Evolution of Basic Equations for Nearshore Wave Field

34 Proc. Jpn. Acad., Ser. B 89 (2013) [Vol. 89,

Review Evolution of basic equations for nearshore wave ﬁeld

† By Masahiko ISOBE*1,

(Communicated by Kiyoshi HORIKAWA, M.J.A.)

Abstract: In this paper, a systematic, overall view of theories for periodic waves of permanent form, such as Stokes and cnoidal waves, is described first with their validity ranges. To deal with random waves, a method for estimating directional spectra is given. Then, various wave equations are introduced according to the assumptions included in their derivations. The mild-slope equation is derived for combined refraction and diffraction of linear periodic waves. Various parabolic approximations and time-dependent forms are proposed to include randomness and nonlinearity of waves as well as to simplify numerical calculation. Boussinesq equations are the equations developed for calculating nonlinear wave transformations in shallow water. Nonlinear mild-slope equations are derived as a set of wave equations to predict transformation of nonlinear random waves in the nearshore region. Finally, wave equations are classified systematically for a clear theoretical understanding and appropriate selection for specific applications.

Keywords: Stokes waves, cnoidal waves, directional spectrum, mild-slope equation, diﬀraction, refraction

permanent form and wave transformation in the 1. Introduction nearshore zone. Theories for periodic waves which The nearshore zone is the transition zone propagate without deformation are reviewed first. between deep ocean and land. The environmental Then, a directional spectrum for describing random gradient is steep in the zone, as found in changes in sea waves and theories for estimating directional waves and currents, salinity and nutrient concen- spectra are introduced. Finally, governing equations trations, and plants and animals, and thus the zone is for calculating wave transformation in the nearshore called an ecotone. As a result, the nearshore zone is zone, from equations for linear periodic waves to one of the zones highest in primary production on the those for nonlinear random waves, are discussed. earth’s surface; on the other hand, it experiences fi severe external forces such as storms and tsunamis, 2. Validities of nite-amplitude wave theories which result in serious natural disasters. Therefore, Fundamental sea waves are periodic waves for conservation and utilization of the nearshore which propagate at a uniform water depth without zone, it is indispensable to understand the various deformation. These waves are called waves of processes occurring there. Waves, especially, are the permanent form. No strict, analytical solution has major external force. In addition, from a scientific been derived for these waves because of nonlinear viewpoint, waves provide us with interesting subjects boundary conditions along the water surface. The to study in relation to the randomness and non- simplest analytical solution was derived by Airy linearity of oscillatory fluid motion. This paper (1841)1) for periodic waves of small amplitude, which describes the framework of theories for waves of allows boundary conditions to linearize. A finite- amplitude wave theory was developed by Stokes *1 Graduate School of Frontier Sciences, The University of (1847)2) for deep water waves using the perturbation Tokyo, Chiba, Japan. † method, which is a technique to derive power series Correspondence should be addressed: M. Isobe, Graduate solutions for nonlinear equations. Small-amplitude School of Frontier Sciences, The University of Tokyo, Environ. fi Bldg. 660, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8563, Japan wave theory (Airy theory) is interpreted as the rst- (e-mail: [email protected]). order solution of the Stokes wave theory. Later, De doi: 10.2183/pjab.89.34 ©2013 The Japan Academy No. 1] Evolution of basic equations for nearshore wave field 35

(1955)3) and Skjelbreia and Hendrickson (1960)4) The water surface elevation is denoted by ðx; tÞ. extended the Stokes wave theory up to the fifth For non-deforming waves, in the coordinate order in deep to shallow water. system moving with the same speed as the wave For very shallow water, cnoidal wave theory was celerity, c, the wave profile is standing still and the developed by Korteweg and de Vries (1895),5) who velocity field becomes steady. The horizontal coor- derived an equation for shallow water waves called dinate, X, in the moving coordinate system is defined the KdV equation and expressed the solution by as using elliptic functions. Here, according to traditional X ¼ x ct: ½2 usage, “deep water”, “shallow water” and “very shallow water” refer respectively to water depths which are Thus, the stream function, , and the water surface large, comparable and small relative to wavelength. elevation, N, in the moving coordinate system are The meaning of “shallow water” is somewhat different not functions of t. These quantities are related to from the general meaning. Systematic derivations of corresponding quantities in the fixed coordinate cnoidal wave theory were developed by Friedrichs system as follows: 6) 7) 8) (1948) and Keller (1948). Laitone (1960) derived ðX; zÞ¼ ðx ct; zÞcz ½3 a second-order solution, and a third-order solution ð Þ¼ ð Þ ½4 was given by Chappelear (1962).9) N X x ct : Since perturbation solutions need huge amounts Now the governing equation for is the Laplace of manipulations of long mathematical formulas, equation, which is obtained from the irrotationality computer codes were developed for systematically condition: deriving Stokes and cnoidal wave solutions of ar- @2 @2 bitrary orders. Schwartz (1974)10) derived higher- þ ¼ 0 ½5 @ 2 @ 2 : order solutions of Stokes waves by applying the X z method of conformal mapping. Fenton (1972)11) and The boundary conditions are 12) Longuet-Higgins and Fenton (1974) studied high- ¼ 0onz ¼h ½6 er-order solutions of a solitary wave, which is in the ¼ const ð¼ Þ on ¼ ½7 shallowest limit of the cnoidal waves. Nishimura : Q z N "# et al. (1977)13) calculated the 24th-order cnoidal 1 @ 2 @ 2 waves as well as the 51st-order Stokes waves. þ þ gN ¼ const: ð¼ PÞ 2 @z @X Based on the above background, Isobe et al. (1982)14) showed mutual relationships among various on z ¼ N ½8 perturbation solutions of finite-amplitude waves, where g is the gravitational acceleration. Equations which confirms that Stokes and cnoidal wave theories [6] and [7] are called the kinematic boundary are the only two useful solutions among an infinite conditions at the bottom and surface, respectively. number of possible perturbation solutions. Then, Since water flows along these boundaries, they must they discussed the accuracies and validities of these be flow lines and hence the values of the stream theories of finite order. The essence will be introduced function must be constant along them. Equation [8] in the following. is the dynamic boundary condition on the surface. Consider periodic waves which propagate at a This means that the pressure on the water surface uniform water depth without deformation. In a should be constant and equal to the atmospheric normal situation, the water is inviscid and incom- pressure. This equation can be obtained from pressible, and hence the fluid motion is irrotational Bernoulli’s equation, which is the integrated form of and solenoidal. This allows us to introduce velocity the equation of motion for an irrotational flow. In potential and stream function to formulate the Eqs. [5] to [8], the unknown (dependent) variables problem. Here, the stream function, ðx; z; tÞ, which are and N. Among these equations, Eqs. [5] and [6] is a function of the horizontal and vertical coor- are linear in terms of the unknown variables, but dinates, x and z, and the time, t, is introduced. The Eqs. [7] and [8] are nonlinear. Although Eq. [7] seems stream function is related to the horizontal and to be linear, the unknown position of the boundary, vertical velocities, u and w, as follows: N, makes it nonlinear. Because of the nonlinearity, @ @ the strict and analytical solution has not been found. u ¼ ;w¼ : ½1 Various methods have been employed to derive @z @x approximate solutions. The following will give a 36 M. ISOBE [Vol. 89,

