Engineering Reports
Volume 7, Issue 4 e70041
RESEARCH ARTICLE
Open Access

An Exact Neumann-to-Dirichlet (NtD) Boundary Condition for Three-Dimensional Wave Motion Over Piecewise Smooth Topographies Near a Vertical Breakwater

Un-Ryong Rim, 

Corresponding Author

Un-Ryong Rim

Institute of Ocean Engineering, Kim Chaek University of Technology, Pyongyang, Democratic People's Republic of Korea

Correspondence: Un-Ryong Rim ([email protected])

Contribution: Supervision

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Jun-Bom Ko, 

Jun-Bom Ko

Department of Marine Management, Rajin University of Marine Transport, Chongjin, Democratic People's Republic of Korea

Contribution: Writing - review & editing

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Chol-Jun Han, 

Chol-Jun Han

Institute of Robotic Engineering, Kim Chaek University of Technology, Pyongyang, Democratic People's Republic of Korea

Contribution: Methodology

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Fuad-Mahfud Assidiq, 

Fuad-Mahfud Assidiq

Department of Ocean Engineering, Universitas Hasanuddin, Makassar, Indonesia

Contribution: ​Investigation

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First published: 02 April 2025
https://doi.org/10.1002/eng2.70041

ABSTRACT

This paper suggests an exact boundary condition, so-called Neumann-to-Dirichlet (NtD) boundary condition for numerical analysis of the three-dimensional wave motion over piecewise smooth topographies near a vertical breakwater. The semi-infinite water domain in front of the breakwater is converted into a horizontally unlimited water domain by use of the mirror-image method by which the vertical breakwater is neglected and an image of the original undulated seabed is generated. An exact NtD map is derived on a virtual cylindrical surface by which the entire water region is separated into an exterior subregion and an interior subregion involving the original undulated topographies and their imaginary topographies. Based on comparison with the DtN approach from the viewpoint of accuracy, the current model is escalated to consider the effects of heading angle of incident wave and submergence of the shoal at different distance from the breakwater.

1 Introduction

Coastal structures such as submerged shoals ([1, 2]) or submerged breakwaters ([3, 4]) are of remarkable interests in the fields of biological oceanography and coastal engineering since they are employed for fisheries and coastal protection. In those cases, it is frequently required to estimate the wave motion over piecewise smooth topographies.

Many analytical studies for hydrodynamics over undulated topographies or submerged structures have been suggested. Athanassoulis and Belibassakis [5] suggested a coupled-mode model for linear water waves over undulated seabed, and they escalated the model to the cases of three-dimensional linear/nonlinear hydrodynamic analysis of floating [6-9]. Bender and Dean [10] and Behera et al. [11] focused on the eigenfunction expansion matching method in which the entire fluid domain is divided into several subregions and a solution of velocity potential in individual subregion is expressed as a series analytically by use of the method of variable separation, where the unknowns in the series type of solution are obtained by a matching condition for continuity of pressure and velocity on an interface between neighboring subregions. Zheng et al. [12] obtained an analytical solution for hydrodynamic interaction between water wave and a submerged bar in front of the breakwater. Koley et al. [13] studied the wave trapping by a porous bar submerged near a vertical breakwater and asseverated that the ocean wave energy can vanish at a specific placement between the breakwater and the bar. Yang et al. [14] applied the eigenfunction expansion matching method to obtain an analytical solution of two-dimensional interaction between a linear wave and a porous bar which is submerged and placed near a partially reflecting wall. The whole water region is separated into four subregions; first subregion toward the sea from the porous bar, second subregion above the porous bar, third subregion occupied by outline of the porous bar, and last subregion between the porous bar and the partially reflecting wall. Each analytical solution on each subregion can be expressed as series composed of eigenfunctions with infinite number of unknown coefficients which are determined by continuity of velocity potentials and their derivatives on the interfaces of adjacent subregions. Later, Yang et al. [15] escalated such a method to the case of an oblique water wave motion over several porous bars submerged near a rigid vertical wall. Meanwhile, Liu et al. [16] and Lopez et al. [17] solved one-dimensional mild-slope equations (MSEs) for linear wave reflection and transmission over piecewise smooth topographies. Liu et al. [18] proposed an analytical solution composed of Taylor series and Fourier series to solve a MSE for simple harmonic water waves incident upon an island over a paraboloidal shoal. Later, Liu et al. [19] suggested another analytical solution composed of Frobenius series in order to solve a modified-type of MSE for a wave field over a circular shoal. Toledo and Agnon [20] focused on nonlinear diffraction and refraction of water waves by use of a frequency-domain model based on a second-order complementary MSE.

Numerical methods and experimental approaches for wave-structure interaction or wave motion over piecewise smooth topographies can be found. Wang and Zou [21] conducted an experiment to consider the effects of beach slope, wave period, and wave height on velocity profile of a longshore current over a barred beach. Liu et al. [22] studied the effects of seabed slope on the slow-drift sway motion of a rectangular barge experimentally and numerically. Finite difference method and boundary element method to solve the MSEs can be found [23-25]. Tsai et al. [26] focused on a numerical study of wave transformation over a porous structure submerged near a breakwater and demonstrated that the wave height was significantly changed by the wall. Reddy and Neelamani [27] simulated the effect of water wave on a vertical seawall protected by a low-crested breakwater experimentally and numerically. Chen et al. [28] studied experimentally the motion of water wave over a porous breakwater submerged near a slope seawall. Jeng et al. [29] observed the wave transformation over a submerged breakwater with porous seabed. The numerical studies mentioned above are confined to the case of a submerged structure mounted on a flat seafloor near a wall in most cases, while there are little numerical studies for the wave motion over piecewise smooth topographies such as undulated seabed or arbitrary-shaped shoals. Also, most of numerical methods related to hydrodynamics over undulated seabed or submerged structures near a wall often extract a finite volume of water subregion from the given infinite water domain, where the finite subregion still surrounds the submerged structures or an undulated seabed. In this case, the formulation of boundary conditions for the extracted subregion has a significant role to obtain numerical results as exactly as possible. To cope with this problems, several boundary conditions were suggested. Oliveira and Anastasiou [30] introduced a reflection coefficient empirically into a shoreward downstream boundary and lateral downstream boundaries to make an efficient numerical model for wave propagation in a coastal region. Silva et al. [31] applied a boundary condition for full transmission of outcoming waves through a virtual boundary as well as a boundary condition for partially reflection to solve the modified time-harmonic MSE. Such open boundary conditions were also applied to another cases [32, 33]. Cheng et al. [34] introduced two damping layers in a nonlinear numerical wave tank to decrease the reflection of waves at an outlet boundary. The boundary conditions mentioned above include some empirical coefficients which affects the accuracy of a given problem.

