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Evolution of Basic Equations for Nearshore Wave Field

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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 À ct: ½2 usage, “deep water”, “shallow water” and “very shallow water” refer respectively to water depths which are Thus, the stream function, É, 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 ÉðX; zÞ¼ ðx À ct; zÞcz ½3 a second-order solution, and a third-order solution ð Þ¼ ð À Þ ½4 was given by Chappelear (1962).9) N X x ct : Since perturbation solutions need huge amounts Now the governing equation for É 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- @2É @2É 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- É ¼ 0onz ¼h ½6 er-order solutions of a solitary wave, which is in the É 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 @É 2 @É 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 É 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 É 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 ÉðX; 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.
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