Availability of a solution of this type means that the problem was solved in a theoretical sense, but the accuracy of the double power series solution of finite order is very low and therefore the solution is of little practical use. However, a major advantage of the solution exists in giving an overall view of the perturbation solutions of finite-amplitude waves, as discussed below. Instead of expanding into the double power series, we first try to find a single power series solution in ": X1 Fig. 1. Definition sketch for waves of permanent form. i FðX; "; Þ¼ " FsiðX; Þ½14 i¼0 X1 ð ; Þ¼ i ð ; Þ ½15 systematic view for the solution to the basic equation N X "; " Nsi X : ¼0 and boundary conditions. i First, the dependence of on z can be solved A physical interpretation of this expansion is that easily by utilizing power series expansion. The and hence the relative water depth, h=L, may not following expression can be verified to satisfy the necessarily be small, being zero-order quantity (in the governing equation [5] only by substituting it into order of unity), but " and hence the wave steepness, Eq. [5]: H=L, must be a small quantity. By expanding 1 2 þ1 Eqs. [7] and [8] in terms of ", then substituting X ðz þ hÞ l ðX; zÞ¼ ð1Þl F ð2lÞðXÞ½9 Eqs. [14] and [15] into them, and rearranging the ð2l þ 1Þ! l¼0 terms in ascending order of ", we obtain a pair of where the superscript, ð2lÞ, represents the 2l-th differential equations for Fsi and Nsi for each order of derivative. In addition, the kinematic bottom bound- i. When we solve these equations in ascending order ary condition [6] is automatically satisfied since from zero, each pair of differential equations becomes ¼ 0 on z ¼h. Now the problem is to find the linear, and thereby a solution can be found without solutions of FðXÞ and NðXÞ which satisfy Eqs. [7] difficulty, although the mathematical manipulation and [8]. However, since these equations are nonlinear, is very long and tedious. The result is summarized as the strict analytical solution cannot be found. follows: "# As depicted in Fig. 1, the independent parame- X1 X1 X1 2i nþ2i ters of wave dimensions are the water depth, h, ¼ c " b0j ðz þ hÞþ " bni wavelength, L, and wave height, H. From these i¼0 n¼1 i¼0 three parameters, we can define various pairs of sinh nkðz þ hÞ cos nkðx ctÞ½16 two independent non-dimensional parameters. Isobe sinh nkh et al. (1982)14) proved that a regular double power X1 X1 nþ2i series solution can be obtained using the following ¼ " ani cos nkðx ctÞ½17 pair of non-dimensional parameters: n¼1 i¼0 ¼ 2 " ¼ HL2=h3 ½10 where k =L is the wave number, and ani and 2 bni are constants which can be determined by the ¼ðh=LÞ : ½11 relative water depth . The theory using this The parameter " is called Ursell’s parameter and expansion is called Stokes wave theory, which was the square of the relative water depth. The double first derived by Stokes (1847).2) Stokes wave theory is power series solution is expressed as valid in deep and shallow water. However, in very X1 X1 shallow water, convergence of the series is very slow i j FðX; "; Þ¼ " FijðXÞ½12 or the series is divergent even for slightly large values i¼0 j¼0 of ". This occurs in cases of small relative water X1 X1 depth, h=L, and finite wave steepness, H=L. In these ð ; Þ¼ i j ð Þ ½13 N X "; " Nij X : cases, the alternative power series solution should be ¼0 ¼0 i j used. No. 1] Evolution of basic equations for nearshore wave field 37

The single power series solution in terms of is expressed as follows: X1 j FðX; "; Þ¼ FcjðX; "Þ½18 j¼0 X1 j NðX; "; Þ¼ NcjðX; "Þ: ½19 j¼0 A physical interpretation of the above is that Ursell’s parameter, ", may not necessarily be small, being 2 zero-order quantity, but the square, ¼ðh=LÞ ,of the relative water depth and hence the relative wave height, H=h, must be a small quantity. Similarly to the Stokes wave expansion, Eqs. [7] and [8] are expanded into the power series in terms of , Eqs. [18] and [19] are substituted into these equations, and terms are rearranged in ascending order of . When solving pairs of differential equations for Fcj and Ncj in ascending order of , the second-order equations Fig. 2. Solution structure of Stokes and cnoidal wave theories as are nonlinear, but the solutions are available by using viewed from the double power series solution. elliptic functions. Since the remaining equations are linear, the solution can be found without difficulty but with huge efforts in calculation. It is summarized and Nsi, of the Stokes wave theory is expressed as as follows: follows: "# X1 X1 j j ¼ c þ boj ðz þ hÞ FsiðX; Þ¼ FijðXÞ½22 j¼1 j¼0 ( 1 X1 2lþ1 X1 X1 X ðz þ hÞ j þ ð1Þl nþj Ns ðX; Þ¼ N ðXÞ: ½23 ð2 þ 1Þ! bnj i ij ¼1 l ¼1 ¼0 j¼0 l n j) ð2 Þ 2 ð Þ l As depicted in Fig. 2, each order solution, Fsi 2n K x ct cn ; ½20 and Ns , includes all elements in the corresponding L i column of the double power series. On the other X1 X1 X1 j nþj hand, the cnoidal wave solution is related to the ¼ a0j þ anj ¼1 ¼1 ¼0 double power series solution as j n j 2 ð Þ X1 2n K x ct i cn ; ½21 FcjðX; "Þ¼ " FijðXÞ½24 L i¼0 where cn is the Jacobian elliptic function, K the X1 ð ; Þ¼ i ð Þ½25 elliptic integral of the first kind, and the modulus. Ncj X " " Nij X i¼0 The parameters anj and bnj are constants to be determined by ". This theory is known as cnoidal which means that the solutions, Fcj and Ncj, include wave theory and is valid for very shallow water, as all elements in the corresponding row. will be understood from the assumption. However, Since the order of summation is arbitrary, we because of the irregularity found in the coefficients can take summation in the oblique direction first, and after the 10th order, the convergence radius becomes then in the direction perpendicular to it. However, very small even in very shallow water. This is because the first summation in the oblique direction includes of singular points in the imaginary number. Care only a small number of elements in the double power should be taken when we use solutions of the 10th or series. For example, if we add the elements in the 45- higher order. degree right-down direction first, only one element is The relationship between the double power included in the zero-order solution, two in the first- series solution, Fij and Nij, and the solution, Fsi order solution, three in the second-order solution, and 38 M. ISOBE [Vol. 89,