Eatock Taylor et al. [35] adopted a cylindrical surface to extract a finite volume of water domain from the sea of finite depth, and assumed that the integral for velocity potential on the cylindrical surface is zero when the floating structure and the cylindrical surface are sufficiently far from each other. Hsu and Wu [36] focused on a two-dimensional interaction between linear water wave and a rectangular structure submerged near a vertical sidewall by use of boundary element method with Sommerfeld radiation condition at a seaward location sufficiently far from the body. Higdon [37, 38] suggested another type of absorbing boundary condition by which the plane water waves with their constant speeds can pass through a certain boundary without reflection. Similar studies can also be found in other papers [39-41]. Liao [42] proposed multi-transmitting formulae (MTF) with an artificial speed ca and combined them with a spatial extrapolation to keep the nonreflection on the boundary in case of one-way incident waves with their own different speeds. Later, Chen and Liao [43] applied such an approach to the case of attenuating waves. Shao and Faltinsen [44] adopted a damping region near a virtual cylindrical boundary and suggested a damping formula with damping coefficients which are set empirically, where the damping region is applied to solve water wave interaction with irregularly shaped bodies in an interior fluid subregion. Xu and Duan [45] proposed a MTF on the damping region to simulate a three-dimensional water wave interaction with several floating structures in the time-domain. Chen and Liang [46] studied wavy properties and free-surface flows around a floating body by using a multi-domain method in which the fluid domain is divided into three subdomains, that is, an internal subdomain, a transit one and an external one. The internal subdomain includes the body and considers the fluid viscosity as well as nonlinearity of free-surface flow while the external subdomain ignores the viscosity and nonlinearity so that linear potential theory can be applied. The free-surface Green function is used in the external subdomain to satisfy the free-surface condition, Sommerfeld condition and other boundary conditions, while the Rankine source function 1/r is adopted in the internal subdomain. Liang and Chen [47] applied such a multi-domain method to study linear and second-order mean drift wave loads on floating bodies. Recently, Rim [48-51] and Rim et al. [52-54] proposed an exact DtN (Dirichlet-to-Neumann) artificial boundary condition to study the wave-structure interactions or water wave motion over piecewise smooth topographies. This paper suggests another type of artificial boundary condition, so-called Neumann-to-Dirichlet (NtD) boundary condition to analyze the motion of water wave over piecewise smooth topographies near a vertical breakwater.

