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

Abstract

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.

1. 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 [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.

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.

2. Theoretical Formulation

The long wave equation for combined refraction and diffraction of waves propagating over porous media is express as ∇_ℎℎ_𝑤+𝛿ℎ_𝑝∇_ℎ𝜙+𝑘²ℎ_𝑤+𝛿ℎ_𝑝𝜙=0, (2.1) 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 𝑛₀/(𝑆+𝐢𝑓_𝑝) with 𝑛₀ the porosity, 𝑆=𝑛₀+(1−𝑛₀)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: 𝜔²𝑔=𝑘𝛿𝑘ℎ_𝑝+𝑘ℎ_𝑤. (2.2) Equations (2.1) and (2.2) can be derived from the mild-slope equation of Rojanakamthorn et al. [19] by neglecting terms of orders 𝑂(𝑘²ℎ²_𝑤,𝑘²ℎ²_𝑝,𝑘²ℎ_𝑤ℎ_𝑝). Note that (2.1) and (2.2) are only valid for long waves, that is, the relative depth is restricted in the shallow water region, that is, ℎ_𝑤/𝐿≤1/20 and ℎ_𝑝/𝐿≤1/20 where 𝐿 is the wave length. More details can be found in the Appendix.

In order to study the wave scattering over a circular porous shoal, we assume that the shoal is symmetric and of the following form: ℎ_𝑤=ℎ_𝑤(𝑟)=ℎ_𝑤𝑟^𝛼₀𝑟^𝛼∝𝑟^𝛼,𝑟≤𝑟₀,ℎ_𝑝=ℎ_𝑝(𝑟)=ℎ_𝑝𝑟^𝛼₀𝑟^𝛼∝𝑟^𝛼,𝑟≤𝑟₀, (2.3) where 𝑟 is the distance of any point on the shoal from the center, 𝑟₁ is the radius of the circular island, 𝑟₀ is the radius of the impermeable circular shoal, ℎ_𝑤 and ℎ_𝑝 are the depths of water and porous layers, respectively, for 𝑟>𝑟₀ and 𝛼 is the power parameter of bottom shape. The definition sketch of the permeable circular island is presented in Figure 1.

Figure 1 

Definition sketch of a circular island.

Using the method of separation variables, the velocity potential is written as 𝜙=^∞_𝑛=0𝑅_𝑛(𝑟)Θ_𝑛(𝜃) (2.4) in which Θ_𝑛 is the eigenfunction given by Θ_𝑛(𝜃)=𝐶_1𝑛cos𝑛𝜃+𝐶_2𝑛sin𝑛𝜃(𝑛=0,1,2,…), (2.5) 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 𝑟²𝑑²𝑅_𝑛𝑑𝑟²𝑟+𝑟1+ℎ𝑑ℎ+𝜔𝑑𝑟²𝑟²𝑔ℎ−𝑛²𝑅_𝑛=0, (2.6) 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: 𝜔𝜌=²𝑟𝑔,𝜉²_𝑤=𝜔²ℎ_𝑤𝑔,𝜉²_𝑝=𝜔²ℎ_𝑝𝑔. (2.8) Substitution of 𝜉²_𝑤 and 𝜉²_𝑝 from (2.8) into (2.3) leads to the resultant equations. 𝜉²_𝑤=𝜉²_𝑤𝜌^𝛼₀𝜌^𝛼,𝜌≤𝜌₀,𝜉²_𝑝=𝜉²_𝑝𝜌^𝛼₀𝜌^𝛼,𝜌≤𝜌₀. (2.9)(2.10) Equation (2.10) is multiplied by 𝛿 and added with (2.9). The following equation is thus obtained, that is 𝜉²=𝜉²_𝑤+𝛿𝜉²_𝑝=𝜉²_𝑤𝜌^𝛼₀𝜌^𝛼+𝛿𝜉²_𝑝𝜌^𝛼₀𝜌^𝛼=𝜉²_𝑤𝜌^𝛼₀+𝛿𝜉²_𝑝𝜌^𝛼₀𝜌^𝛼=1𝛽²𝜌^𝛼. (2.11)