Fig. 3. Curves of relative error, 1%, in the dynamic free surface Fig. 4. Diagram for selection of an appropriate wave theory. boundary condition for various orders of Stokes wave theory (St.), cnoidal wave theory (cn.) and Stream Function Method (SFM). parameter is defined by considering shallow water 2 2 approximation as Ur ¼ gHT =h , in which wave so on. On the other hand, each order solution of period is used instead of wavelength in the original Stokes or cnoidal wave theory includes an infinite definition. Figure 5 shows examples of wave profiles number of elements in the vertical or horizontal calculated by Stokes and cnoidal wave theories as direction. Therefore, although theories other than well as by 19th-order Stream Function Method Stokes and cnoidal wave theories exist and can be (SFM 19; Dean 196517)), which is an almost exact derived, their accuracies are lower than the two wave numerical solution. As can be expected, Stokes wave theories and they are less useful. theory is accurate for small Ursell’s parameters, Finally, Nishimura et al. (1977)13) examined the whereas cnoidal wave theory is accurate for large accuracy of finite-order Stokes and cnoidal wave Ursell’s parameters. With this, the relationships and theories by calculating errors for the nonlinear background of various finite-amplitude wave theories boundary conditions, [7] and [8], which are satisfied have been clarified. only approximately by finite-order solutions. As an example, Fig. 3 shows relative error curves for the 3. Estimation of directional spectrum dynamic boundary condition, [8], for various orders Although it is essential to study periodic waves of Stokes and cnoidal waves. In the horizontal axis, to understand fundamental characteristics of sea 2 Loð¼ gT =2; T: wave periodÞ is the deep-water waves, as done in the previous section, real sea waves wavelength. Also drawn in the figure is the curve are random since they are generated by wind blowing for the highest waves of permanent form obtained over the sea surface. Goda (2008)18) reviewed by Yamada and Shiotani (1968)15) by numerical fundamental concepts for analyzing random waves, calculation. The curves of breaking wave height on i.e., spectral analysis and individual wave analysis, sloping bottoms proposed by Goda (1970)16) are and their applications to engineering practices. Here, added by chain lines. These are higher than those of as an essential element for analyzing random wave the highest waves of permanent form. As can clearly transformation in the subsequent sections, a method be seen, Stokes wave theory is valid in deep to to estimate directional spectra is introduced. shallow water and cnoidal wave theory in shallow Frequency spectrum is normally calculated from to very shallow water. There is an overlap region. the time series of water surface fluctuation during a From this and other similar figures, an appropriate certain period. The total number of data is normally wave theory can be selected for practical use as in the order of hundreds to thousands. When we need shown in Fig. 4. Roughly speaking, Stokes wave a directional spectrum, several kinds of time series of theory should be used if the shallow-water Ursell’s oscillating quantities such as water surface fluctua- parameter is smaller than 25, and for larger values tion and water particle velocity at various locations we should switch to cnoidal wave theory. Here, are used. However, the number of these oscillating for the sake of convenience in practical use, Ursell’s quantities is very small compared to the number of No. 1] Evolution of basic equations for nearshore wave field 39

Directional spreading of the energy of random sea waves is indispensable to predict wave transformation in the nearshore region, as shown by Goda (2008).18) A component of random waves has a certain wave period, T, wavelength, L, and wave direction, . By changing the variables into the wave number vector, k ¼ðk cos ; k sin Þ (k ¼ 2=L: wavenumber), and the angular frequency, ! ¼ 2=T , the amplitude of the component waves representing the energy in the interval between k and k þ dk, and ! and ! þ d! is expressed as Zðdk;d!Þ. Then the random water surface displacement, ,is obtained by integrating all components: Z Z ðx;tÞ¼ eiðkx!tÞZðdk;d!Þ: ½26 ! k The wave number–frequency spectrum is defined as the energy density function in terms of the wave number vector and frequency: Sðk;!Þdk d! ¼hZðdk;d!ÞZðdk;d!Þi ½27 where hi denotes the ensemble mean and the complex conjugate. Thus, the right-hand side represents the expected value of squared amplitude, which is proportional to the energy of component waves; so Sðk;!Þ represents the energy density. Since the wave number, k, and angular frequency, !, are uniquely related to each other through an equation called the dispersion relation, S becomes a function of ! and as Sð; !Þ, which is called the directional spectrum. Now, a method to estimate the wave number– frequency spectrum from measured time series of oscillating quantities is introduced. Based on the linear (small-amplitude) wave theory, any oscillating quantity, ðx;tÞ, is expressed by using the transfer function, Hðk;!Þ, from the water surface elevation: Z Z ðx;tÞ¼ Hðk;!Þeiðkx!tÞZðdk;d!Þ: ½28 k Some representative transfer functions are shown in Table 1. Suppose M kinds of time series of various quantities at various locations are obtained. From Eq. [28], the cross-power spectra matrix, ð!Þ Fig. 5. Comparison of finite-amplitude wave profiles in deep, mn ðm; n ¼ 1toMÞ, is expressed as shallow and very shallow water calculated by the 5th-order Z Stokes wave theory, the 3rd-order cnoidal wave theory, small- kðx x Þ ð Þ¼ H ðk ÞH ðk Þei n m ðk Þ k amplitude wave theory and the 19th-order Stream Function mn ! m ;! n ;! S ;! d : Method (SFM). k ½29 data to calculate frequency spectra. Therefore, a The symbol mn represents power spectra for m ¼ n, theory is needed to estimate directional spectra with and cross spectra for m 6¼ n. The problem now is how very high accuracy. The following is an example to to estimate the wave number–frequency spectrum improve accuracy. as a function of mn. Many methods have been 40 M. ISOBE [Vol. 89,

Table 1. Examples of transfer functions from the water surface elevation to various oscillating quantities

