1.Introduction

Deep water waves can undergo dramatic changes as they propagate towards a shoreline over an irregular seabed.The processes of refraction,shoaling,diffraction,and breaking can significantly alter wave heights and directions near the shore,and must be adequately accounted for in developing beach protection and flood defence.The wave period is also an important parameter in the design of coastal structures,particularly in relation to the process of shoaling.Long-period waves,such as tsunamis,storm surges,and swells,have greater wave lengths than local storm waves,and are therefore more susceptible to the effects of shoaling.Shoaling in particular can lead to reductions in wave length and speed,as well as amplification of wave height.As long waves propagate into the coastal zone over a shelving seabed,some of the wave energy will be reflected and some will be transmitted shoreward.The focus of this paper is on wave shoaling,as well as the transmission and reflection coefficients.

This topic has been the subject of extensive research through experiments,analytical methods,and numerical modeling.Approaches can be categorized broadly into the following three categories:two horizontal dimensions(2DH);vertical slices(2DV);and resolving the vertical and horizontal structures of the wave field(3D).Early analytical treatments,such as those of Carrier(1966)and Synolakis(1987),were 2DVand focused on predicting the run-up level of long waves.These have been supplemented by other analytical solutions accounting for refraction and friction(Nielson,1983,1984)and cnoidal wave theories such as those in Sadeghian and Peyman(2012).A wide range of 2DV computational models have been developed and are based predominantly on the Boussinesq equations,the shallow water equations,or Reynolds-averaged Navier-Stokes (RANS) formulations.RANS modeling is the most computationally expensive of these equations and for that reason is limited to describing waves in a small domain such as the surf zone(e.g.,Lin and Liu,1998).There are comparatively fewer computational requirements for solving the Boussinesq equations.As a consequence,various forms of Boussinesq equations have been developed for both 2DV and 2DH applications.Early attempts were able to capture some nonlinear effects but encountered difficulties with accurate representation of linear dispersion properties at all water depths(e.g.,Nwogu,1993).More recent developments have allowed this deficiency to be addressed,at the cost of additional complexity(e.g.,Madsen et al.,2002).

Early 2DH wave models were based on a form of the elliptic mild slope equation proposed by Berkhoff(1972),which account for refraction and diffraction and provide the variation of wave amplitude over a 2DH spatial domain.A more general time-dependent form was subsequently proposed by Smith and Sprinks(1975).The computational challenges of solving an elliptical wave equation led to the development of various parabolic approximations,such as those in Radder(1979),Kirby(1986),Ebersole(1985),and Panchang et al.(1988),as well as methods for solving the transient problem via a hyperbolic formulation(Copeland,1985).An efficient means of solving the elliptic mild slope equation was proposed by Li and Anastasiou(1992),and extended by Li et al.(1993)to treat irregular wave propagation.Further developments of this theme have seen the inclusion of nonlinear effects by Beji and Nadoaka(1997);description of the wave field through an angular spectrum by Dalrymple et al.(1989),Suh et al.(1990),and Reeve(1992);and alternative derivations by Kim et al.(2009).

Here,we considered long waves occurring on a shelving foreshore and used the shallow water equations(SWEs)in a 2DV domain.Bautista et al.(2011)derived an analytical solution for long wave propagation over a linear transition shelf using a singular perturbation analysis based on the Wentzel-Kramers-Brillouin(WKB)technique.In this study,we derived analytical solutions to the linear SWEs based on the variable separation method and also developed a staggered finite volume method for their numerical solution.

This paper is organized as follows:Section 2 contains a brief description of the SWEs for waves propagating over linear and parabolic transition topography,and analytical solutions for the wave reflection and transmission coefficients are derived using the variable separation method,leading to solutions in terms of Bessel functions and polynomials;a staggered finite volume method for solving the SWEs over arbitrary topography is presented in Section 3;and in Section 4,we describe the implementation of our scheme to simulate wave propagation over linear and parabolic transition shelves.Moreover,to validate the numerical model,the computational solutions are compared with the analytical solutions and other results from previous studies.The validated numerical model is then applied to a topographic transect of the coast of Aceh,Indonesia.Conclusions are presented in the final section.