And, (2.6) is rearranged with the help of the dimensionless variable 𝜌=𝜔²𝑟/𝑔 (2.8) and the result reads 𝜌²𝑑²𝑅_𝑛𝑑𝜌²+(𝛼+1)𝜌𝑑𝑅_𝑛+𝛽𝑑𝜌²𝜌^2−𝛼−𝑛²𝑅_𝑛=0. (2.12) For the special case of 𝛼=2, (2.12) is solved by making use of the Frobenius series, the solution is given by 𝑅_𝑛=𝐶_3𝑛𝜌^{−1+𝜇𝑛}+𝐶_4𝑛𝜌^{−1−𝜇𝑛}if1+𝑛²−𝛽²𝐶≠0_3𝑛𝜌⁻¹+𝐶_4𝑛𝜌⁻¹ln𝜌if1+𝑛²−𝛽²=0, (2.13) where 𝐶_3𝑛 and 𝐶_4𝑛 are constants, and 𝜇_𝑛 is represented by 𝜇_𝑛=1+𝑛²−𝛽². (2.14) We introduce two dimensionless parameters 𝑄_𝑛=𝜌^𝛼/2𝑅_𝑛 and 𝜒=(2𝛽/|2−𝛼|)𝜌^1−𝛼/2 for 𝛼≠2, (2.6) is rewritten in the form 𝜒²𝑑²𝑄_𝑛𝑑𝜒²+𝜒𝑑𝑄_𝑛+𝜒𝑑𝜒²−𝜈²_𝑛𝑄_𝑛=0, (2.15) where 𝜈_𝑛=(𝛼²+4𝑛²)/(|2−𝛼|).

The general solution of (2.15) is the Bessel function expressed as 𝑄_𝑛=𝐶_3𝑛𝐽_𝜈𝑛𝜒+𝐶_4𝑛𝑌_𝜈𝑛𝜒(𝑛=0,1,2…), (2.16) where 𝐽_𝜈𝑛 and 𝑌_𝜈𝑛 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 𝑅_𝑛𝜌=𝜌^−𝛼/2𝐶_3𝑛𝐽_𝜈𝑛2𝛽||𝜌2−𝛼^1−(𝛼/2)+𝐶_4𝑛𝑌_𝜈𝑛2𝛽||𝜌2−𝛼^1−(𝛼/2)(𝑛=0,1,2,...). (2.17) For an arbitrary bathymetry of a circular island, the solution is thus written as follows 𝑅_𝑛𝜌=𝐶_3𝑛𝑅_1𝑛𝜌+𝐶_4𝑛𝑅_2𝑛𝜌, (2.18) where 𝑅_1𝑛𝜌=𝜌^{−1+𝜇𝑛}if𝛼=2&1+𝑛²−𝛽²𝜌≠0,₋₁if𝛼=2&1+𝑛²−𝛽²𝜌=0,_−𝛼/2𝐽_𝜈𝑛2𝛽||𝜌2−𝛼^1−(𝛼/2)𝑅if𝛼≠2,_2𝑛𝜌=𝜌^{−1−𝜇𝑛}if𝛼=2&1+𝑛²−𝛽²𝜌≠0,₋₁ln𝜌if𝛼=2&1+𝑛²−𝛽²𝜌=0,_−𝛼/2𝑌_𝜈𝑛2𝛽||𝜌2−𝛼^1−(𝛼/2)if𝛼≠2. (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 𝜙=^∞_𝑛=0𝐶_3𝑛𝑅_1𝑛𝜌+𝐶_4𝑛𝑅_2𝑛𝜌cos𝑛𝜃. (2.20) Considering an incident wave expressed in a nondimensional form: 𝜙₀=𝐴_𝐼𝑒^{𝐢𝑘0𝑥}=𝐴_𝐼^∞_𝑛=0𝐢^𝑛𝜀_𝑛𝐽_𝑛𝑘₀𝑟cos𝑛𝜃=𝐴_𝐼^∞_𝑛=0𝐢^𝑛𝜀_𝑛𝐽_𝑛𝜅₀𝜌cos𝑛𝜃, (2.21) where 𝐴_𝐼 is the reference wave height defined as 𝐴_𝐼=𝜙₀(𝜌=0). In addition, the wavenumber 𝑘₀ for 𝑟>𝑟₀ can be obtained from the dispersion relation (2.2), and the dimensionless wave number 𝜅₀ is defined as 𝑘₀𝑔/𝜔². In addition, 𝜀_𝑛 is the Jacobian parameter given by 𝜀_𝑛=1𝑛=0,2𝑛≥1. (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 (𝑟₁<𝑟<𝑟₀) and semi-infinite far region with a porous constant depth (𝑟>𝑟₀). The general solution of the complex velocity potential subject to the Somerfield condition as well as the symmetric condition is written in the form 𝜙₁=𝜙₀+^∞_𝑛=0𝐷_𝑛Ψ_𝑛𝜌,Ψ_𝑛𝜌≡𝐻_𝑛𝜅₀𝜌, (2.23)(2.24) 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 𝜙₂=^∞_𝑛=0𝐶_3𝑛𝑅_1𝑛𝜌+𝐶_4𝑛𝑅_2𝑛𝜌cos𝑛𝜃. (2.25) Because the present investigation assumes the symmetric condition, the corresponding terms of the odd function sin𝑛𝜃 are negligible.

