinesq-like equations Formally the Boussinesq-like equations are derived under the condition that 2 both h/A and a/h << I, with (a/h)(A/h) = 0(1). For applications to non- breaking periodic waves especially the condition h/A << is the limiting one. When the amplitude of the wave under consideration is very small, one should expect that the linearized equations (7.1) would yield solutions close to the usual linear solution. The difference is now in first instance due to the different dispersion relation. Whereas in the usual linear case one has 2 w = gktanh kh, (EI) Eqs. (7.i) yield the linear dispersion relation (for horizontal bottom): 2 (Jj (E2) B with the phase- and group velocities I 2 Cp /(gh) (E3) B Notice that it follows from (E2) that the dispersion relation can also be written as (E4) It follows from (El) that 2 2 T g/h = 4n /(kho) a= tanh kh. (ES) It is seen from (5.23a) that c can be written as g I 2 C /(gh ) = (E6) g 0 An heuristic criterium for applicability of Boussinesq-like equations can be obtained by considering the difference between the phase velocities as - 141 - resulting from the two dispersion relations. In the table below we list some (dimensionless) values for T,TB and cg,cgB obtained for fixed values of A/h. It is noted that (T-TB)/T = (cB-c)/cB =(TB/T).(cB-c)/c and thus T-T c-cB T B --- (E7) C TB T ! ! A/h T/g/h TB/g/h C /(gh) 2 CgB/(gh)2 (T-TB)/T (c -cg )/c g g B g 5 6.0790 6 .1773 .5798 .5303 -I .62% 8.54% 6 6.9490 7 .0114 .6578 .6267 - .90% 4.73% 7 7.8424 7.8841 . 7200 .6999 - .53% 2.79% 8 8.7549 8.7840 .7687 .7554 - .33% I. 73% 10 10.6219 10.6376 .8371 .8307 . i5% .76% 14 14.4564 14.4623 .9090 .9071 - .04% .21% Notice that, for A/h = 5 and 6 one obtains (c-cB)/c = 1.59% and 0.89% respectively. Demanding that the relative error in the phase velocities is less than 1% results into A/h ~ 6 or Tlg/h ~ 6.9 but the group velocities differ about 5% then. Demanding a difference of less than 1% in the group velocities would result in about T/g/h > 10. - 142 - APPENDIX F - Sketch of the derivation of Shen and Keller The relations (7.21) are used to non-dimensionalize the governing equations which consist of the continuity equation, the three equations of motion, the kinematic free surface and bottom condition, the dynamic free surface condi tion and the adiabatic condition. After introducing the dependence on the phase variable in the t, w, p, sand p, and changing the governing equations accordingly, the quantities are expanded as power series of the small para meter v: • • • • • u(s,X,T,z;v) u (s,X,T,z) + vul + ... 0 etc .. The zeroth-order quantities are supposed to be the solution of the state of 2 2 rest; that is, t = 0 w O, s = (Dr) /2, p = p (Z), p = ! p (z )dz + c 0 ' 0 0 0 0 2 I 2 0 2 0 • 0 with r = x 1 + x 2 and Z = -z +(Dr) /2. With p = constant and ~ = 0 we would have only p -z + C, where p = C is the dynamic condition. It is noted that 0 the choice of the state of rest as the zeroth-order solution means in fact that the waves to be considered have a small amplitude. Upon substitution of the expansions into the equations, the terms with V yield , , , , , a set of equations for p 1 u 1 v 1 w1 p 1 s 1 and S (t = (u,v)). Elimination of p , u , v and s yields a set of equations and boundary conditions for w 1 1 1 1 1 and p • These equations are solved by seeking a solution of the form 1 • • p (s,X,T,z) = A(~,T,X)~(X,z) 1 (FI) -I • • -w s As(s,T,X) The equation 2 2 2 2 8 = ST/(Sx +Sx ), (F2) 1 2 yields S corresponding to each mode. By introducing the frequency wand the wave number k. by J w = -clS/clT k. = clS/clX., J J Eq. (F2) becomes the dispersion relation 2 w (F3) It is thus seen that to this order of approximation the waves have no fre quency dispersion. The solution of this equation by means of the method of characteristics yields the rays. The modes and the rays are now the same as found from linear theory. For the determination of A the next order of approximation