Oscillating quantity Symbol Transfer function, Hðk;!Þ cosh kðz þ hÞ Pressure fluctuation pg cosh kh @=@xikcos Surface slope @=@yiksin cosh kðz þ hÞ u! cos sinh kh Fig. 6. Comparison among true and estimated directional Water particle velocity cosh kðz þ hÞ spectra. v! sin sinh kh ^ m ¼ m p ½36 1 ¼ tan ðm01=m10Þ½37 m proposed to improve accuracy. Isobe et al. (1984)19) 1 2 ¼ tan1 m11 ½38 applied the maximum likelihood method to derive an p 2 m20 m02 explicit formula. The result shows that the estimated qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 2 2 spectrum, Sðk;!Þ, is expressed as ðm20 þ m02Þ ðm20 m02Þ þ 4m11 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½39 ^ðk Þ 2 2 S ;! ðm20 þ m02Þþ ðm20 m02Þ þ 4m "# 11 XM XM 2 3 1 kðx x Þ ¼ ð ÞH ðk ÞH ðk Þei n m m00 m10 m01 a mn ! m ;! n ;! 6 7 m¼1 n¼1 mn ¼ 4 m10 m20 m11 5: ½40 ½30 m01 m11 m02 where a is a constant for adjusting the value of total In the above definitions, m is called the average energy. The above equation has a high resolution and direction, which is the gravitational center of the can detect peaks of discrete spectrum. The method directional spectrum; p the principal direction in is called Extended Maximum Likelihood Method which the mean square value of the velocity becomes (EMLM). maximum; and the long-crestedness parameter, In the practical ocean wave observation, the which represents the peakedness of the directional water surface fluctuation (or pressure fluctuation) spectrum. For this case, the same equation was and two-component horizontal velocities are often derived by Oltman-Shay and Guza (1984).20) measured, since these can be measured by installing a Figure 6 shows an example of a comparison between wave gage (or pressure gage) and a two-component true and estimated directional spectra from numer- current meter at the same location. For this case, ical simulation. The EMLM gives estimation close to Eq. [30] is simplified to yield the following equation: the true spectrum only from the three components, ^ð Þ whereas the method derived by Longuet-Higgins S ; ! 21) "#et al. (1963) has less resolution. ð 2 cos2 ^þ sin2 ^Þ 2 sin2ð^ ^ Þ M0M2 M1 m Other methods have been proposed by ¼ a 2 ^ ^ ^ ^ 2 2 22) 2M1M2ð cos cos m þ sin sin mÞþM2 Hashimoto and Kobune (1988) based on the ½31 Bayesian approach, and by Hashimoto et al. (1994)23) based on the maximum entropy principle. where For a combined incident and reflected wave field, the phase difference between a component of incident M0 ¼ m00 ½32 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi waves and the corresponding component of reflected 2 2 M1 ¼ m10 þ m01 ½33 waves is not random but deterministic depending rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on the location of the reflective boundary. Isobe þ 2 m20 m02 m20 m02 2 and Kondo (1984)24) and Hashimoto et al. (1997)25) M2 ¼ þ þ m11 ½34 2 2 proposed methods to separate incident and reflected ^ ¼ p ½35 waves by modifying the maximum likelihood method and maximum entropy method. No. 1] Evolution of basic equations for nearshore wave field 41

Unlike the methods introduced above, directional spectra can be estimated from the instanta- neous two-dimensional water surface profile as measured by remote sensing technology. An example is given by Hashimoto and Tokuda (1999).26) Based on directional spectra estimated from field data, Mitsuyasu et al. (1975)27) proposed a standard form of directional spectra by introducing the Mitsuyasu-type directional function. Once an incident directional spectrum is given, the time series of the water surface elevation can be generated by superposing component periodic waves. In the next section, equations for predicting the transformation Fig. 7. Sketch of components of nearshore wave transformation. of periodic waves as well as random waves in the nearshore zone are introduced. From the periodic small-amplitude wave theory 4. Equations for wave transformation at a uniform water depth, the velocity potential, , in the nearshore zone is expressed as follows: When waves enter the nearshore region, water g cosh kðz þ hÞ i motion is restricted by the sea bottom, shoreline and ¼ ê !t ½41 i! cosh kh coastal structures, and hence the waves transform. There are many elements of wave transformation, where i is the unit of imaginary number. The water such as shoaling, refraction, diffraction, reflection and particle velocity is obtained by taking the gradient of breaking. To understand nearshore processes such as the velocity potential, . The above expression is wave-induced current, sediment transport, wave run- equivalent to the first-order (i.e., small amplitude) up and overtopping on coastal structures, and wave solution of the Stokes wave theory expressed by forces on breakwaters, we need to predict the wave Eq. [16]. The symbol, ^,defined by the following field in the nearshore region. Historically, each equation, denotes the complex amplitude of the element of wave transformation has been treated water surface elevation, : separately, such as by refraction diagrams or diffrac- i ¼ ê !t: ½42 tion diagrams. Refraction means wave transformation involving a change in wave direction and height, Since we deal with refraction-diffraction problems, due to a change in water depth. This includes wave the complex amplitude, ^, is not constant but is a shoaling, which is the transformation of waves inci- gradually varying function of the horizontal coor- dent perpendicular to the coast with parallel bottom dinates, ðx; yÞ. contours. Diffraction is caused mainly by barriers The mild-slope equation is obtained by multi- which interrupt wave propagation and create a shel- plying a vertical distribution function, cosh kðz þ hÞ, tered region behind them. Figure 7 depicts the vari- with the governing Laplace equation and integrating ous components of nearshore wave transformation. from the bottom to the mean water surface in the These elements occur simultaneously in the vertical direction: nearshore zone. The mild-slope equation was derived 2 rðcc r^Þþk cc ^ ¼ 0 ½43 to analyze combined refraction and diffraction of g g periodic waves of small amplitude. Since then, where r¼ð@=@x; @=@yÞ is the two-dimensional hor- various wave equations have been proposed to take izontal gradient operator, c the wave celerity and more elements into consideration. In the following, cg the group velocity (i.e., energy transfer velocity). the mild-slope equation is first introduced and then For a given set of boundary conditions at all equations extended for randomness and nonlinearity boundaries of the computational domain, the above of waves are explained. elliptic partial differential equation can be solved 4.1 Mild-slope equation. The mild-slope uniquely. equation was derived by Berkhoff (1972)28) to analyze Berkhoff et al. (1982)29) proved that the mild- combined refraction and diffraction of small-ampli- slope equation includes the eikonal equation with tude periodic waves. diffraction effect and the energy conservation equa- 42 M. ISOBE [Vol. 89, tion between two wave rays. This implies that the mild-slope equation includes a refraction process with modification due to diffraction. Hence, wave fields such as those in a harbor of varying water depth and in a nearshore zone with coastal structures can be predicted. 4.2 Parabolic approximation. The mild-slope equation requires boundary conditions on all boundaries of the computational domain, along both the offshore boundary and shoreline. All boundaries, including the shoreline, have more or less effects on the wave field. This means that each point in the wave field interacts with all other points and the solution must be calculated simultaneously for the whole wave field. However, from the physical viewpoint, waves entering a nearshore region transform Fig. 8. Definition sketch of the curvilinear ray-front coordinate due to refraction and diffraction, lose energy due to system. breaking, and finally disappear near the shoreline. This implies that the shoreline boundary has little and hence a coarse grid system can be used in effect on the major wave field as long as reflection is the numerical calculation with sufficient accuracy. negligible. This understanding reminds us to calcu- Equation [44] includes the second-order derivative of late the wave field step by step from the offshore with respect to , but the first order with respect to boundary to the onshore-ward direction. From a . Therefore, the elliptic-type mild-slope equation is mathematical viewpoint, a parabolic equation has approximated by the parabolic-type partial differ- such characteristics. Radder (1979)30) approximated ential equation. The second derivative with respect the mild-slope equation by a parabolic equation using to expresses a diffraction effect in the wave front Cartesian coordinates. To extend applicability, Isobe direction during wave propagation. Because of the (1986)31) used curvilinear coordinates which approx- parabolic nature of the partial differential equation, imate wave rays and fronts. it can be solved from the offshore incoming wave A curvilinear coordinate system, ð; Þ, of which boundary to the onshore direction step by step. This the directions of and correspond to those of wave remarkably reduces the required computer storage rays and fronts, respectively, is introduced as shown and computational time. in Fig. 8. On splitting the velocity potential into Figure 9 compares analytical and numerical incident and reflected wave components and assum- solutions of wave height distribution along the ing the latter is much smaller than the former, we sheltered side of a semi-infinite breakwater. Due to can eliminate the second derivative in terms of and diffraction, incoming wave energy penetrates into the obtain the following parabolic equation: sheltered region, although the diffraction coefficient, Kd, which is the ratio of wave height in the sheltered 1 @ 1 1 @ h @ 2ik þ ccg region to the incident wave height, is smaller than h @ cc h h @ h @ g unity. Good agreement is found especially for a small i 1 @ þ ð Þþ2 ð Þ ¼ 0 ½44 angle of incidence o. It should be noted that wave @ kccgh k k K ccgh h energy is proportional to the square of the wave where height, and hence the diffraction in terms of energy is Z not very large. This means diffraction occurs to a ¼ êxp i Khd : ½45 certain degree, but diffracted wave energy may often be neglected. The quantities h and h are the scale factors of the Figure 10 is an example to compare calculated curvilinear coordinate system and taken to be 1=K, and (laboratory) experimental results on the distri- in which K is the approximated value of the wave butions of the relative wave height, H=HI (the ratio number at each location. Since the spatial change of the local wave height to the incident wave height). of phase in ^ is approximately removed in by Since wave reflection is negligible due to the sloping definition [45], the change in is very gradual beach in the harbor, the assumption of the parabolic No. 1] Evolution of basic equations for nearshore wave field 43 approximation is satisfied and hence good agreement 1983;32) Liu and Tsay, 198433)) and breaking waves can be found. (Dalrymple et al., 1984).34) Diffraction-refraction of Parabolic equation models are extended to random waves can be treated by superimposing weakly nonlinear waves (Kirby and Dalrymple, results for component waves with various frequencies and directions. A refraction, diffraction and breaking model of random waves was proposed by Isobe (1987)35) by introducing energy dissipation due to wave breaking. The model can be used in general situations of wave transformation in the nearshore zone as long as wave nonlinearity and multi-reflection are not critical factors. Figure 11 shows a comparison of calculated and measured significant wave height, H1=3, and mean direction, m. 4.3 Time-dependent mild slope equations for random waves. When multi-reflection is important, we need to solve for the mild-slope equation of the elliptic type. Time-dependent equations are proposed as a numerical solution technique. Since the wave profile is calculated time by time, a dissipation term due to wave breaking can be incorporated easily. Watanabe and Maruyama (1986)36) proposed time- dependent mild-slope equations for calculating the nearshore wave field due to refraction, diffraction and breaking. For random waves, Isobe (1994)37) derived Fig. 9. Comparison of diffraction coefficients behind a semi- a set of time-dependent equations as introduced in infinite breakwater between numerical solution calculated by the the following. parabolic equation and analytical solution.