The following boundary conditions are used for solving the problem: 𝜕𝜙₂𝜕𝜌=0for𝜌=𝜌₁or𝑟=𝑟₁. (2.26) Using (2.25), the relation between two coefficients 𝐶_3𝑛 and 𝐶_4𝑛 is given by 𝐶_𝑛=𝐶_3𝑛𝑅^′_2𝑛𝜌₁=𝐶_4𝑛𝑅^′_1𝑛𝜌₁. (2.27) The prime denotes the first derivative with respect to 𝜌. The rearrangement of (2.25) yields 𝜙₂=^∞_𝑛=0𝐶_𝑛𝑅^′_2𝑛𝜌₁𝑅_1𝑛𝜌+𝑅^′_1𝑛𝜌₁𝑅_2𝑛𝜌cos𝑛𝜃. (2.28) Two matching boundary conditions of continuity of flow mass and pressure are again adopted which are given by 𝜙₁=𝜙₂for𝜌=𝜌₀,𝜕𝜙₁=𝜕𝜌𝜕𝜙₂𝜕𝜌for𝜌=𝜌₀. (2.29) Substituting (2.29) into (2.23) and (2.25), we have 𝐶_𝑛Γ_𝑛=𝐴_𝐼𝐢^𝑛𝜀_𝑛𝐽_𝑛𝜅₀𝜌₀+𝐷_𝑛Ψ_𝑛𝜌₀,𝐶_𝑛Λ_𝑛=𝐴_𝐼𝐢^𝑛𝜀_𝑛𝐽^′_𝑛𝜅₀𝜌₀+𝐷_𝑛Ψ^′_𝑛𝜌₀, (2.30) where Γ_𝑛=𝑅^′_2𝑛𝜌₁𝑅_1𝑛𝜌₀−𝑅^′_1𝑛𝜌₁𝑅_2𝑛𝜌₀,Λ_𝑛=𝑅^′_2𝑛𝜌₁𝑅^′_1𝑛𝜌₀+𝑅^′_1𝑛𝜌₁𝑅^′_2𝑛𝜌₀. (2.31)

The coefficients 𝐶_𝑛 and 𝐷_𝑛 are solved by (2.30) and are expressed by 𝐶_𝑛=−𝐴_𝐼𝐢^𝑛𝜀_𝑛Ψ_𝑛𝜌₀𝐽^′_𝑛𝜅₀𝜌₀−Ψ^′_𝑛𝜌₀𝐽_𝑛𝜅₀𝜌₀Γ_𝑛Ψ^′_𝑛𝜌₀−Λ_𝑛Ψ_𝑛𝜌₀,𝐷_𝑛=−𝐴_𝐼𝑖^𝑛𝜀_𝑛Γ_𝑛𝐽^′_𝑛𝜅₀𝜌₀−Λ_𝑛𝐽_𝑛𝜅₀𝜌₀Γ_𝑛Ψ^′_𝑛𝜌₀−Λ_𝑛Ψ_𝑛𝜌₀. (2.32)

3. Results and Discussions

In the following numerical results, we have typically set 𝜔²/𝑔=1, the inertial coefficient 𝑆=1 and ℎ_𝑝=ℎ_𝑤=𝜉²_𝑝=𝜉²_𝑤=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 cross-section 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 𝑛₀, but the variation of wave height distribution behind the island decreases monotonically with the increasing 𝑛₀ for 𝛼=1, 𝑟₀=0.2𝜋. Note that the variation of wave height distribution at 𝛼=2 shows the same feature shown in Figure 2(b).

(a) 𝛼=1

(b) 𝛼=2

Figure 2 

Effects of the porosity of the permeable material (𝜌0=0.2𝜋, 𝜌1=0.1𝜋, 𝑓𝑝=1).

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.

(a) 𝛼=1

(b) 𝛼=2

Figure 3 

Effects of linearized friction coefficient (𝜌0=0.2𝜋, 𝜌1=0.1𝜋, 𝑛0=0.5).