has to be con sidered, The resulting equations in the second-order quantities have the same structure as those in the first-order quantities, however, they are now in homogeneous. In order that the solutions of the second-order equations remain bounded, the right-hand members yield a solvability cond.ition in the usual way. Thi~ solvability condition can now be written as a differential equation + for A(~,X,T) along the ray for the particular 8 . The equation for A is of n the form (F4) + where f , f and f are complicated expressions as function of X and T and 1 2 3 in f occur also terms from the Jacobian of the t~ansformation of ray coor- 3 dinated (T,y ,y ) to (T,X ,X ). 1 2 1 2 The term f 3A in Eq. (F4) can be eliminated by a transformation of the form + + + T ~ B(~,X,T) = f (X)A(~,X,T)exp{-J f dT}. (F5) 4 0 3 - 144 - For the precise form of this transformation, see Eq. (39) of Shen and Keller (1973). The equation for Bis then of the form (F6) which is seen to be the KdV equation with varying coefficients. Essentially the method for finding the complete solution of the first-order quantities consists of the following steps: First a solution of the eigenvalue problem is to be found, yielding the 2 eigenvalues and the eigenfunctions~ and~ . e-n n n - Secondly the rays are to be determined. - Thirdly the KdV-type of equation for B has to be solved along the particu- lar ray for a certain mode; the amplitude A then follows. - 145 - APPENDIX G - The smallness of bottom slope in case of cnoidal wave refraction Equations (7.1) describe the evolution equations of fairly long waves over an uneven bottom and are derived under the condition that the derivatives of h to x and x are at most O(h/A) of magnitude. (Note that hand have 1 2 1 dimension here.) In order to investigate the conditions imposed on h(*) for a locally given cnoidal wave to remain of cnoidal wave shape when progressing, • it is easiest to consider only one spatial dimension x = x 1 Equations (7.1) then become 2 au au ai;; 1 a a 1 2 ,:;-t + u "'x + g - = - h --(h ~) - - h 0 0 ax 2 ax2 at 6 (Gl) ~~ + ~x {(h+i;;)u} = 0 These equations can be written as au au g ~ _ J_ h2 at: + u ax + ax 3 (G2) = -h u X ' and h dh/dx. X = Introducing non-dimensional quantities by using the relations (7.8), Eqs. (G2) become in non-dimensional form 2 aa a~ at 1 ~2 ~~ a u 1 ~~ au +£ii~+ - - JJh = µhh_ -- + - µhh-- - + o(E,µ) 3 2 ai: ax ax x ai:ax xx ai: (G3) ~ au + h - ax Note that the (dimensionless) bottom slope h~, and also h~~, is 0(1) in (G3). X XX Cnoidal wave solutions can be obtained from Eqs. (G3) in an approximate way (see, e.g., Dingemans (1974, Appendix)) whenever the right-hand sides of (G3) are zero, and thus for the case h = constant. - 146 - For a locally given cnoidal wave to remain cnoidal in shape on an uneven bottom, the right-hand sides of (G3) should therefore be perturbations to the left-hand sides, or, otherwise stated, the right-hand sides should be o(E,µ). The term h u in the second of Eqs. (G3) is 0(1) in (G3) and should X thus become o(E,µ); because u = 0(1) this results into the condition fi~ = o(s,µ), (G4) X and thus fi_ << s or h~ <<µ.Note that hand x are dimensionless here. X X Choosing fi = h(ox), o the inhomogeneity parameter, the condition (G4) result into 0 << E or 8 << µ, (GS) . dfi with - - h' = 0(1). dox In variables with dimension the condition (GS) can be written as follows; 8 << s = I %. Therefore, (h/A)oh' << a/A, or dh dx << a/A. (G6) The condition that the bottom slope Jdh/dxl is much smaller than the wave slope a/A does of course not imply that the bottom becomes horizontal in the linear approximation (vanishing amplitude) because here cnoidal waves are 2 considered for which a/h ~ (h/A) (i.e., the Stokes number is 0(1». Keeping · 3 3 the Stokes number order one, one has a/A~ (h/A) and thus Jdh/dxj < (h/A) . - 147 - APPENDIX H - The mean energy flux for cnoidal waves The energy flux per unit length along the wave crest is given by (HI) One has for cnoidal waves to leading order _!