2 Formulation of Problem

Consider an incident water wave with amplitude A $$ A $$ , angular velocity ω $$ \omega $$ , water depth h $$ h $$ , and incident angle α $$ \alpha $$ over piecewise smooth topographies near a vertical breakwater as shown in Figure 1a, where the Oxy plane is defined at the location of still water surface, Oz axis is pointing upward, Ox axis is pointing the breakwater and vertical to it, and Oy axis lies on the breakwater surface. The present problem can be considered as a hydrodynamically equivalent problem for water wave motion over the original topographies and their imaginary topographies under the bidirectional incident water waves with angles α $$ \alpha $$ and ( π − α ) $$ \left(\pi -\alpha \right) $$ by use of the mirror-image method as seen in Figure 1b. Assuming that the fluid is incompressible and inviscid and the flow is irrotational, the water wave field can be expressed by time-periodic velocity potential as follows:
ϕ ( x , y , z , t ) = Re Φ ( x , y , z ; ω ) e − i ω t $$ \phi \left(x,y,z,t\right)=\operatorname{Re}\left[\Phi \left(x,y,z;\omega \right){e}^{-\mathrm{i}\omega t}\right] $$ (1)
where i = − 1 $$ i=\sqrt{-1} $$ is an imaginary unit, t $$ t $$ is time and Φ ( x , y , z ; ω ) $$ \Phi \left(x,y,z;\omega \right) $$ denotes an amplitude of ϕ $$ \phi $$ and can be again decomposed into
Φ ( x , y , z ; ω ) = Φ I ( x , y , z ; ω ) + Φ S ( x , y , z ; ω ) $$ \Phi \left(x,y,z;\omega \right)={\Phi}_I\left(x,y,z;\omega \right)+{\Phi}_S\left(x,y,z;\omega \right) $$ (2)
where Φ I $$ {\Phi}_I $$ and Φ S $$ {\Phi}_S $$ are the amplitudes of velocity potentials for the incident water wave and the scattered wave due to the piecewise topographies, respectively.
Details are in the caption following the image
FIGURE 1
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(a) Definition sketch and (b) imaginary system.
The velocity potential Φ I $$ {\Phi}_I $$ by the incident water wave is well-known as [55]
Φ I ( x , y , z ; ω ) = − i gA ω cosh k 0 ( z + h ) cosh k 0 h e i k 0 y sin α e i k 0 x cos α + e − i k 0 x cos α $$ {\Phi}_I\left(x,y,z;\omega \right)=-\frac{\mathrm{i} gA}{\omega}\frac{\cosh {k}_0\left(z+h\right)}{\cosh {k}_0h}{e}^{\mathrm{i}{k}_0y\sin \alpha}\left({e}^{\mathrm{i}{k}_0x\cos \alpha }+{e}^{-\mathrm{i}{k}_0x\cos \alpha}\right) $$ (3)
where g $$ g $$ denotes the magnitude of gravitational acceleration, k0 is a wave number of progressive mode satisfying the dispersion relation ω 2 = gk 0 tanh k 0 h $$ {\omega}^2={gk}_0\tanh {k}_0h $$ .
The potential Φ S $$ {\Phi}_S $$ for the scattered water wave is specified by a governing equation and boundary conditions as follows:
∂ 2 Φ S ∂ x 2 + ∂ 2 Φ S ∂ y 2 + ∂ 2 Φ S ∂ z 2 = 0 everywhere in the water domain $$ \frac{\partial^2{\Phi}_S}{\partial {x}^2}+\frac{\partial^2{\Phi}_S}{\partial {y}^2}+\frac{\partial^2{\Phi}_S}{\partial {z}^2}=0\ \mathrm{everywhere}\ \mathrm{in}\ \mathrm{the}\ \mathrm{water}\ \mathrm{domain} $$ (4)
∂ Φ S ∂ z − ω 2 g Φ S = 0 on still water surface z = 0 $$ \frac{\partial {\Phi}_S}{\partial z}-\frac{\omega^2}{g}{\Phi}_S=0\ \mathrm{on}\ \mathrm{still}\ \mathrm{water}\ \mathrm{surface}\ z=0 $$ (5)
∂ Φ S ∂ n = − ∂ Φ I ∂ n at undulated seafloor $$ \frac{\partial {\Phi}_S}{\partial \boldsymbol{n}}=-\frac{\partial {\Phi}_I}{\partial \boldsymbol{n}}\ \mathrm{at}\ \mathrm{undulated}\ \mathrm{seafloor} $$ (6)
r ∂ Φ S ∂ r − i k 0 Φ S = 0 at r → ∞ $$ \sqrt{r}\left(\frac{\partial {\Phi}_S}{\partial r}-\mathrm{i}{k}_0{\Phi}_S\right)=0\ \mathrm{at}\ r\to \infty $$ (7)
where r $$ r $$ means a horizontal distance from the origin point to the considered point in the water domain and n = n x , n y , n z $$ n=\left({n}_x,{n}_y,{n}_z\right) $$ denotes a unit vector which is directed into the water and normal to the undulated seafloor.
To focus on a wave motion in a confined water subdomain with finite volume and solve it numerically, a virtual circular cylinder of radius R is adopted as seen in Figure 1b (see the rose-pink part of online figure), where the radius R has to be chosen so that the virtual cylinder can include the undulated seafloor region. Then the water domain is separated into an interior subregion and an exterior subregion, where the interior subregion Ω i $$ {\varOmega}_i $$ is enclosed by the still water surface Γ F $$ {\Gamma}_F $$ , the undulated seafloor Γ D $$ {\Gamma}_D $$ , and the virtual cylindrical surface Γ R $$ {\Gamma}_R $$ as seen in Figure 1b while the exterior subregion Ω e $$ {\varOmega}_e $$ is the other rest part unbounded horizontally. The boundary conditions on Γ F $$ {\Gamma}_F $$ and Γ D $$ {\Gamma}_D $$ are already described in Equations (5) and (6), therefore, it is needed to set a boundary condition only on Γ R $$ {\Gamma}_R $$ to solve the wave motion in the interior subregion, and it can be set as Φ S = Λ ∂ Φ S ∂ r $$ {\Phi}_S=\Lambda \frac{{\mathrm{\partial \Phi}}_S}{\partial r} $$ with an operator Λ $$ \Lambda $$ which is called a NtD map because it maps the Neumann-type data into a Dirichlet-type value on on Γ R $$ {\Gamma}_R $$ . The derivation of the NtD operator Λ $$ \Lambda $$ is described in the following section. Then the problem for the interior subregion is again formulated as follows:
∂ 2 Φ S ∂ x 2 + ∂ 2 Φ S ∂ y 2 + ∂ 2 Φ S ∂ z 2 = 0 in Ω i $$ \frac{\partial^2{\Phi}_S}{\partial {x}^2}+\frac{\partial^2{\Phi}_S}{\partial {y}^2}+\frac{\partial^2{\Phi}_S}{\partial {z}^2}=0\ \mathrm{in}\ {\varOmega}_i $$ (8)
∂ Φ S ∂ z − ω 2 g Φ S = 0 on Γ F $$ \frac{\partial {\Phi}_S}{\partial z}-\frac{\omega^2}{g}{\Phi}_S=0\ \mathrm{on}\ {\Gamma}_F $$ (9)
∂ Φ S ∂ n = − ∂ Φ I ∂ n on Γ D $$ \frac{\partial {\Phi}_S}{\partial \boldsymbol{n}}=-\frac{\partial {\Phi}_I}{\partial \boldsymbol{n}}\ \mathrm{on}\ {\Gamma}_D $$ (10)
Φ S = Λ ∂ Φ S ∂ r on Γ R $$ {\Phi}_S=\Lambda \frac{{\mathrm{\partial \Phi}}_S}{\partial r}\ \mathrm{on}\ {\Gamma}_R $$ (11)