Fig. 10. Comparison of calculated and measured relative wave heights in a harbor with a sloping bottom. 44 M. ISOBE [Vol. 89,

Fig. 11. Comparison of calculated and measured signiﬁcant wave height and mean direction (Oarai Beach, Ibaraki Prefecture, Japan, October 1 to 2, 1987).

~ pffiffiffiffiffiffi In the mild-slope equation [43], the complex transformed to f ¼ =~ ccg, which simplifies the amplitude, ^,isdefined as Eq. [42] by using the mild-slope equation into the Helmholtz equation: angular frequency, !, of the periodic waves. However, r2f~ þ k2f~ ¼ 0: ½49 for random waves, angular frequencies are different per component waves. In addition, the values of the To include energy dissipation due to breaking, the 2 coefficients, ccg and k , are dependent on the angular energy dissipation coefficient, D, is introduced in the frequency. Therefore, we cannot use the mild-slope equation: equation directly to calculate random wave trans- r2f~ þ k2ð1 þ iDÞf~ ¼ 0: ½50 formation. Here, we consider narrow-banded frequency Since the values of the coefficients change with spectra with a representative angular frequency, !, the angular frequency, we cannot deal with random around which random wave energy is concentrated, waves by the above equation. If we introduce time and define ~ by derivatives of f~ into a similar equation: ! ¼ ~ei!t ½46 ~ : 2 ~ 2 @f ~ r f ia1r þðb0 þ ic0Þf Comparison with Eq. [42] yields @t 0 ~ 2 ~ ~ ¼ êi! t ½47 @f @ f þ iðb1 þ ic1Þ b2 ¼ 0 ½51 @t @t2 where !0 ¼ ! !, which is small compared to ! by assumption. So, ~ is defined independently of the and compare with Eq. [50], the following constraints angular frequency of each component of random are obtained for the constants, a1, b0, b1, b2, c0 and c1: waves and a slowly varying function of time, t. 2 0 02 0 k ¼ðb0 þ b1! þ b2! Þ=ð1 a1! Þ½52 On considering Eq. [47], the mild-slope equation 0 must also be satisfied by ~: D ¼ c0 þ c1! : ½53 2 By determining the values which approximate the rðcc r~Þþk cc ~ ¼ 0: ½48 g g above two equations, all the coefficients in Eq. [51] Now for the sake of mathematical simplicity, ~ is become independent of frequency and thus the No. 1] Evolution of basic equations for nearshore wave field 45

of the relative water depth and hence the relative wave height, H=h, are small. By taking the terms up to the second order of , the following Boussinesq equations were derived: @ þ r½ðh þ Þu¼0 ½54 @t @u þðurÞu þ r @ g t h @u h2 @u ¼ rr h rr ½55 2 @t 6 @t where u ¼ðu; vÞ is the horizontal velocity averaged over the water depth. Boussinesq equations include terms up to the second order of the relative wave height and square of the relative water depth, which are indicators of wave nonlinearity and dispersivity, respectively. Therefore, Boussinesq equations are weakly nonlinear and weakly dispersive wave equations, which implies the Fig. 12. Comparison between calculated and measured water equations are valid for moderately high waves in surface fluctuation due to random waves in shoaling water. shallow water. The right-hand side of Eq. [55] represents the equation can be used for component waves with any effect of wave dispersion. For very shallow water frequency. Thus by inputting f~ along the offshore waves of small amplitude, the wave celerity is boundary defined from incident random water sur- independent of the wave period, and hence all face fluctuation, which is obtained by superimposing componentpffiffiffiffiffiffi waves propagate by the same wave component waves for a given wave spectrum, Eq. [51] celerity, gh, which means waves propagate by can be used to analyze refraction, diffraction and keeping the same profile. However, since the two breaking of random waves. Since temporal evolution terms on the right-hand side of Eq. [55] result in a is obtained in the numerical calculation, the equation slight difference in celerity depending on the wave is called time-dependent mild-slope equation for period of each component wave, each component random waves. wave propagates with slightly different celerity, the Figure 12 shows an example to compare calcu- total wave profile changes gradually, and finally the lated and measured water surface fluctuation due to component waves separate from each other from shoaling random waves. The incident wave profile front to back due to the difference in wave celerity. which is shown in the middle figure was measured on This phenomenon is called wave dispersion, which is a horizontal bottom. As seen in the bottom figure, the reason why Boussinesq equations are dispersive good agreement can be found as long as the equations. If these terms are neglected, the resultant small-amplitude assumption is satisfied. Moreover, equations are called nonlinear shallow water equa- although discrepancies especially in the steepness tions, which do not include dispersion caused by the of wave crests become prominent just prior to the effect of finite water depth. breaking point, integral properties such as wave To extend the applicable range of Boussinesq energy and radiation stress can be accurately equations, modified Boussinesq equations were calculated. However, to predict properties which are proposed. As an example, Madsen and Sørensen strongly affected by nonlinearity, nonlinear theory (1992)39) introduced additional terms to improve the must be used. dispersion relation in deep water. In addition, apart 4.4 Boussinesq equations. For nonlinear from the theoretical assumption and derivation, the waves, Boussinesq equations were derived by nonlinear term in the expression of Boussinesq Peregrine (1967).38) The basic assumption is the equations is the same as the fully nonlinear term. same as in the case of cnoidal waves, explained in Therefore, the modified version of Boussinesq equa- section 2. Ursell’s parameter, ", is assumed to be in tions is valid for most of the practical situations to 2 the order of unity, whereas the square, ¼ðh=LÞ , calculate wave field, such as wave height distribution. 46 M. ISOBE [Vol. 89,