__(p+gz) ~ g/'; p 2 2 l2 (u +u ) ;; c/';/h (H2) 1 2 2 2 2 w << u +u. 1 2 Substitution of these approximation into (HI) yields as leading order approximation• • for •F: 2 • S C S = pgc(l+ h)(l+ gh h). Because r;/h << I for cnoidal waves (in fact r;/h is the expansion parameter of the perturbation series), one obtains to leading order the following expression• for the mean energy flux •Ef =TI JT •F dt: O (H3) Using the solution (7.24) for /';(t,~) one finds for Ef(~) = IEfl the following expression: 2 Ef = pgcBH (H4) - 148 - APPENDIX J - Derivation of spectral evolution equation from the D&S equations In this Appendix the derivation of Alber is followed for the transport equa tion of the spectrum for an ensemble of nearly monochromatic waves, It is assumed that the Davey and Stewartson equations (5.27) remain valid when the complex amplitude A(t,n,,) is considered to be a random function of (s,n). For shortness of writing we introduce the notation i = Cs,n). (JI) Note that* is a stretched coordinate here. A two-point correlation function p(i,r,T) is defined by • • (• I • ) *(• I • ) p(r,x,T) = (J3) the first of Eqs. (5.27) can be written at~ as, 8A(~ ) 1 2 i---+;\ + 1--1 = V jA(* )j A(~) + V AQ, (J4) dT 1 1 l l l 2 where simply is written A(~) for A("'k. ,,). 1 l An equation for p can be obtained in the following way. Multiplying (J4) by ) ) A*(~2 and adding the resulting equation to the equation for A*(~2 multiplied by A(1 1 ) results in - 149 - 8 + •+ 8 2 8 2 ) + •+ i dT < A(x )A (x) >+A - - - < A(x )A (x) > + 1 2 1 ( ~2 ~2 1 2 8 1 8 2 (JS) - V < A(i )A*(i) > {Q(i )-Q(i )} + 2 1 2 1 2 As is usual in equations with cubic non-linearity, the evolution equation for second-order moments involves also terms with fourth-order moments. When the fourth-order cumulant is set equal to zero, it is seen from (9.3) that, with< A>= 0, there can be written < A(i )A*(~ )A(~ )A*(~)>= 2 < ACt. )A*Ct) > • < A(i )A*(i) >. 1 1 1 2 1 1 1 2 (J6) • • 2 • = 2p(r,x,T).a (x ), 1 2 where a (i ) = < A(i )A*(i ) > is the ensemble mean square amplitude. 1 1 1 • • (• • • • • • • It is remarked that a term< A(x )A x ) > is neglected in 1 2 1 2 the right-hand member of (J6) because the phases of both expressions before averaging are large quantities and yield therefore upon averaging negligibly covariances, in just the same way as when = \i -i \ is large. \t\ 1 2 When also a change is made back to the averaged coordinates i = (~,n) and the separation coordinates t = (u,v) Eq. (JS) can now be written as: V p. ~LQ(x+ • I •r)-Q(x- • I •]r) + 2 2 2 (J7) - 2V p. 2a (x++ I •r)-a 2 (x-+ I •]r) = O, 1 [ 2 2 with. 2(a •)x = P(O,x,T). Notice that (J7) is a differential/difference equation for p. A differential equation of infinite order is obtained by expanding the terms wrthin square - 150 - brackets in Taylor series about 1 = 0. One obtains (JS) The wave envelope power spectral density F(K,x,T)-+ -+ is defined as the Fourier transform of the correlation function p: ,-+ -+ I F ( -+K,x,T + ) = --- !Joo p (-+-+r,x,T ) e -1K.r d•r. (J9) (2TT) 2 -oo When 1 O, the mean square wave amplitude is given as 2 • • 00 • • • a (x,T) p(O,x,T) = f[ F(K,x,T)dK. (JIO) 00 The Fourier transform of Eq. (JS) then yields the sought for transport equa tion for the spectrum F(k,1,T): aF , n I I 2 ) -+ 2 aF 2 aF . ( a a a a ) (4 2 = 0 dT /\1)0 di;+ µ1m an - srnzar:aI + 2 an am \\a+ V2Q F (JI I) where k = (9,,m) and the spatial derivatives in the sine operator act only on 2 a and Q, whereas the wave number derivatives act on F. The second of Eqs. (5.27) yields upon averaging the equation for Q: 2 a2q a2q a2 2 (gh-c )- + gh - = K - (a ) • (Jl2) 2 2 1 2 g aE; an an Taking only the first term of the expansion of the sine operator, one obtains from (JJI): -+aF dT - 151 - The spectral counterpart of the Stokes' wave solution is now given by the basic solution of (JI]) and (J12) F = F (k) Q = Q = constant, (JI 4) 0 0 which is independent of (s,n,T). F (t) represent a homogeneous and stationary 0 spectrum and Q represents a uniform current. 