3 NtD Boundary Condition

The NtD operator on Γ R $$ {\Gamma}_R $$ can be obtained based on an analytical solution for the exterior subregion in which the velocity potential is specified by a governing equation and boundary conditions as follows:
∂ 2 Φ S ∂ x 2 + ∂ 2 Φ S ∂ y 2 + ∂ 2 Φ S ∂ z 2 = 0 in Ω e $$ \frac{\partial^2{\Phi}_S}{\partial {x}^2}+\frac{\partial^2{\Phi}_S}{\partial {y}^2}+\frac{\partial^2{\Phi}_S}{\partial {z}^2}=0\ \mathrm{in}\ {\varOmega}_e $$ (12)
∂ Φ S ∂ z − ω 2 g Φ S = 0 on z = 0 and r ≥ R $$ \frac{\partial {\Phi}_S}{\partial z}-\frac{\omega^2}{g}{\Phi}_S=0\ \mathrm{on}\ z=0\ \mathrm{and}\ r\ge R $$ (13)
∂ Φ S ∂ z = 0 on z = − h and r ≥ R $$ \frac{\partial {\Phi}_S}{\partial z}=0\ \mathrm{on}\ z=-h\ \mathrm{and}\ r\ge R $$ (14)
r ∂ Φ S ∂ r − i k 0 Φ S = 0 at r → ∞ $$ \sqrt{r}\left(\frac{\partial {\Phi}_S}{\partial r}-\mathrm{i}{k}_0{\Phi}_S\right)=0\ \mathrm{at}\ r\to \infty $$ (15)
∂ Φ S ∂ r = ∂ Φ S ( R , θ , z ) ∂ r on Γ R $$ \frac{{\mathrm{\partial \Phi}}_S}{\partial r}=\frac{{\mathrm{\partial \Phi}}_S\left(R,\theta, z\right)}{\partial r}\ \mathrm{on}\ {\Gamma}_R $$ (16)
where ( R , θ , z ) $$ \left(R,\theta, z\right) $$ in Equation (16) denotes the circular cylindrical coordinates of Γ R $$ {\Gamma}_R $$ . Equation (16) is called a Neumann boundary condition, that is, it is assumed that the normal derivative of spatial potential on Γ R $$ {\Gamma}_R $$ is already given as a specifc function Ξ ( R , θ , z ) $$ \varXi \left(R,\theta, z\right) $$ or ∂ Φ S ( R , θ , z ) ∂ r $$ \frac{{\mathrm{\partial \Phi}}_S\left(R,\theta, z\right)}{\partial r} $$ intelligibly.
A physically admissible solution satisfying Equations (12-15) is expressed as
Φ S ( r , θ , z ) = ∑ m = 0 ∞ a m 0 H m k 0 r cosh k 0 ( z + h ) + ∑ n = 1 ∞ a mn K m k n r cos k n ( z + h ) cos m θ + b m 0 H m k 0 r cosh k 0 ( z + h ) + ∑ n = 1 ∞ b mn K m k n r cos k n ( z + h ) sin m θ $$ {\displaystyle \begin{array}{cc}& {\Phi}_S\left(r,\theta, z\right)=\sum \limits_{m=0}^{\infty}\left\{\left[{a}_{m0}{H}_m\left({k}_0r\right)\cosh {k}_0\left(z+h\right)+\sum \limits_{n=1}^{\infty }{a}_{mn}{K}_m\left({k}_nr\right)\right.\right.\\ {}& \kern1em \left.\cos {k}_n\left(z+h\right)\right]\cos m\theta +\left[{b}_{m0}{H}_m\left({k}_0r\right)\cosh {k}_0\left(z+h\right)\right.\\ {}& \kern1em \left.\left.+\sum \limits_{n=1}^{\infty }{b}_{mn}{K}_m\left({k}_nr\right)\cos {k}_n\left(z+h\right)\right]\sin m\theta \right\}\end{array}} $$ (17)
and its radial derivative is expressed as
∂ Φ S ( r , θ , z ) ∂ r = ∑ m = 0 ∞ a m 0 k 0 H m ′ k 0 r cosh k 0 ( z + h ) + ∑ n = 1 ∞ a mn k n K m ′ k n r cos k n ( z + h ) cos m θ + b m 0 k 0 H m ′ k 0 r cosh k 0 ( z + h ) + ∑ n = 1 ∞ b mn k n K m ′ k n r cos k n ( z + h ) sin m θ $$ {\displaystyle \begin{array}{cc}& \frac{{\mathrm{\partial \Phi}}_S\left(r,\theta, z\right)}{\partial r}=\sum \limits_{m=0}^{\infty}\left\{\left[{a}_{m0}{k}_0{H}_m^{\prime}\left({k}_0r\right)\cosh {k}_0\left(z+h\right)\right.\right.\\ {}& \kern1em \left.+\sum \limits_{n=1}^{\infty }{a}_{mn}{k}_n{K}_m^{\prime}\left({k}_nr\right)\cos {k}_n\left(z+h\right)\right]\cos m\theta \\ {}& \kern1em \left.+\left[{b}_{m0}{k}_0{H}_m^{\prime}\left({k}_0r\right)\cosh {k}_0\left(z+h\right)+\sum \limits_{n=1}^{\infty }{b}_{mn}{k}_n{K}_m^{\prime}\left({k}_nr\right)\cos {k}_n\left(z+h\right)\right]\sin m\theta \right\}\end{array}} $$ (18)