Figures 4 and 5 show different bottom slopes for different values of 𝛼, 𝜌₀, 𝜉₀ and 𝜌₁. 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.

(a) 𝜌1=0.02𝜋

(b) 𝜌1=0.15𝜋

Figure 4 

Effects of island on wave scattering for 𝛼=1 (𝜌0=0.2𝜋, 𝑓𝑝=1, 𝑛0=0.5).

(a)  𝜌1=0.02𝜋

(b) 𝜌1=0.15𝜋

Figure 5 

Effects of island on wave scattering for 𝛼=2 (𝜌0=0.2𝜋, 𝑓𝑝=1, 𝑛0=0.5).

4. 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. [19], the mild-slope equation for waves propagating over a permeable media is described by ∇_ℎ𝐼∇_ℎ𝜙+𝑘²𝐼𝜙=0, (A.1) where 𝐼=⁰_−ℎ𝑤𝑊²1𝑑𝑧+𝛿^−ℎ𝑤_{−ℎ𝑤+ℎ𝑝}𝑃²𝑑𝑧, (A.2) with 𝑊𝑧,ℎ_𝑤𝑥,𝑦,ℎ_𝑝=()ℎ𝑥,𝑦𝛿−1sinh𝑘_𝑤+ℎ_𝑝+𝑧tanh𝑘ℎ_𝑝+cosh𝑘ℎ_𝑤+ℎ_𝑝+𝑧1−𝛿tanh²𝑘ℎ_𝑝(𝛿−1)sinh𝑘ℎ_𝑤+ℎ_𝑝tanh𝑘ℎ_𝑝+cosh𝑘ℎ_𝑤+ℎ_𝑝1−𝛿tanh²𝑘ℎ_𝑝𝑃𝑧,ℎ_𝑤𝑥,𝑦,ℎ_𝑝=ℎ𝑥,𝑦𝛿cosh𝑘_𝑤+ℎ_𝑝+𝑧cosh²𝑘ℎ_𝑝(𝛿−1)sinh𝑘ℎ_𝑤+ℎ_𝑝tanh𝑘ℎ_𝑝+cosh𝑘ℎ_𝑤+ℎ_𝑝1−𝛿tanh²𝑘ℎ_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: 𝜔²𝑔()ℎ=𝑘𝛿−1sinh𝑘_𝑤−ℎ_𝑝−(𝛿+1)sinh𝑘ℎ_𝑤+ℎ_𝑝(𝛿−1)cosh𝑘ℎ_𝑤−ℎ_𝑝−(𝛿+1)coshℎ_𝑤+ℎ_𝑝. (A.4) In order to consider the long wave limit, (A.3) should be expanded in polynomials of 𝑘ℎ_𝑤 and 𝑘ℎ_𝑝 as 𝑊𝑧,ℎ_𝑤𝑥,𝑦,ℎ_𝑝=()𝑥,𝑦𝛿−1𝑘ℎ_𝑝sinh𝑘𝑧+cosh𝑘𝑧+𝑘ℎ_𝑤+ℎ_𝑝sinh𝑘𝑧+𝑂𝑘²ℎ²_𝑤,𝑘²ℎ²_𝑝,𝑘²ℎ_𝑤ℎ_𝑝,𝑃𝑧,ℎ_𝑤𝑥,𝑦,ℎ_𝑝𝑥,𝑦=𝛿²cosh𝑘𝑧+𝑘ℎ_𝑤+ℎ_𝑝sinh𝑘𝑧+𝑂𝑘²ℎ²_𝑤,𝑘²ℎ²_𝑝,𝑘²ℎ_𝑤ℎ_𝑝. (A.5)(A.6) Substituting (A.4) and (A.5) into (A.2) results in 𝐼=ℎ_𝑤+𝛿ℎ_𝑝+𝑂𝑘²ℎ²_𝑤,𝑘²ℎ²_𝑝,𝑘²ℎ_𝑤ℎ_𝑝. (A.7) 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 𝜔²𝑔=𝑘𝛿𝑘ℎ_𝑝+𝑘ℎ_𝑤+𝑂𝑘²ℎ²_𝑤,𝑘²ℎ²_𝑝,𝑘²ℎ_𝑤ℎ_𝑝. (A.8)Equation (A.8) is sufficient to obtain the dispersion relation of long waves as given in (2.2).

Acknowledgments

This work was supported by the National Science Council of Taiwan, NSC 101-3113-P-006-016 and Headquarters of University Advancement, National Cheng Kung University.