0 - 152 - APPENDIX K - Some comments on modulational instability K.l Introduction In this Appendix the problem of the stability of modulated finite amplitude waves will be discussed. A modulated wave is a wave of which the parameters are slowly varying. It was seen in Chapter 2 that such waves, in absence of currents, were unstable when w (k).w~(k) < O, where the non-linear dispersion 2 relation w(k,a) was written as 2 w(k, a) w (k) + w (k)a + ... 0 2 By taking higher order terms (in ka) into account, it was seen that above some value of ka the waves were stable again. Another bound for instability is that waves are only rnodulational unstable for kh > I .363. Because of (5.18c), the V V condition w2 (k).w~(k) < 0 is equivalent with A1 1 < O, where A1 and 1 are the coefficients in the NLS equation 0 (Kl) V or in the D&S equations (5.27); A1 1 < 0 for kh > 1,363. A physical explanation for the onset of instability of weakly non-linear waves consists of the following (Lighthill (1965, 1967)). Consider a pulse of weakly non-linear waves in deep water which initially contains waves of uniform wave lengths. Since the non-linearity causes the 2 crests of waves of larger amplitude to travel foreward more quickly (w = 2 2 gk(l+k a)), wave numbers will tend to increase in front of the pulse and decrease in the back of the pulse. Because de /dk < O, the shorter waves in g front of the pulse and the longer waves behind the pulse cause energy to approach the center of the pulse and this causes the amplitudes at the center of the pulse to increase. This in turn accelerates the instability. - 153 - Water o[ finite_d(!p_th For finite amplitude waves in water of finite depth the spatial variation of the waves induces a mean flow and a change in mean water level. These mean flow and mean water level variations have a stabilizing effect because they cause wave numbers to increase behind the pulse and decrease in front of the pulse. In the following the question of stability of Stokes' wave trains due to sub harmonic perturbations is discussed. Primarily only longitudinal perturbations are considered so that wave propagation in one spatial dimension may be con sidered. At first, the stability analysis based on the non-linear Schrodinger equation will be given; this is carried out in some detail because the analysis is rather easy and the results are the same as these obtained from the discrete side-band disturbances approach of Benjamin and Feir. Benjamin and Feir (1967) were the first to consider the question of the stability of Stokes' waves and therefore the phenomenon of modulational instability of wave trains is some times also called Benjamin-Feir type of instability. Because the analysis of Benjamin and Feir is quite involved, only a few remarks on their method are made. Prediction of instability from the NLS equation Consider the NLS equation (Kl). The stability of the uniform Stokes' wave train 2 A(T) = a exp(-iv a T) (K2) 0 1 0 is now to be studied. By putting A(~,T) = a(~,T)exp{iX(~,T)} , (K3) the NLS equation can be written as the following system for the two real functions a(~,T) and X(~,T): - 154 - 2 aa -- + dT (K4) 2 ax ax 2 3 - ~} = a - + A1 {a(~) + V 1 a 0 dT a,2 Note that when A measures the amplitude of the free surface elevation, A1 and 2 \) l are for the case of deep water given by A = -w /(8k) and V = w k /2 1 0 0 1 0 0 where w and k are the frequency and wave number of the primary wave. For 0 0 the case that A measures the amplitude of the velocity potential, A and V 1 1 are given by (5.28). Consider now the following perturbation of the uniform Stokes' wave: + b(s,T) x (T) + ecs,T) 0 with b << a , 0 << X and X (T) = - V a~T. 0 0 0 ½ 1 Substitution of (KS) in Eqs. (K4) yields upon linearization with respect to the perturbations the following set of linear equations in band 8: 2 db I a e -+-Aa --=0 dT 2 1 0 dS2 (K6) 2 ae a h 2 a - - A -- + 2V a b 0 0 dT 1 dS2 l 0 Elimination of 0 yields the single differential equation for b: (K7) Assume the perturbations of the form b,0 (b,8)exp[i(Ks-S"h)]. (KS) Then the dispersion relation between~ and K follows from (K6) or from (K7) as - 155 - (K9) 2 and thus instability is obtained when ~ < 0, or (KIO) For the case of water of finite depth hone has V1 A1 < 0 for kh > 1.363 and instability might occur for V A < O; for deep water one has always V A < 0. 