where K m ( ⋅ ) $$ {K}_m\left(\cdot \right) $$ is a modified-type of second-kind Bessel function with order m (m = 0, 1, 2, …), H m ( ⋅ ) $$ {H}_m\left(\cdot \right) $$ is a first-kind Hankel function with order m, the superscript (   ′ $$ {}^{\prime } $$ ) denotes a derivative of function with r $$ r $$ , k n $$ {k}_n $$ (n > 0) is an eigenvalue for order n (n > 0) calculated by ω 2 = − g k n tan k n h $$ {\omega}^2=-g{k}_n\tan {k}_nh $$ , and the unknown coefficients a mn $$ {a}_{mn} $$ and b mn $$ {b}_{mn} $$ are determined in the following paragraph.
Equation (18) is written in the form of a Fourier series with θ $$ \theta $$ in which the Fourier coefficients are again expressed in the form of a series composed of orthonormal functions with z in an interval [ − h , 0 ] $$ \left[-h,0\right] $$ as follows:
Z k n ( z ) = N k 0 − 0.5 cosh k 0 ( z + h ) , n = 0 N k n − 0.5 cos k n ( z + h ) , n ≠ 0 $$ {Z}_{k_n}(z)=\left\{\begin{array}{l}{N}_{k_0}^{-0.5}\cosh {k}_0\left(z+h\right),n=0\\ {}{N}_{k_n}^{-0.5}\cos {k}_n\left(z+h\right),n\ne 0\end{array}\right. $$ (19)
where
N k n = h 2 1 + sinh 2 k 0 h 2 k 0 h , n = 0 h 2 1 + sin 2 k n h 2 k n h , n ≠ 0 $$ {N}_{k_n}=\left\{\begin{array}{l}\frac{h}{2}\left(1+\frac{\sinh 2{k}_0h}{2{k}_0h}\right),n=0\\ {}\frac{h}{2}\left(1+\frac{\sin 2{k}_nh}{2{k}_nh}\right),n\ne 0\end{array}\right. $$ (20)
Substituting r = R $$ r=R $$ into Equation (18) and from the Neumann boundary condition Equation (16) on r = R $$ r=R $$ , the unknown coefficients a mn $$ {a}_{mn} $$ and b mn $$ {b}_{mn} $$ can be expressed by use of Equation (19) as follows:
a mn = N k n − 0.5 π ε m k n F m ′ k n R ∫ − h 0 ∫ 0 2 π ∂ Φ S R , θ ′ , z ′ ∂ r Z k n z ′ cos m θ ′ d θ ′ d z ′ $$ {a}_{mn}=\frac{N_{k_n}^{-0.5}}{{\pi \varepsilon}_m{k}_n{F}_m^{\prime}\left({k}_nR\right)}\int_{-h}^0\int_0^{2\pi}\frac{{\mathrm{\partial \Phi}}_S\left(R,{\theta}^{\prime },{z}^{\prime}\right)}{\partial r}{Z}_{k_n}\left({z}^{\prime}\right)\cos m{\theta}^{\prime }d{\theta}^{\prime }d{z}^{\prime } $$ (21)
b mn = N k n − 0.5 π ε m k n F m ′ k n R ∫ − h 0 ∫ 0 2 π ∂ Φ S R , θ ′ , z ′ ∂ r Z k n z ′ sin m θ ′ d θ ′ d z ′ $$ {b}_{mn}=\frac{N_{k_n}^{-0.5}}{{\pi \varepsilon}_m{k}_n{F}_m^{\prime}\left({k}_nR\right)}\int_{-h}^0\int_0^{2\pi}\frac{{\mathrm{\partial \Phi}}_S\left(R,{\theta}^{\prime },{z}^{\prime}\right)}{\partial r}{Z}_{k_n}\left({z}^{\prime}\right)\sin m{\theta}^{\prime }d{\theta}^{\prime }d{z}^{\prime } $$ (22)
where
ε m = 2 , m = 0 1 , m ≠ 0 $$ {\varepsilon}_m=\left\{\begin{array}{l}2,m=0\\ {}1,m\ne 0\end{array}\right. $$ (23)
F m k n r = H m k 0 r , n = 0 K m k n r , n ≠ 0 $$ {F}_m\left({k}_nr\right)=\left\{\begin{array}{l}{H}_m\left({k}_0r\right),n=0\\ {}{K}_m\left({k}_nr\right),n\ne 0\end{array}\right. $$ (24)