2.Shallow water model

2.1.Background

In this study,we considered a layer of ideal fluid,bounded above by a free surface and below by an impermeable bottom topography z=-h（x）,where x,z,and h are independent variables denoting the horizontal direction,vertical direction,and water depth,respectively.The following shallow water model for long waves in a relatively shallow region was adopted,following Kowalik(1993):

where u is the horizontal velocity,η is the free surface elevation measured from the still water level,g is the gravitational acceleration,and t is time.Eq.(1)represents the conservation of mass and momentum balance.The physical domain considered is shown in Fig.1.

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The physical domain is divided into three different regions:Ω1, Ω2,and Ω3,with prescribed depths defined by the following relationships:

Approaches similar to those explained in Subsection 2.2 are used to find the wave reflection and transmission coefficients for a parabolic transition shelf.The free surface elevation and horizontal velocity in domains Ω1 and Ω3 are expressed as follows:

The scheme leads to the numerical solutions of Eq.(1):

(2)Parabolic transition shelf:

These equations may be solved within domain Ω2 in exactly the same manner as before.Seeking solutions of η（x，t）and u（x，t）of the form given in Eq.(5)and substituting the results into Eq.(22)yield

In the following two subsections,we derive the reflection and transmission coefficients of the free surface over the linear and parabolic transition shelves,respectively.When waves propagate over a shelf,they undergo a scattering process,with some of the wave energy being transmitted shoreward and some reflected back towards the deep sea.Analytical solutions of the SWEs can be derived based on multiple-scale asymptotic expansion(Mei et al.,2005;Noviantri and Pudjaprasetya,2010)and the well-known WKB perturbation techniques(Bender and Orszag,1978;Bautista et al.,2011).Here,solutions of the linear SWEs are derived using the variable separation method.

2.2.Reflected and transmitted waves for linear transition shelf

In domain Ω1 the seabed is flat and the governing equations are as follows:

Using the variable separation method,the general solutions of η（x，t）and u（x，t）in domain Ω1 are written for a monochromatic wave with unknown spatial dependence,F（x）and G（x）,and arbitrary frequency ω:

where a i is the amplitude of the incoming wave,and a r is the amplitude of the reflected wave.The wave number,k0,in domain Ω1 obeys the following dispersion relation:

Substituting Eq.(5)into Eq.(4)yields the following:

Next,substituting Eq.(7)into Eq.(5)yields the following expressions for the free surface and velocity:

In domain Ω2 the governing equations for the linear transition region become

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As in domain Ω1,a general solution in the form given in Eq.(5)is sought.Substituting Eq.(5)into Eq.(10)yields

With the solution of F（x）,and by differentiating F（x）with respect to x and substituting the resultinto Eq.(6),the solutions to Eq.(6)are obtained:

where J1 and Y1 are the first-order Bessel functions of the first and second kinds,respectively.

同时由图2得知在热处理4 h时样品柱型图最低，即粗糙度最小;热处理1 h的样品柱型图最高，即粗糙度最大;两者差值为3.28 μm。而2 h和未处理样品的粗糙度居中，但热处理2 h样品的粗糙度数值与未处理样品的粗糙度数值接近，但2 h的粗糙度略小于未处理样品。同时可以看到，热处理过程中，样品的粗糙度下降很大，说明马尾松受热处理影响较大。

Considering an incoming wave from the deep-waterarea with depth h2 propagating shoreward,the wave is partially reflected and partially transmitted while passing through the linear transition shelf.In the nearshore region Ω3,where the water depth is shallow,there is only a transmitted wave(propagating to the right,as shown in Fig.1),with an amplitude a t,and the solution is given by

where k1 obeys the following dispersion relation:

In general,accounting for the transmitted and reflected parts of the incident wave train,the free surface η（x，t）and horizontal velocity u（x，t）throughout the domain can be formulated as follows:

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with

The solution of the equations above is x=A-1b.This leads to the wave reflection and transmission coefficients that are respectively defined as

At the points x=0 and x=L,where the three regions connect,matching conditions must be used to conserve the free surface and horizontal momentum across the whole domain.There are,therefore,four equations with four unknown variables in the matrix form,which may be written as Ax=b,where

where N11,N12,N21,N22,M1,and M2 are defined as follows:

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2.3.Reflected and transmitted waves for parabolic transition shelf

(1)Linear transition shelf:

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In domain Ω2,the governing equations for the shelf with parabolic transition topography h（x）=ax2 read as follows:

where h2=,h3=and a is an arbitrary parameter.

The solution of F（x）in Eq.(23)is

whereand the constant coefficients C1 and C2 remain to be determined.Substituting F（x）into Eq.(23)yields

The matching conditions are also used to conserve the free surface η（x，t）and the horizontal momentum h（x）u（x，t）along the discontinuity points,x=x1 and x=x2,which give us four equations with four unknown variables,a r,a t,C1,and C2.By determining a r and a t and the coefficients C1 and C2,we obtain the following reflection and transmission coefficients for wave propagation over a parabolic transition shelf:

where μ=x1/x2.

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For the linear transition case,Fig.2(a)and(b)shows the variations of C R with L and h2/h3,respectively,and Fig.2(c)and(d)shows the variations of C T with L and h2/h3,respectively.C R and C T are virtually independent of L for the range 1 m＜L＜10 m.C R rises as h2/h3 increases to about 3.5,and then decreases slightly as h2/h3 increases further.In contrast,C T rises monotonically with h2/h3.

Corresponding results for C R and C T,as functions of ω2/（ga）,for the parabolic transition case are shown in Fig.3(a)and(b).The trend is that C T increases with ω2/（ga）,μ,and h2/h3,meaning that more wave energy propagates towards the coast with the increases of ω2/（ga）,μ,and h2/h3.Fig.4 shows that C T,as a function of h2/h3,exhibits similar variations for both the parabolic and linear transition cases,which is also in close agreement with the analytical result of Kajiura(1961).

3.Computational model

In this section,the numerical solution of Eq.(1),using a staggered conservative scheme,is presented.The scheme was described in detail in Pudjaprasetya and Magdalena(2014)and Magdalena et al.(2015).For the sake of concision,the scheme is described briefly here.We consider a computational domain Ω,consisting of a spatial domain ΩL= ［L0，L1］and a time domain ΩT= ［0，T］.The spatial domain ΩL is partitioned in a staggered way into several cells with a spatial step Δx.The time domain ΩT is discretized into finite time steps with a constant interval Δt.Momentum and mass conservation equations(Eq.(1))are discretized with cells centred at xi and xi+1/2,respectively.Furthermore,the water surface,η,and velocity,u,are defined at full-and half-grid points,respectively,as given in Fig.5.

where x1 and x2 are the locations of the left and right borders of the second domain,respectively;and h2 and h3 are the depths in the first and third domains,respectively.For the linear transition shelf,we assumed that x1=0 and x2=L,where L is the horizontal extent of the sloping bottom.

where superscripts and subscripts denote the time step and spatial grid point,respectively.As seen from Eq.(28),the water depth,h*,is required at half-grid points and is unknown.The value of h*is approximated using a first-order upwind scheme and is chosen depending on the flow velocity as follows:

Note that on the right-hand side of the momentum equation(Eq.(1)),η is calculated at tn+1,instead of at tn,as this choice is required to maintain the stability of the scheme.Using Von Neumann stability analysis,the system is stable if|λ|＜ 1,where λ is the eigenvalue.The detailed derivation of the stability criteria for the numerical scheme can be found in Appendix A.Both the wave elevation and velocity are set to zero initially.As time passes,the wave is incident from the left,propagates over the sloping topography,and radiates through the right boundary.The initial and boundary conditions,respectively,are as follows:

where A is the wave amplitude.