1 1 1 1 V Assume now 1 A1 < O, otherwise condition (KIO) can not be fulfilled. Then for instability, (Kl I) and thus another boundary for the possibility of instability is obtained: the wave number of perturbation (or side-band disturbance) has to be small enough. I 2 2 For deep water one has, with V = - w k, A = -w /(8k ), 1 2 O O l O O K < 2/2 k a • (Kl 2) k 0 0 0 The growth rate b /b or 8 /8 of the modulation 1.s given by -i~ = Im(Q) and T T thus by 2 2 b /b = K(lv A _!_ A (Kl3) T l l ia2O - 2 lK )½ ' 2 2 The maximum growth rate is obtained for K = /A la and is Iv l 1 O (Kl4) 2 For deep water the maximum growth rate is w (k a ) /12. obtained for the side o O 0 band wave number K given by K/k = 2k a. The growth rate Im(~)= -i~ of the 0 0 0 disturbances has the following general behaviour. , - 156 - Im(n) t I Jv1 la!~------.;;;-..-,--. 12 ½ ½ \) 1 \) 1 a 12 a ½ 0 r-~ 0 0 Ill> K Figure Ki It is noted that A and v can readily be obtained from the non-linear disper 1 1 2 2 2 2 sion relation w(k,a) = w (k) + w (k)a as A = { a w /3k and \! = 3w/3(a ) = 0 2 1 0 1 w2 (k), see (5.18c). The dispersion relation valid for the NLS equation approxi- 2 mation is then given as w = w + \! a - A a /a. 0 1 1 XX As conditions for the modulational instability of the Stokes' wave train we have now I) > \! A < the water has to be deep enough, kh 1.363 or 1 1 0 which is equivalent tow w" < 0 and 2 0 2) the wave numbers k + K of the side-band disturbances have to be close o l enough to k, or K/k < (-w /w") 2 a /k . 0 O 2 O O O It is noted that the third boundary for instability, i.e. the waves become stable again for large values of ka can not be obtained from the NLS equation < because this equation is only valid for small ka, say ka = 0.15. For these values of ka the growth is reasonable accurate, at least compared to the exact analysis of Longuet-Higgins (1978b). Benjamin and Feir Benjamin and Feir (1967) considered the stability of a second order Stokes wave on deep water; Benjamin (1967) extended the analysis for water of finite depth. The basic solution (deep water) is - 157 - 1 2 = acosX + z ka cos2X -I kz . wk ae sinX 2 2 2 with X = kx - wt and w = gk(I+k a ). This solution is perturbed: s = n + n and these expressions for l; and (2.12), (2.13), (2. 15) without y-dependence and cl 2 2 n. a.cosX. + kaa.{A.cos(X+X.) + B.cos(X-X.)} + O(k a a.) i i i i i i i i i where (k+K)x -(w+D)t + Y1 (k+K)x -(w-D)t + Y2 and K << k, D << w, a. << a. The amplitudes a. and the phase-shifts y. are i i i supposed to be slowly varying. Due to non-linear interaction of the side-bands with the second harmonic com ponent of the primary wave, we have the following picture (Benjamin (1967)): Xl~ ~2X - X1 x2 + (y 1+y 2) 2X'-----.._ x_,-- 2X - X2 X1 +(y1+Y2) 2 ' That means that if (Y 1+Y 2)-+ constant (# O, # ,TT) when time evolves, resonant interaction becomes possible and the side bands are growing without bound. Benjamin and Feir derive differential equations for dai/dt and d(y1+y 2)/dt and from its solution it follows that instability occurs when (K12) is ful filled. In order that the side bands fulfil the linearized dispersion relation approximately, one obtaines D = c K. g - 158 - The average Lagrangian approach Whitham (1967a) considered the Eqs. (4.13) and (4.14). The four characteristics consist of two which tend to 'fvglland two which tend to c as the amplitude g goes to zero. These last two