and F m ′ ( ⋅ ) $$ {F}_m^{\prime}\left(\cdot \right) $$ denotes a derivative of F m ( ⋅ ) $$ {F}_m\left(\cdot \right) $$ while θ ′ $$ {\theta}^{\prime } $$ and z ′ $$ {z}^{\prime } $$ denote another independent variables distinguished from θ $$ \theta $$ and z $$ z $$ .

Therefore, the solution for the exterior subregion (12-16) is finally obtained from Equations (17), (21), and (22) as follows:
Φ S ( r , θ , z ) = ∑ m = 0 ∞ ∑ n = 0 ∞ ∫ − h 0 ∫ 0 2 π F m k n r π ε m k n F m ′ k n R Z k n ( z ) Z k n z ′ cos m θ − θ ′ ∂ Φ S R , θ ′ , z ′ ∂ r d θ ′ d z ′ $$ {\displaystyle \begin{array}{cc}& {\Phi}_S\left(r,\theta, z\right)=\sum \limits_{m=0}^{\infty}\sum \limits_{n=0}^{\infty}\int_{-h}^0\int_0^{2\pi}\frac{F_m\left({k}_nr\right)}{{\pi \varepsilon}_m{k}_n{F}_m^{\prime}\left({k}_nR\right)}{Z}_{k_n}(z){Z}_{k_n}\left({z}^{\prime}\right)\\ {}& \kern1em \cos m\left(\theta -{\theta}^{\prime}\right)\frac{{\mathrm{\partial \Phi}}_S\left(R,{\theta}^{\prime },{z}^{\prime}\right)}{\partial r}d{\theta}^{\prime }d{z}^{\prime}\end{array}} $$ (25)
Substitution of r = R $$ r=R $$ into Equation (25) will result in a NtD artificial boundary condition on Γ R $$ {\Gamma}_R $$ as follows:
Φ S ( R , θ , z ) = Λ ∂ Φ S ( R , θ , z ) ∂ r = ∑ m = 0 ∞ ∑ n = 0 ∞ ∫ − h 0 ∫ 0 2 π λ mn z , z ′ , θ − θ ′ ∂ Φ S R , θ ′ , z ′ ∂ r d θ ′ d z ′ $$ {\displaystyle \begin{array}{cc}& {\Phi}_S\left(R,\theta, z\right)=\Lambda \frac{\partial {\Phi}_S\left(R,\theta, z\right)}{\partial r}=\sum \limits_{m=0}^{\infty}\sum \limits_{n=0}^{\infty}\int_{-h}^0\int_0^{2\pi }{\lambda}_{mn}\left(z,{z}^{\prime },\theta -{\theta}^{\prime}\right)\\ {}& \kern1em \frac{{\mathrm{\partial \Phi}}_S\left(R,{\theta}^{\prime },{z}^{\prime}\right)}{\partial r}d{\theta}^{\prime }d{z}^{\prime}\end{array}} $$ (26)
where λ mn z , z ′ , θ − θ ′ $$ {\lambda}_{mn}\left(z,{z}^{\prime },\theta -{\theta}^{\prime}\right) $$ is a NtD kernel as follows:
λ mn z , z ′ , θ − θ ′ = F m k n R π ε m k n F m ′ k n R Z k n ( z ) Z k n z ′ cos m θ − θ ′ $$ {\lambda}_{mn}\left(z,{z}^{\prime },\theta -{\theta}^{\prime}\right)=\frac{F_m\left({k}_nR\right)}{{\pi \varepsilon}_m{k}_n{F}_m^{\prime}\left({k}_nR\right)}{Z}_{k_n}(z){Z}_{k_n}\left({z}^{\prime}\right)\cos m\left(\theta -{\theta}^{\prime}\right) $$ (27)

4 Results and Discussion

4.1 Implementation With NtD Boundary Condition

From Green's theorem or Green's second identity, the following integral equation for potential Φ S $$ {\Phi}_S $$ is constructed:
2 π Φ S ( P ) = ∫ Γ Φ S ( Q ) ∂ G ( P , Q ) ∂ n Q − ∂ Φ S ( Q ) ∂ n Q G ( P , Q ) d Γ Q $$ 2\pi {\Phi}_S(P)=\underset{\Gamma}{\int}\left[{\Phi}_S(Q)\frac{\partial G\left(P,Q\right)}{\partial {\boldsymbol{n}}_Q}-\frac{\partial {\Phi}_S(Q)}{\partial {\boldsymbol{n}}_Q}G\Big(P,Q\Big)\right]d{\Gamma}_Q $$ (28)
where Γ $$ \Gamma $$ denotes an entire boundary surface of Ω i $$ {\varOmega}_i $$ (i.e., Γ = Γ F + Γ D + Γ R $$ \Gamma ={\Gamma}_F+{\Gamma}_D+{\Gamma}_R $$ ), P $$ P $$ and Q $$ Q $$ are field point and source point on Γ $$ \Gamma $$ , respectively, n Q $$ {\boldsymbol{n}}_Q $$ means a unit normal vector which is starting from a point Q $$ Q $$ and normal to the surface Γ $$ \Gamma $$ , and G $$ G $$ denotes a Rankine-source type of Green's function described by
G ( P , Q ) = 1 r = 1 | P − Q | $$ G\left(P,Q\right)=\frac{1}{r}=\frac{1}{\mid P-Q\mid } $$ (29)
From the boundary conditions (9-11) and (26), Equation (28) can be rearranged as
2 π Φ S ( P ) − D Γ F + Γ D Φ S ( P ) − ω 2 g S Γ F Φ S ( P ) − S Γ R ∂ Φ S ∂ r ( P ) − ∑ m = 0 ∞ ∑ n = 0 ∞ ∫ − h 0 ∫ 0 2 π λ mn z , z ′ , θ − θ ′ D Γ R ∂ Φ S ∂ r ( P ) d θ ′ d z ′ = S Γ D ∂ Φ I ∂ n ( P ) $$ {\displaystyle \begin{array}{cc}& 2\pi {\Phi}_S(P)-\left({D}_{\Gamma_F+{\Gamma}_D}{\Phi}_S\right)(P)-\frac{\omega^2}{g}\left[\left({S}_{\Gamma_F}{\Phi}_S\right)(P)\right]-\left({S}_{\Gamma_R}\frac{{\mathrm{\partial \Phi}}_S}{\partial r}\right)(P)\\ {}& \kern1em -\sum \limits_{m=0}^{\infty}\sum \limits_{n=0}^{\infty}\int_{-h}^0\int_0^{2\pi }{\lambda}_{mn}\left(z,{z}^{\prime },\theta -{\theta}^{\prime}\right)\left[\left({D}_{\Gamma_R}\frac{{\mathrm{\partial \Phi}}_S}{\partial r}\right)(P)\right]d{\theta}^{\prime }d{z}^{\prime }=\left({S}_{\Gamma_D}\frac{\partial {\Phi}_I}{\partial \boldsymbol{n}}\right)(P)\end{array}} $$ (30)
where S δ ψ $$ {S}_{\delta}\psi $$ and D δ ψ $$ {D}_{\delta}\psi $$ denote the functionals or operators of a spatial function ψ $$ \psi $$ defined on arbitrary boundary surface δ $$ \delta $$ as follows:
S δ ψ ( P ) = ∫ δ G ( P , Q ) ψ ( Q ) d δ Q $$ \left({S}_{\delta}\psi \right)(P)=\underset{\delta }{\int }G\left(P,Q\right)\psi (Q)d{\delta}_Q $$ (31)
D δ ψ ( P ) = ∫ δ ∂ G ( P , Q ) ∂ n Q ψ ( Q ) d δ Q $$ \left({D}_{\delta}\psi \right)(P)=\underset{\delta }{\int}\frac{\partial G\left(P,Q\right)}{\partial {\boldsymbol{n}}_Q}\psi (Q)d{\delta}_Q $$ (32)