4.Results and discussion

To validate the numerical scheme,the numerical results were compared with the analytical solutions and the solution of Bautista et al.(2011).To study the influence of the sloping topography on the transmission and reflection of waves,the following parameter values were chosen.The constant depth h2 was set as 10 m,the gravitational acceleration as g=9.81 m·s-2,the length of the sloping bottom as L=10 m,the time step as Δt=0.001 s,and the spatial step length as Δx=0.025 m.For the calculation,we used a monochromatic wave with a single crest.To calculate the amplitude of the reflected wave,the wave elevation at a certain point before the sloping bottom was observed,and then the maximum amplitude of the wave moving leftward,which was the reflected wave,was recorded.Comparisons of the transmission and reflection coefficients of those solutions are given in Figs.6 and 7,respectively.Fig.6 shows that the analytical and numerical solutions of C T are in agreement.The solutions of C T obtained in this study converge to that of Bautista et al.(2011)as ε→1,where ε=h3/h2.It can also be seen in Fig.7 that the solutions of C R obtained in this study show the same trend,converging to that of Bautista et al.(2011)as ε→1.This is as expected as the WKB solution being valid for ε→1.The analytical and numerical solutions obtained in this study are in agreement with each other and extend the solution of Bautista et al.(2011)to the regime in which ε is not close to 1.

The differences between solutions,obtained by the analytical and numerical methods,for different values of ε are compared in Table 1,where E T and E R are the absolute errors of the numerical results of the transmission and reflection coefficients,compared with their analytical results,respectively,with E T=|C Tn-C Ta|and E R=|C Rn-C Ra|.C Tn and C Ta are the transmission coefficients obtained by the numerical and analytical methods,respectively;and C Rn and C Ra are the reflection coefficients obtained by the numerical and analytical methods,respectively.

It can be clearly seen from the table that the errors are very small.To investigate the accuracy of the numerical scheme proposed in this study,repeated simulations were conducted,with a fixed value of Δx=0.025 m and Δt ranging from 0.000131 to 0.001831 s,to calculate the error E T j corresponding to Δtj and the rate of convergence τt j=lg（E T（j+1）/E T j）/lg（Δtj+1/Δtj）.The results are shown in Table 2.Thein the range of 0-1,indicating that the order of accuracy in the time domain is one.

Explicit expressions for the coefficients C1 and C2 are given in the matrix x in Eq.(17).

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Next,to examine the accuracy of the numerical scheme in the spatial domain,repeated simulations were conducted,with Δt=0.001 s and Δx varying between 0.010 m and 1.000 m,to calculate E T j corresponding to Δxj and the rate of convergence Table 3.The calculated value ofis in the range of 1.8-2.4,indicating that the order of accuracy in the spatial domain is two.We therefore concluded that the numerical scheme has an accuracy of O（Δt，Δx2）(Pudjaprasetya and Magdalena,2014;Magdalena et al.,2015).

生态理念在阿克陶县木吉乡农村供水工程中的应用……………………………………………………… 肖怀玺（3-75）

For a further test,results for a parabolic transition shelf were compared with those of Kajiura(1961).The x-ordinate ran from-100 m to 0 m with x1=-60 m and x2=-40 m.The parameter a=0.00625 m-1.The undisturbed water depths in deep and shallow areas were described by h2=and h3=,respectively.The wave propagated from the left with an amplitude of a i=0.5 m and a wave frequency of ω=5.9610 s-1.On the right side of the domain,an absorbing boundary condition was applied.The numerical result of the wave transmission coefficient was compared with those obtained by the analytical formula of Eq.(27)and from the literature,as shown in Tables 4 and 5,where C Tl is the wave transmission coefficient calculated from the formula derived by Kajiura(1961)and reproduced by Mei et al.(2005)using asymptotic expansions.The formula is valid only for x2-x1≫h2 or h3,or when the water depth in the transition region with a sloping bottom strictly decreases in the shoreward direction.For example,in the case for x2=-50 m in Table 4,where x2-x1=10 m is less than h2=22.5 m and h3=15.625 m,the value of C Tl is far from the analytical and numerical results,demonstrating that Kajiura's formula is invalid.However,when all the parameters were chosen for the transition region with decreasing water depth under that condition,we could calculate the shoaling coefficient by inversion of the wave transmission coefficient given by Kajiura(1961)(Fig.4).

