An Extended Analytic Solution of Combined Refraction and Diffraction of Long Waves Propagating over Circular Island

An analytic solution of long waves scattering by a cylindrical island mounted on a permeable circular shoal was obtained by solving the linear long wave equation LWE . The solution is in terms of the Bessel function expressed by complex variables. The present solution is suitable for arbitrary bottom configurations described by a power function with two independent parameters. For the case of the paraboloidal shoal, there exists a singular point α 2 which can be removed using Frobenius series, where α is a real constant. The present solution is reduced to Yu and Zhang’s 2003 solution for impermeable circular shoal. The numerical results show some special features of the combined effect of wave refraction and diffraction caused by a porous circular island. The effect of key parameters of the island dimension, the shoal slope, and permeability on wave scattering was discussed based on the analytic solution.


Introduction
The combined effect of wave refraction and diffraction is caused by the varying bottom topography or the presence of islands and structures.The mild-slope equation MSE was first derived by Berkhoff 1 and then was extended by Porter and Staziker 2 , Hsu and Wen 3 to the rapidly varying sea bottom configuration.Up to now, the MSE still plays an important role in coastal and ocean engineering practice, as it can be widely implemented to solve the problems of combined refraction and diffraction based on linear wave theory.
Although numerical models are commonly used in practical applications, it is still limited by the problems of accuracy and numerical stability.Numerical results are often needed to be verified by comparing with measured data or analytic solutions.In some cases, it takes time and expensive cost for laboratory experiments because of the complexity in completion of the model and facilities setup.An alternate is to look for analytic solution that could provide more clear expressions in which the physical properties of wave motion can be used to examine the predictability and validity of the numerical models.
A typical example of the analytic study of combined refraction and diffraction is the case when the long-wave approximation is applied to the MSE in one dimension or axisymmetric bathymetry in two dimensions.Homma  The present investigation is to extend Yu and Zhang's 18 analytic solution to long wave propagation over a porous circular island with arbitrary topography.The combined effect of refraction and diffraction of long wave motion as well as key features of the intensity of wave ray focus behind the shoal is presented.A general solution was derived based on the mild slope equation MSE .The wave motion with combined refraction and diffraction was described by the Bessel function α / 2 and Frobenius series.The effect of key parameters on wave scattering is intensively discussed based on the analytic solution.

Theoretical Formulation
The long wave equation for combined refraction and diffraction of waves propagating over porous media is express as where φ is the velocity potential, ∇ h ∂/∂x, ∂/∂y the horizontal gradient operator, h w the thickness of the water, h p the thickness of porous layer, k is the wavenumber and δ is defined as n 0 / S if p with n 0 the porosity, S n 0 1 − n 0 C M , the inertial coefficient, C M the virtual mass coefficient, i √ −1 the unit complex number, and f p the friction coefficient.In the derivation of 2.1 , the time harmonic e iωt is assumed with ω being the angular frequency of the incident wave.Here, the angular frequency ω is related to the wavenumber k by the following dispersion relation: In order to study the wave scattering over a circular porous shoal, we assume that the shoal is symmetric and of the following form: where r is the distance of any point on the shoal from the center, r 1 is the radius of the circular island, r 0 is the radius of the impermeable circular shoal, h w and h p are the depths of water and porous layers, respectively, for r > r 0 and α is the power parameter of bottom shape.The definition sketch of the permeable circular island is presented in Figure 1.
Using the method of separation variables, the velocity potential is written as in which Θ n is the eigenfunction given by where C 1n and C 2n are coefficients to be determined.C 2n is set to be zero due to the symmetric property of the present problem with respect to the coordinate x.For convenience of computational procedure, the coefficient C 1n is specified as C 1n 1. Substituting 2.4 and 2.5 into 2.1 gives where h is an equivalent water depth defined by h h w δh p .

2.7
The following dimensionless variables are introduced for convenience of mathematical formulation:

Substitution of ξ 2
w and ξ 2 p from 2.8 into 2.3 leads to the resultant equations.
Equation 2.10 is multiplied by δ and added with 2.9 .The following equation is thus obtained, that is And, 2.6 is rearranged with the help of the dimensionless variable ρ ω 2 r/g 2.8 and the result reads For the special case of α 2, 2.12 is solved by making use of the Frobenius series, the solution is given by where C 3n and C 4n are constants, and μ n is represented by

2.14
We introduce two dimensionless parameters Q n ρ α/2 R n and χ 2β/|2 − α| ρ 1−α/2 for α / 2, 2.6 is rewritten in the form where ν n α 2 4n 2 / |2 − α| .The general solution of 2.15 is the Bessel function expressed as where J ν n and Y ν n are the Bessel functions of the first and second kind of v n order.Substitution the corresponding boundary conditions in 2.16 results in the particular solution for α / 2 given by

2.17
For an arbitrary bathymetry of a circular island, the solution is thus written as follows where

2.19
It is noted that the LWE of 2.1 has been solved for long waves propagating over a permeable circular island.The velocity potential is finally expressed as

2.20
Considering an incident wave expressed in a nondimensional form: where A I is the reference wave height defined as A I φ 0 ρ 0 .In addition, the wavenumber k 0 for r > r 0 can be obtained from the dispersion relation 2.2 , and the dimensionless wave number κ 0 is defined as k 0 g/ω 2 .In addition, ε n is the Jacobian parameter given by