However, because of the approximation included @ Z þrðA rf ÞB f þðC C Þrf rh in the theory, detailed wave properties such as @t vertical distribution of velocity cannot be described @Z accurately. þ Z f rrh ¼ 0 ½58 @h 4.5 Time-dependent nonlinear mild slope @ 1 1 @Z @Z equations for random waves. A strongly nonlinear þ f þ r r þ g Z Z Z f f ff wave equation is derived in a somewhat different @t 2 2 @z @z 40) @Z way. It was shown by Luke (1967) that the velocity þ Zf rf rh ¼ 0 ½59 potential, , and water surface fluctuation, , which @h terminate the following Lagrangian, satisfy the where Z governing equation and boundary conditions for waves on a varying water depth: A ¼ ZZdz ½60 Z ZZ Z h t2 Z @ 1 2 @ @ L½; ¼ þ ðrÞ Z Z @ 2 B ¼ dz ½61 t1 A h t h @z @z 2 Z 1 @ @Z þ þ gz dz dA dt ½56 C ¼ Z dz ½62 2 @z @ Zh z where t1 and t2 are the limits of time, t, and A @ @ ¼ Z Z ½63 denotes the area of concern in the horizontal plane. D @ @ dz h h h In general, wave equations are two-dimensional equations which are obtained by integrating the and the superscript, , denotes the value at the water governing three-dimensional equations in the vertical surface, z ¼ . direction. For the vertical integration, vertical Since Eq. [58] is a vector equation consisting of distribution functions are given theoretically or M components and Eq. [59] is a scalar equation, empirically. Massel (1993)41) derived an extended ðM þ 1Þ equations are included in these equations. mild slope equation by introducing a vertical On the other hand, the number of unknown functions distribution function of the hyperbolic cosine type is ðM þ 1Þ : M for f and 1 for . Hence we can solve and integrating the governing equation. A clear and these simultaneous equations step by step if the generalized concept of this procedure was proposed appropriate initial and boundary conditions are by Nadaoka et al. (1994)42) in deriving a strongly given. nonlinear, strongly dispersive wave equation by All wave equations such as the mild-slope applying the Galerkin method to the Euler equations equation and Boussinesq equations are obtained by of motion. Isobe (1994)37) derived a general form of integrating three-dimensional basic equations in the nonlinear, dispersive wave equations by using the vertical direction for a corresponding vertical distri- Lagrangian defined by Eq. [56] as follows. bution function. In this sense, the nonlinear time- The velocity potential, , which is a function in dependent mild-slope equations explained above are three dimensions, including the vertical direction, is a general form of wave equations, since the set of expanded into a series in terms of a certain set of vertical distribution functions can be given arbitra- vertical distribution functions, ZðzÞ ( ¼ 1toM): rily. For example, if we give only one component of a XM hyperbolic cosine-type vertical distribution function ðx;z;tÞ¼ Zðz; hðxÞÞfðx;tÞZf ½57 and neglect nonlinear terms, we obtain the same ¼1 result as the mild-slope equation. If we give a where x ¼ðx; yÞ is the two-component horizontal parabolic vertical distribution function, the result is coordinate and the summation convention is applied equivalent to Boussinesq equations. Finally, if we give when the same subscript appears twice in a term. The asufficient number of vertical distribution functions function, Z, may be dependent on the local water which can accurately approximate the strict solution depth, hðxÞ, and f is the weighting function for Z of the vertical profile of the velocity potential, we and a function of x and t but not of z. can calculate an accurate numerical solution for On substituting Eq. [57] into Eq. [56] and nonlinear wave transformation in the nearshore area. applying the variational principle with respect to f The accuracy of the dispersion relation, i.e., the and , we obtain the following set of time-evolutional relation between the wave frequency and wavelength partial differential equations: or wave celerity, is often used to examine the validity No. 1] Evolution of basic equations for nearshore wave field 47

Fig. 13. Dispersion relation for polynomial vertical distribution functions of various orders. of the approximation to linear wave theory. The linear dispersion relation is strictly satisfied by assuming a hyperbolic cosine-type vertical distribution function. As an alternative, a set of polynomial functions can satisfy the dispersion relation with sufficient accuracy. This set has an advantage in Fig. 14. Comparison between calculated and measured water that it can give an accurate vertical distribution surface elevation and water particle velocity at the bottom in to analyze nonlinear wave transformation in very shoaling water. shallow water. Figure 13 compares the linear dispersion relations of the strict solution and series waters, respectively, according to the assumption on expressions in terms of polynomial functions. The water depth. To deal with wave transformation in agreement is good in very shallow water even for a the nearshore region, the effect of changing water small number of terms, such as M ¼ 2, and also good depth, i.e., the bottom slope, must be taken into even in deep water if we use a sufficient number of account. The mild-slope equation includes the effect terms, such as M ¼ 4. This implies that if we use of the bottom slope on the basis of the first-order asufficient number of polynomial functions, trans- Stokes wave theory, i.e., small-amplitude wave formation of nonlinear waves can accurately be theory. Parabolic equations have been derived from predicted from deep water to very shallow water. the mild-slope equation for simpler numerical calcu- Figure 14 compares calculated and measured water lation. Time-dependent mild-slope equations have surface fluctuation, , and horizontal water particle been proposed mainly to incorporate with a wave velocity, ub, at the bottom. Although the profile has breaking model. Furthermore, time-dependent mild- steep crests resulting from strong nonlinearity, the slope equations for random waves can deal with agreement is good. Figure 15 is for waves travelling refraction, diffraction and breaking of random waves. over a submerged breakwater. Although both non- Nonlinear shallow water equations have been linearity and dispersion are strong, a good agreement derived for nonlinear waves in very shallow water. can be obtained. Such a high accuracy cannot be The equations do not include the effect of wave found if Boussinesq equations are used because they dispersion. Boussinesq equations, which in the case of are weakly nonlinear and weakly dispersive wave a horizontal bottom reduce to the first-order cnoidal equations. These imply that the time-dependent wave equations, include weakly dispersive terms in nonlinear mild-slope equations are valid if an addition to the terms in nonlinear shallow water appropriate set of vertical distribution functions are waves. Modified versions can represent strong dis- selected. persivity in relatively deep water. 4.6 Classification of wave equations. Funda- Time-dependent nonlinear mild-slope equations mental wave theories are for waves which propagate for random waves can represent both strong non- at a uniform water depth without deformation. linearity and strong dispersivity and deal with Stokes and cnoidal wave theories have been derived nonlinear random waves in a time-series numerical for deep to shallow, and shallow to very shallow calculation. 48 M. ISOBE [Vol. 89,