For a final test of the numerical scheme,a case in which the seabed transition was neither linear nor parabolic was chosen.The topography was determined as follows:

Finally,the numerical scheme was applied to simulation of the wave shoaling phenomena over a more complex bathymetry:a cross-sectional bathymetry of the coast of Aceh,in Sumatra,Indonesia,where the 2014 Boxing Day tsunami occurred.The cross-section was taken from (95.0278°E,3.2335°N)to(96.6583°E,3.6959°N)with Δx=0.4333 m.The wave propagating over the bathymetry along this crosssection is depicted in Fig.10.The incident wave with a 1-m wave height and a wave period of 17 min propagated to the shallow-water area.As the water depth decreased,wave

Monochromatic waves entered from the left with a wave amplitude of 0.5 m,and water depths were set as h2=22.5 m and h3=10 m.As time passed,the waves propagated to the shallow area and flowed through the right boundary.The computation was carried out in the spatial domain ranging from 0 to 250 m,with a spatial grid size of Δx=0.01 m.Fig.8 shows a comparison of the free surface elevation of the numerical solution in this study and the WKB solution described in Kristina et al.(2013).It is hard to distinguish between the two solutions for the free surface elevation in Fig.8.The WKB and numerical solutions are in close agreement.

The wave transmission coefficient was calculated for different values of ε.Comparison of the calculated results with those of the WKB solution in Kristina et al.(2013)is shown in Fig.9.shoaling phenomena occurred,with the wavelength shortening and the amplitude increasing.For the domain shown,the water depth was 1047 m in the deep-water region and approximately 30 m in the shallow-water region.The wave height was 2.44 m in the shallow-water region,corresponding to a numerical solution of the wave transmission coefficient of 2.44.The analytical solution gave a transmission coefficient of 2.43 in this study,showing a reasonable agreement with the numerical solution.

5.Conclusions

In this paper,new analytical solutions for transient long waves shoaling over coastal shelves have been presented.Specifically,expressions for the surface elevation,horizontal velocity,and wave reflection and transmission coefficients have been derived for linear and parabolic transition shelves.Novel results from a numerical scheme have also been presented based on a staggered conservative formulation with upwind updating.The analytical and numerical solutions for several cases show agreement.The solutions obtained in this study have also been compared with previous results from WKB approximations,showing that this study provides a means of extending solutions beyond the normal WKB limits.The numerical scheme was applied to the case of tsunami propagation towards the coast of Aceh,Indonesia.Results obtained from the numerical scheme suggest that waves with a period of 17 min,close to values quoted by Stosius et al.(2010)for the Boxing Day tsunami,would experience amplification by a factor of 2.44 when traveling into the coastal waters where the depth was 30 m.This coincides with the scale of amplification proposed by Kowalik(2012),suggesting that the relatively simple approach proposed in this paper can provide solutions that are useful in practice,in either an operational or strategic context.

Appendix A.Stability analysis

（4）重砂异常标志：矿区主要的铌钽矿脉与铌钽铍重砂异常晕圈长轴方向基本吻合, 铌钽铍等稀有金属重砂异常可以作为重要的找矿标志。

where ρ =eiωnΔt,with ωn being the angular frequency of η;a′ =kΔx,with k being the wave number;r=eiωrΔt,with ωr being the angular frequency of u;and h max is the maximum water depth.Eq.(A.1)can be rewritten as

备选试验点集Pbi包含众多Pareto最优意义下的试验点，这些试验点兼具改进目标函数和刻画可行域边界的能力，在为试验者提供极大选择空间的同时，也提高了试验点选取的目的性。新试验点刻画可行域边界的能力十分有助于提高优化算法的优化精度和效率。

The eigenvalue λ of matrix B satisfies the following equation:

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