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Waterloopkundig Laboratorium Delft Hydraulics Laboratory T @.1,1 71 ' ' ( 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 </< I one has Stokes' waves, or short waves. Another subdivision can be made between heuristic methods and more formal methods to solve non-linear wave propagation problems. 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 </< l). One approach therefore is to investigate the properties of long wave propagation over a varying bottom; a distinction will be made between "fairly" long waves (for which the effect of vertical' accelerations, although small, is taken into account), and long waves (described by the classical shallow water equa­ tions in which the effect of vertical acceleration is neglected); see Chap­ ters 7 and 8. Because fairly long waves have periods which are larger than the peak periods of the spectra as they occur in the Southern North 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
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