Table 2. Assumptions behind various wave equations

Bottom slope Relative water depth Nonlinearity Wave equations Multi reflection 2 Randomness Ljrhj=h ðh=LÞ H=L or H=h MSE1) small yes arbitrary small (finite8)) no (yes10)) Parabolic eq. small no6) arbitrary small (finite8)) no (yes10)) T-D2) MSE small yes arbitrary small no (yes10)) T-D2) MSE for random waves small yes arbitrary small yes NL3) shallow water eqs. arbitrary4) yes very small arbitrary yes Boussinesq eqs. arbitrary4) yes small (arbitrary7)) finite9) yes NL3) MSE arbitrary5) yes arbitrary arbitrary yes 1) MSE: mild slope equation, 2) T-D: time-dependent, 3) NL: nonlinear, 4) within restriction of shallow water assumption, 5) dependent on vertical distribution functions, 6) improved version includes incident and reflected waves, 7) improved version, 8) improved version, 9) in spite of theoretical assumption, resultant equations includes a fully nonlinear term, 10) by linear superposition of component waves.

Table 2 summarizes the classification of various wave equations according to the assumptions in deriving the equations. Since the more strict and general equations need more computer storage and computational time, appropriate selection of wave equations is essential in practice. In this section, we systematically reviewed various wave equations which can be obtained by integrating the three-dimensional governing equations in the vertical direction. If vertical irregularity is essential, such as in wave transformation due to three-dimensional structures, direct numerical calculation of three-dimensional governing equations is indispensable. The progress in this field is also remarkable with the aid of improved computer performance (JSCE, 2012).43) 5. Concluding remarks In this paper, an overall view of wave equations to analyze the nearshore wave field has been introduced as systematically as possible. Fundamental wave theories are for periodic waves which propagate at a uniform water depth without deformation, i.e., waves of permanent form. As analytical solutions of this type of waves, Stokes and cnoidal wave theories were developed for deep to shallow water and for shallow to very shallow water, respectively. Independent parameters for waves of permanent form are the water depth, h, wavelength, L, and wave height, H, from which two independent non-dimensional parameters can be defined arbitra- 2 rily. If we select the square, ¼ðh=LÞ , of relative water depth and Ursell’s parameter, " ¼ HL2=h3,a regular double power series solution can be derived. Fig. 15. Comparison between calculated and measured water Based on this solution, a systematic understanding of surface elevation on and behind a submerged breakwater. various single power series solutions, i.e., perturba- No. 1] Evolution of basic equations for nearshore wave field 49 tion solutions, can be achieved. Stokes and cnoidal 4) Skjelbreia, L. and Hendrickson, J. (1960) Fifth order gravity wave theory. Proc. 7th Int. Conf. on wave theories have been proved to be the only two – ff fi Coastal Eng., 184 196. practically e ective solutions among an in nite 5) Korteweg, D.J. and de Vries, G. (1895) On the number of possible perturbation solutions. Further- change of form of long waves advancing in a more, validity ranges for these wave theories of rectangular canal and on a new type of long various orders were determined by calculating the stationary waves. Phil. Mag. Ser. 5 39, 422–443. relative error included in the solutions. In practice, 6) Friedrichs, K.O. (1948) Appendix. On the derivation of the shallow water theory. Commun. Pure Appl. we have only to use Stokes and cnoidal waves for Math. 1,81–85. Ursell’s parameters of less than and larger than 25, 7) Keller, J.B. (1948) The solitary wave and periodic respectively. waves in shallow water. Commun. Pure Appl. To deal with random waves, the concept of a Math. 1, 323–339. directional spectrum and methods to estimate direc- 8) Laitone, E.V. (1960) The second approximation to cnoidal and solitary waves. J. Fluid Mech. 9, 430– tional spectra were introduced. The maximum like- 444. lihood method, Bayesian approach and maximum 9) Chappelear, J.E. (1962) Shallow water waves. J. entropy method were among the latter. By applying Geophys. Res. 67, 4693–4704. these methods to field data, we can accumulate 10) Schwartz, L.W. (1974) Computer extension and ’ samples of directional spectra, from which standard analytic continuation of Stokes expansion for gravity waves. J. Fluid Mech. 62, 553–578. forms have been proposed. From the standard 11) Fenton, J.D. (1972) A ninth-order solution for the directional spectra, we can give incident waves to solitary wave. J. Fluid Mech. 53, 257–271. analyze transformation of random waves in the 12) Longuet-Higgins, M.S. and Fenton, J.D. (1974) On nearshore zone. the mass, momentum, energy and circulation of a Various wave equations to calculate wave trans- solitary wave II. Proc. Roy. Soc. Lond., Ser. A 340, 471–493. formation in the nearshore zone were introduced 13) Nishimura, H., Isobe, M. and Horikawa, K. (1977) based on the assumption in deriving the equations. Higher order solutions of the Stokes and the The mild-slope equation initiates numerical calcula- cnoidal waves. J. Fac. Eng. Univ. Tokyo, Ser. B tion of combined refraction and diffraction. Parabolic 2, 267–293. approximations were proposed to save computer 14) Isobe, M., Nishimura, H. and Horikawa, K. (1982) Theoretical considerations on perturbation solu- storage and computational time, and time-dependent tions for waves of permanent type. Bull. Fac. Eng., mild-slope equations were proposed to simplify Yokohama Natl. Univ. 31,29–57. numerical calculation and introduce a dissipation 15) Yamada, H. and Shiotani, T. (1968) On the highest term due to wave breaking. Time-dependent mild- water waves of permanent type. Bull. Disas. Prev. – slope equations were further extended for random Res. Inst. Kyoto Univ. 18, No. 135, 1 22. 16) Goda, Y. (1970) A synthesis of breaker indices. waves. As a set of nonlinear wave equations, Trans. Japan Soc. Civil Engrs. 2, 227–230. Boussinesq equations were derived for weakly non- 17) Dean, R.G. (1965) Stream function representation linear and weakly dispersive waves. Finally, non- of nonlinear ocean waves. J. Geophys. Res. 70, linear mild-slope equations were derived to analyze 4561–4572. shoaling, refraction, diffraction, reflection and break- 18) Goda, Y. (2008) Overview on the applications of random wave concept in coastal engineering. Proc. ing of nonlinear random waves. Jpn. Acad., Ser. B 84, 374–385. As introduced above, wave transformation in 19) Isobe, M., Kondo, K. and Horikawa, K. (1984) the nearshore zone is of theoretical and practical Extension of MLM for estimating directional wave interest. More accurate and detailed analysis meth- spectrum. Proc. Symp. on Description and Model- – ods will be developed with the help of advanced ling of Directional Seas A-6, 1 15. 20) Oltman–Shay, J. and Guza, R.T. (1984) A data- computer performance. adaptive ocean wave directional–spectrum estima- tor for pitch and roll type measurements. J. Phys. References Oceanogr. 14, 1800–1810. 21) Longuet-Higgins, M.S., Cartwright, E.D. and Smith, 1) Airy, G.B. (1841) Tides and waves. In Encyclopaedia N.D. (1963) Observation of the directional spec- Metropolitana (eds. Rose, H.J. et al.) Mixed trum of sea waves using motions of a floating buoy. Sciences. 3, 1817–1845. In Ocean Wave Spectra, Prentice Hall, Inc., New 2) Stokes, G.G. (1847) On the theory of oscillatory Jersey, pp. 111–136. waves. Trans. Cambridge Phil. Soc. 8, 441–455. 22) Hashimoto, N. and Kobune, K. (1988) Directional 3) De, S.C. (1955) Contributions to the theory of Stokes spectrum estimation from a Bayesian approach. waves. Proc. Camb. Philos. Soc. 51, 713–736. Proc. 21st Int. Conf. on Coastal Eng., 62–76. 50 M. ISOBE [Vol. 89,