Since Φ I $$ {\Phi}_I $$ in the right side of Equation (30) is given already in Equation (3), therefore, the unkowns in the left side of Equation (30), that is, Φ S $$ {\Phi}_S $$ on Γ D + Γ F $$ {\Gamma}_D+{\Gamma}_F $$ and ∂ Φ S ∂ r $$ \frac{{\mathrm{\partial \Phi}}_S}{\partial r} $$ on Γ R $$ {\Gamma}_R $$ can be solved by dividing the entire boundary Γ $$ \Gamma $$ into finite panels and by reducing the series in Equation (30) to finite terms.

Once the unknown potential Φ S $$ {\Phi}_S $$ on Γ F $$ {\Gamma}_F $$ is determined from Equation (30), the time-periodic wave elevation from SWL (still water level) can be expressed by
η time ( x , y , ω , t ) = Re η ^ ( x , y , ω ) e − i ω t $$ {\eta}_{time}\left(x,y,\omega, t\right)=\operatorname{Re}\left[\hat{\eta}\left(x,y,\omega \right){e}^{-\mathrm{i}\omega t}\right] $$ (33)
where the complex amplitude η ^ ( x , y , ω ) $$ \hat{\eta}\left(x,y,\omega \right) $$ is calculated as follows:
η ^ ( x , y , ω ) = i ω g Φ ( x , y , 0 ; ω ) $$ \hat{\eta}\left(x,y,\omega \right)=\frac{\mathrm{i}\omega }{g}\Phi \left(x,y,0;\omega \right) $$ (34)

4.2 Model Validation

The present NtD model is compared with the DtN model presented in Rim [49]. Figure 2a shows a geometry of paraboloidal shoal with distance e $$ e $$ from a vertical wall, where r 0 $$ {r}_0 $$ and d $$ d $$ denotes a radius and a submergence of the shoal, respectively, and the test points O $$ O $$ , A $$ A $$ , B $$ B $$ , C $$ C $$ , D $$ D $$ , and E $$ E $$ are placed at specific locations of the shoal. The water depth is given by ζ ( r ) = d + ( h − d ) r 2 / r 0 2 $$ \zeta (r)=d+\left(h-d\right){r}^2/{r}_0^2 $$ over the shoal, where r $$ r $$ denotes a horizontal distance from a central axis of the shoal. Figure 2b shows the bird and plan views of water wave elevation η time ( x , y , ω , t ) $$ {\eta}_{\mathrm{time}}\left(x,y,\omega, t\right) $$ over the shoal with d / h = $$ d/h= $$  0.2, r 0 / h = $$ {r}_0/h= $$  0.8, and e / r 0 = $$ e/{r}_0= $$  1.5 at time t = 2 k π / ω $$ t=2 k\pi /\omega $$ (k = 0, 1, …) under the incident water wave with A / h = $$ A/h= $$  0.1, r 0 / λ = $$ {r}_0/\lambda = $$  0.55, and α = 15 ° $$ \alpha ={15}^{{}^{\circ}} $$ , where λ $$ \lambda $$ denotes a wavelength given by λ = 2 π / k 0 $$ \lambda =2\pi /{k}_0 $$ . Figure 2c shows the result comparison between the precedent DtN model ([49]) and the present NtD model for the amplitudes of wave elevations at specific locations for different r 0 / λ $$ {r}_0/\lambda $$ and the other parameters which are given in Figure 2b, where the wave amplitude is nondimensionalized by η = η ^ ( x , y , ω ) / h $$ \eta =\hat{\eta}\left(x,y,\omega \right)/h $$ . The fluctuations of plots in Figure 2c are almost same at three test points (i.e., point A $$ A $$ , point C $$ C $$ , and point E $$ E $$ ) on a straight line parallel to the wall. As seen in Figure 2c, the relative error of the wave elevation results at five test points O $$ O $$ , A $$ A $$ , B $$ B $$ , C $$ C $$ , D $$ D $$ , and E $$ E $$ caculated by the present NtD model to the results obtained by the precedent DtN model ([49]) is less than 1.5% in the range of 0.02 ≤ r 0 / λ ≤ 0.8 $$ 0.02\le {r}_0/\lambda \le 0.8 $$ , especially, the maximum relative error is estimated to be 0.96% at r 0 / λ = $$ {r}_0/\lambda = $$  0.35 for point C $$ C $$ and 1.5% at r 0 / λ = $$ {r}_0/\lambda = $$  0.685 for point A $$ A $$ . It is noted that the results presented in Figure 2c are obtained for M = 15 $$ M=15 $$ and N = 14 $$ N=14 $$ , where M $$ M $$ and N $$ N $$ denote the numbers of truncated terms in the series for m $$ m $$ and n $$ n $$ in Equation (30), respectively. The relative errors of present NtD results to the precedent DtN model ([49]) with different M $$ M $$ and N $$ N $$ are shown in Table 1. As seen in Table 1, the relative error will be reduced when the truncated number of terms of the series in Equation (30) is increased and the boundary of interior subdomain is meshed more densely. The results in Figure 2c and Table 1 show the validity of the present NtD approach.