characteristics are given approximately as kw" gh + 2B c dx 0 0 g C + -- ! {khD dt g h C 0 2 gh - C g I 4 2 4 where c = w /k, c = w' (k), B = c - - c, D = (9cr -!Ocr +9)/(8cr ) , a= tanh kh. 0 g O g 2 0 In order that these characteristics are real, the terms between square brackets have to be positive. This is found to be true for kh < 1.36, and thus the Stokes' wave are stable for kh < 1.36 and unstable for kh > 1.36, as was also found by Benjamin (1967) and from the· NLS equation approach. The difference between Whitham's approach and the NLS equation approach is that the wave length of the perturbations in Whitham's case has to be much larger than in the second case. Lighthill (1967) gave a discussion on the use of average Lagrangians for the study of wave stability and found also that the waves become stable again for high enough ka; see also page 7. The growth-rate as function of ka is (see Lighthill ( 196 7, 1978)) as given in Figure K2. Longuet-Higgins (1978b) gave a precise analysis of subharmonic perturbations of steady symmetric waves of any order on water of finite depth. His theory is valid for all values of ka, unlike the Benjamin-Feir approach which is only valid asymptotically for small ka. For values of ka = 0.346 the instability disappears and near values of ka = 0.41 another kind of instability appears, which is instability of superharmonic perturbations (Longuet-Higgins (1978a)). The latter type of instability occurs in the region of ka values for which the various integral invariants (such as potential and kinetic energy) of the wave motion obtain their maximum value, The maximum growth of the subharmonic perturbations occurs for ka = 0.32 and is about 14% per wave length. The instability region is given in Figure K3, which is taken from Longuet-Higgins (1978b). - 159 - 1.5 1.0 0,5 0.025 0.05 0.075 a/;\ Figure 115. Waves generated by a wavemaker oscillating with fixed frequency w0, and with amplitude departing from a large constant value by a small amount which varies periodically at a frequency iJ far smaller than Wo• The rate ,1 of exponential increase of modulation with distance is plotted, together with the effectiv" group velocity U0, as functions of the amplitude a. Above a = 0.054,\ the effective group velocity splits into two (U1 and Ui) while fl vanishes, Figure K2 (from Lighthill (1978)) 0.3------~------ stable b I C: I ~ "' 0.2 - ..Q... - i ....8, -- 0 unstable ..... ri' -- - ~ --- .... §. 0.1 -- - ~ ------I I 1unstable 0 O.i 0.2 (13 U4 0.4434 wa.ve steepness, ak FIGURE 7. Stability diagram for gravity waves in deep water. The tangent at the origin corresponds to the asymptotic theory of Benjamin & Feir (1967), (The resemblance to a. breaking wave is coincidental.) Figure K3 - 160 - The growth rate of the subharmonic perturbations as function of ka, as calculated by Longuet-Higgins (1978b) becomes much closer to experimental values as given by Benjamin and Feir than resulting from (Kl3), The relation between the modulational instability and the reoccurrence behaviour of wave groups has been discussed by Yuen and Ferguson (1978a, 1978b). Measurements of the instability are reported by Feir (1967) and by Yuen and Lake (1975, 1978) and Lake et al (1978). Fornberg and Whitham (1978) reported numerical calculations on Korteweg-de Vries and other model equations in which modulational instability occurred; it was seen that there was need to introduce the perturbations explicitely because these already were generated because of the number representation of the boundary conditions in a computer, In t·wo spatial dimensions it is al,vays possible to choose the components (2,m) + of k such that instability occurs in the NLS equation. Therefore, for oblique subharmonic perturbations, a Stokes' wave is always instable. See, e.g., Benney and Roskes (1967), Zakharov and Kharitonov (1970), Hayes (1973) and Davey and Stewartson (1974). 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