23) Hashimoto, N., Nagai, T. and Asai, T. (1994) 33) Liu, P.L.-F. and Tsay, T.-K. (1984) Refraction- Extension of the maximum entropy principle diffraction model for weakly nonlinear water method for directional wave spectrum estimation. waves. J. Fluid Mech. 141, 265–274. Proc. 24th Int. Conf. on Coastal Eng., 232–246. 34) Dalrymple, R.A., Kirby, J.T. and Hwang, P.A. 24) Isobe, M. and Kondo, K. (1984) Method for (1984) Wave diffraction due to areas of energy estimating directional wave spectrum in incident dissipation. J. Waterway Port Coastal Ocean Eng. and reflected wave field. Proc. 19th Int. Conf. on 110,67–79. Coastal Eng., 467–483. 35) Isobe, M. (1987) A parabolic equation model for 25) Hashimoto, N., Nakagawa, Y., Nagai, T. and Ito, K. transformation of irregular waves due to refraction, (1997) Modification of extended maximum entropy diffraction and breaking. Coastal Eng. in Japan principle method (EMEP) for estimating direc- 30,33–47. tional spectrum in incident and reflected wave 36) Watanabe, A. and Maruyama, K. (1986) Numerical field. Proc. Int. Conf. on Offshore Mechanics and modeling of nearshore wave field under combined Arctic Eng., OMAE 2,1–7. refraction, diffraction and breaking. Coastal Eng. 26) Hashimoto, N. and Tokuda, M. (1999) A Bayesian in Japan 29,19–39. approach for estimation of directional wave spectra 37) Isobe, M. (1994) Time-dependent mild-slope equa- with HF radar. Coastal Eng. J. 41, 137–149. tions for random waves. Proc. 24th Int. Conf. on 27) Mitsuyasu, H., Tasai, F., Suhara, T., Mizuno, S., Coastal Eng., 285–299. Ohkusu, M., Honda, T. and Rikiishi, K. (1975) 38) Peregrine, D.H. (1967) Long waves on a beach. J. Observation of the directional spectrum of ocean Fluid Mech. 27, 815–827. waves using a cloverleaf buoy. J. Phys. Oceanogr. 39) Madsen, P.A. and Sørensen, O.R. (1992) A new form 5, 750–760. of the Boussinesq equations with improved linear 28) Berkhoff, J.C.W. (1972) Computation of combined dispersion characteristics. Part 2. A slowly-varying refraction-diffraction. Proc. 13th Int. Conf. on bathymetry. Coastal Eng. 18, 183–204. Coastal Eng., 471–490. 40) Luke, J.C. (1967) A variational principle for a fluid 29) Berkhoff, J.C.W., Booy, N. and Radder, A.C. (1982) with a free surface. J. Fluid Mech. 27, 395–397. Verification of numerical wave propagation models 41) Massel, S.R. (1993) Extended refraction-diffraction for simple harmonic linear water waves. Coastal equation for surface waves. Coastal Eng. 19,97– Eng. 6, 253–279. 126. 30) Radder, A.C. (1979) On the parabolic equation 42) Nadaoka, K., Beji, S. and Nakagawa, Y. (1994) A method for water-wave propagation. J. Fluid fully-dispersive nonlinear wave model and its Mech. 72, 373–384. numerical solutions. Proc. 24th Int. Conf. on 31) Isobe, M. (1986) A parabolic refraction-diffraction Coastal Eng., 427–441. equation in the ray-front coordinate system. Proc. 43) Subcommittee on Numerical Wave Flume, Coastal 20th Int. Conf. on Coastal Eng., 306–317. Engineering Committee, JSCE (ed.) (2012) Nu- 32) Kirby, J.T. and Dalrymple, R.A. (1983) A parabolic merical Wave Flume, JSCE (in Japanese). equation for the combined refraction-diffraction of Stokes waves by mildly varying topography. J. Fluid Mech. 136, 453–466. (Received Sep. 26, 2012; accepted Nov. 20, 2012)

Proﬁle

Masahiko Isobe was born in 1952 in Tokyo. He graduated from the Faculty of Engineering at the University of Tokyo, specializing in civil engineering. He continued his study at the Graduate School of Engineering and obtained master’s and doctoral degrees in the ﬁeld of coastal engineering under the supervision of Prof. Kiyoshi Horikawa. He began his professional career as Research Associate in 1978 at the University of Tokyo. Since then he has experienced Assistant Professor (1981 to 1983) and Associate Professor (1983 to 1987) at the Yokohama National University, and Associate Professor (1987 to 1992) and Professor (1992 to date) at the University of Tokyo. He was one of the core founders of the Graduate School of Frontier Sciences which was established at Kashiwa Campus, the newly developed and third major campus of the University of Tokyo, in 1998. He served as the Dean (2005 to 2007) of the graduate school and the Vice President (2009 to 2011) of the university. His contribution in research is mainly on the wave hydrodynamics and coastal environment. He has been working on the nonlinear wave dynamics in the nearshore area and water quality problems in the enclosed sea. As a result of his achievements, he was awarded annual incentive prize and annual research award of the Japan Society of Civil Engineers in 1986 and 1997, respectively.