Details are in the caption following the image
FIGURE 2
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Result comparison between DtN boundary condition ([49]) and NtD boundary condition (present); (a) geometry of paraboloidal shoal near a wall; (b) three-dimensional field of wave elevation for α = 15 ° $$ \alpha ={15}^{{}^{\circ}} $$ (bird and plan views); and (c) plots of wave elevation for α = 15 ° $$ \alpha ={15}^{{}^{\circ}} $$ at test points.
TABLE 1. Relative error according to the different values of M $$ M $$ and N $$ N $$ .
M $$ M $$ N $$ N $$ Relative error, %
3 2 11.95
5 3 7.12
7 4 4.77
7 6 3.18
9 7 2.37
11 9 1.89
15 14 1.50
18 15 1.31
21 17 1.25

4.3 Effects of Some Parameters

Figure 3 shows an effect of α $$ \alpha $$ on amplitudes of wave elevations at specific locations of the shoal given in Figure 2b,c. The amplitudes of water wave elevations in Figure 3 are approximately between 0 and 2 A $$ 2A $$ by reason of the superposition of two waves, that is, the incident water wave and its mirror image fully reflected by the wall. Furthermore, the fluctuations of the plots in Figure 3b–f are decreasing with an increase of α $$ \alpha $$ .

Details are in the caption following the image
FIGURE 3
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Effects of α $$ \alpha $$ on wave elevations at specific locations of a shoal; (a) at point O $$ O $$ ; (b) at point A $$ A $$ ; (c) at point B $$ B $$ ; (d) at point C $$ C $$ ; (e) at point D $$ D $$ ; and (f) at point E $$ E $$ .

Figure 4 shows an effect of d $$ d $$ on amplitudes of wave elevations at specific locations of the shoal given in Figure 2b,c except that the heading angle is fixed as α = 30 ° $$ \alpha ={30}^{{}^{\circ}} $$ and d / h $$ d/h $$ is differently chosen from 0.15 to 0.9. The plot in Figure 4a is not only less fluctuating but also closer to 2 A $$ 2A $$ with an increase of d / h $$ d/h $$ , because the undulated seabed due to the shoal grows to be a flat seabed with an increase of d / h $$ d/h $$ and therefore the point O $$ O $$ placed on the wall becomes an antinode for any wavelength or r 0 / λ $$ {r}_0/\lambda $$ which can be explained by the principle of superposition in case of d / h $$ d/h $$  = 1. The plots in Figure 4b–f are fluctuating almost between 0 and 2 A $$ 2A $$ in the interval of 0 <  r 0 / λ $$ {r}_0/\lambda $$  < 0.8, which indicates that all the test points except the point O $$ O $$ can be approximately nodes or antinodes for specific values of r 0 / λ $$ {r}_0/\lambda $$ and their valley or peak values can vary according to d / h $$ d/h $$ .

Details are in the caption following the image
FIGURE 4
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Effects of d $$ d $$ on wave elevations at specific locations of a shoal; (a) at point O $$ O $$ ; (b) at point A $$ A $$ ; (c) at point B $$ B $$ ; (d) at point C $$ C $$ ; (e) at point D $$ D $$ ; and (f) at point E $$ E $$ .

Figure 5 shows an effect of e $$ e $$ on amplitudes of wave elevations at specific locations of the shoal given in Figure 2b,c except that the heading angle is fixed as α = 30 ° $$ \alpha ={30}^{{}^{\circ}} $$ and e / r 0 $$ e/{r}_0 $$ is differently chosen from 1.5 to 3.0. All the plots in Figure 5 are fluctuating more and more with an increase of e / r 0 $$ e/{r}_0 $$ . Furthermore, the first peaks for e / r 0 $$ e/{r}_0 $$ = 1.5 and e / r 0 $$ e/{r}_0 $$ = 3.0 in Figure 5b can be seen at about r 0 / λ $$ {r}_0/\lambda $$ = 0.38 and r 0 / λ $$ {r}_0/\lambda $$ = 0.19, respectively, which indicates that the plot for point A $$ A $$ will be twofold fluctuating with a twofold value of e / r 0 $$ e/{r}_0 $$ . Such a feature can also be seen for the other points in Figure 5c–f.

Details are in the caption following the image
FIGURE 5
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Effects of e $$ e $$ on wave elevations at specific locations of a shoal; (a) at point O $$ O $$ ; (b) at point A $$ A $$ ; (c) at point B $$ B $$ ; (d) at point C $$ C $$ ; (e) at point D $$ D $$ ; and (f) at point E $$ E $$ .

5 Conclusions

This paper suggests a NtD boundary condition to analyze the three-dimensional water wave motion over piecewise smooth topographies near a vertical breakwater. The water domain in front of the breakwater is converted into a horizontally unlimited water domain by use of the mirror-image method by which the vertical breakwater is neglected and an image of the original undulated seabed is generated. A virtual circular cylindrical surface which envelopes the original undulated topographies and their imaginary topographies is chosen as an artificial boundary by which the entire water domain is separated into an interior subregion and an exterior subregion. An exact NtD map on the virtual cylindrical surface is derived analytically from a solution for the exterior subregion, which is chosen as a NtD boundary condition to focus on the interior problem. The present NtD model is compared with the DtN approach suggested by Rim [49] in case of a paraboloidal shoal and shows good consistency. The present NtD approach is extended to consider the effects of heading angle of incident wave and submergence of the shoal at different distance from the breakwater. The present paper is applicable to the three-dimensional motion analysis of linear water wave over piecewise topographies with arbitrary shapes near a vertical breakwater, and a nonlinear analysis of water wave will be studied in the next time.

Author Contributions

Un-Ryong Rim: supervision. Jun-Bom Ko: writing – review and editing. Chol-Jun Han: methodology. Fuad-Mahfud Assidiq: investigation.

Acknowledgments

The authors are grateful to the handling editor and reviewers for their helpful comments.

    Conflicts of Interest

    The authors declare no conflicts of interest.

    Data Availability Statement

    Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

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