2.22
The physical problem is presented in Figure 1.Following Yu and Zhang 18 , the method of eigenexpansion is adopted to solve the problem.The study domain is divided into two subregions on the horizontal plane: the finite region with varying depth r 1 < r < r 0 and semi-infinite far region with a porous constant depth r > r 0 .The general solution of the complex velocity potential subject to the Somerfield condition as well as the symmetric condition is written in the form 23 where H n is the Hankel function of the first kind of the nth order, D n are complex coefficients to be determined.
In the finite region, the permeable bottom configuration is represented by 2.3 .The complex velocity potential is given by

2.25
Because the present investigation assumes the symmetric condition, the corresponding terms of the odd function sin nθ are negligible.
The following boundary conditions are used for solving the problem:

2.26
Using 2.25 , the relation between two coefficients C 3n and C 4n is given by

2.27
The prime denotes the first derivative with respect to ρ.The rearrangement of 2.25 yields

2.29
Substituting 2.29 into 2.23 and 2.25 , we have where

2.31
The coefficients C n and D n are solved by 2.30 and are expressed by 2.32

Results and Discussions
In the following numerical results, we have typically set ω 2 /g 1, the inertial coefficient 0.02.The infinite series in the solution of the velocity potential should be truncated, indicating that the number of N must be large enough to represent the infinity so that the infinite series in 2.23 and 2.28 can effectively accounts for the wave motion by a prescribed accuracy.For the case of α 1 and α 2, respectively.Figure 2 presents the variation of the relative wave runup height A/A I around the crosssection along the y 0. The result demonstrates the wave height distribution at different values of α for different permeable materials with variant porosities.In Figure 2 a , the wave height distribution before the island increases monotonically with the increase of n 0 , but the variation of wave height distribution behind the island decreases monotonically with the increasing n 0 for α 1, r 0 0.2π.Note that the variation of wave height distribution at α 2 shows the same feature shown in Figure 2 b .
Figure 3 demonstrates the variation of wave height distribution at different values of α as a linearized friction coefficient changes.We notice that the wave height increases with the increase of the friction factor f p in the front of the island but opposite property behind the island.
Figures 4 and 5 show different bottom slopes for different values of α, ρ 0 , ξ 0 and ρ 1 .It is interesting to note that a higher value could produce a steeper bottom geometry,  the intensity of wave ray focusing increases with the decrease of the bottom slope.Consequently, a significant increase of wave runup height is evident.

Conclusions
An analytic solution of the long-wave equation for waves propagating over a porous circular island was derived.The solution is in terms of the Bessel functions.The singularity was removed by the Frobenius series for the case of a parabolic island.The solution is able to account for the combined effect of refraction and diffraction around a circular island on the   top of a shoal for arbitrary geometry described by two independent parameters.The effect of key parameters on waves scattering was investigated based on numerical calculations from the analytic solution.

Appendix
According to Rojanakamthorn et al.
4 is a pioneer to obtain the solution of long-wave equation LWE by considering long waves propagating over the top of a parabolic shoal.The solution was further extended and applied by Vastano and Reid 5 and Jonsson et al. 6 .Different geometries of circular or conical island were explored by Fujima and Goto 7 , Fujima et al. 8 , Zhang and Zhu 9 , and Zhu and Zhang 10 .The research field is still under rapid developments.Details can be referred to recent articles of Jung et al. 11 , Niu and Yu 12 , Liu and Xie 13 , and Jung and Lee 14 .Furthermore, there are also studies, such as Liu et al. 15 and Niu and Yu 16 , using Hunt's 17 approximate solution of the implicit dispersion equation to solve the MSE without the long-wave approximation.
≤ 1/20 and h p /L ≤ 1/20 where L is the wave length.More details can be found in the Appendix.
− 1 sinh k h w h p z tanh kh p cosh k h w h p z 1 − δ tanh 2 kh p δ − 1 sinh k h w h p tanh kh p cosh k h w h p 1 − δ tanh 2 kh p P z, h w x, y , h p x, y δ cosh k h w h p z cosh 2 kh p δ − 1 sinh k h w h p tanh kh p cosh k h w h p 1 − δ tanh 2 kh p A.3 being the depth functions in the water and porous layers respectively.The angular frequency ω is related to the wavenumber k by the following dispersion relation: In order to consider the long wave limit, A.3 should be expanded in polynomials of kh w and kh p as W z, h w x, y , h p x, y δ − 1 kh p sinh kz cosh kz k h w h p sinh kz O k 2 h 2 w , k 2 h 2 p , k 2 h w h p , A.5 P z, h w x, y , h p x, y δ 2 cosh kz k h w h p sinh kz O k 2 h 2 w , k 2 h 2 p , k 2 h w h p .Equations A.1 and A.7 give the long wave equation 2.1 of the present study.In addition, the dispersion relation A.4 can also be expanded in polynomials of kh w and kh p as ω 2 g k δkh p kh w O k 2 h 2 w , k 2 h 2 p , k 2 h w h p .A.8 Equation A.8 is sufficient to obtain the dispersion relation of long waves as given in 2.2 .
w − h p − δ 1 sinh k h w h p δ − 1 cosh k h w − h p − δ 1 cosh h w h p .A.4