Water wave propagation over an infinite step in the presence of a thin vertical barrier

Abstract

Problems of water wave propagation over an infinite step in the presence of a thin vertical barrier of four different geometrical configurations are investigated in this paper. For each configuration of the barrier, the problem is reduced to solving an integral equation or a coupled integral equation of first kind involving horizontal component of velocity below or above the barrier and above the step. The integral equations are solved employing Galerkin approximation in terms of simple polynomials multiplied by appropriate weight functions whose forms are dictated by the edge conditions at the corner of the step and at the submerged end(s) of the barrier. The reflection and transmission coefficients are then computed and depicted graphically against the wave number.

Appendices

Appendices

1.1 A: Expansion of water wave potentials in different regions

For the case of incidence of waves from the finite depth water region, use of Havelock’s [12] expansion for water wave potential produces

$$\begin{aligned} \phi _{1}(x,y)= & {} \left\{ \begin{array}{lr} \left( \mathrm{e}^{\mathrm{i}k_{0}x}+R_{1}\mathrm{e}^{-\mathrm{i}k_{0}x}\right) I_{0}(y)+\sum _{n=1}^{\infty }A_{n}I_{n}(y)\mathrm{e}^{k_{n}x} ~\quad \text{ for }~x<0,~0<y<h,\\ T_{1}\mathrm{e}^{-Ky+\mathrm{i}Kx}+\int _{0}^{\infty }A(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k\quad \quad \quad \quad ~~~\text{ for }~x>0,~0<y<\infty ; \end{array} \right. \end{aligned}$$

(A.1)

$$\begin{aligned} \phi _{2}(x,y)= & {} \left\{ \begin{array}{lr} \left( \mathrm{e}^{\mathrm{i}k_{0}x}+R_{2}\mathrm{e}^{-\mathrm{i}k_{0}x}\right) I_{0}(y)+\sum _{n=1}^{\infty }B_{n}I_{n}(y)\mathrm{e}^{k_{n}x} ~\text{ for }~x<-b,~0<y<h,\\ \\ \left( C_{0}\cos k_{0}x+D_{0}\sin k_{0}x\right) I_{0}(y)+\sum _{n=1}^{\infty }\left( C_{n}\cosh k_{n}x+D_{n}\sinh k_{n}x\right) I_{n}(y) \\ ~\text{ for }~-b<x<0,~0<y<h,\\ \\ T_{2}\mathrm{e}^{-Ky+\mathrm{i}Kx}+\int _{0}^{\infty }B(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k~~~\quad \quad \quad \text{ for }~x>0,~0<y<\infty ; \end{array} \right. \end{aligned}$$

(A.2)

$$\begin{aligned} \phi _{3}(x,y)= & {} \left\{ \begin{array}{lr} \left( \mathrm{e}^{\mathrm{i}k_{0}x}+R_{3}\mathrm{e}^{-\mathrm{i}k_{0}x}\right) I_{0}(y)+\sum _{n=1}^{\infty }E_{n}I_{n}(y)\mathrm{e}^{k_{n}x} ~\text{ for }~x<0,~0<y<h,\\ \\ \left( F_{0}\mathrm{e}^{\mathrm{i}Kx}x+G_{0}\mathrm{e}^{-\mathrm{i}Kx}\right) \mathrm{e}^{-Ky}+\int _{0}^{\infty }\left( F(k)\mathrm{e}^{-kx}x+G(k)\mathrm{e}^{kx}\right) S(k,y)\mathrm{e}^{-kx}\mathrm{d}k\\ ~\text{ for }~0<x<b,~0<y<\infty ,\\ \\ T_{3}\mathrm{e}^{-Ky+\mathrm{i}Kx}+\int _{0}^{\infty }E(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k~~~\quad \quad \text{ for }~x>b,~0<y<\infty ; \end{array} \right. \end{aligned}$$

(A.3)

$$\begin{aligned} \phi _{4}(x,y)= & {} \left\{ \begin{array}{lr} \left( \mathrm{e}^{\mathrm{i}k_{0}x}+R_{4}\mathrm{e}^{-\mathrm{i}k_{0}x}\right) I_{0}(y)+\sum _{n=1}^{\infty }H_{n}I_{n}(y)\quad ~\text{ for }~x<0,~0<y<h,\\ \\ T_{4}\mathrm{e}^{-Ky+\mathrm{i}Kx}+\int _{0}^{\infty }H(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k~~~\quad \quad \text{ for }~x>0,~0<y<\infty , \end{array} \right. \end{aligned}$$

(A.4)

where $S(k,y)=k\cos ky-K\sin ky$ and $k_{n}(n=1,2,\ldots )$ are the roots of the transcendental equation $k\tan kh+K=0$,

$$\begin{aligned} I_{n}(y)=\frac{\cos k_{n}(h-y)}{\cos k_{n}h},~~~n=1,2,\ldots , \end{aligned}$$

(A.5)

A(k) , B(k) , E(k) , F(k) , G(k) , H(k) are unknown functions and $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, $E_{n}$, $H_{n}~(n=1,2,\ldots )$, $C_{0}$, $D_{0}$, $F_{0}$, $G_{0}$ are unknown constants.

For the case of incidence of waves from the deep water region, use of Havelock’s [12] expansion for water wave potential produces

$$\begin{aligned} \phi ^{*}_{1}(x,y)= & {} \left\{ \begin{array}{lr} (\mathrm{e}^{-\mathrm{i}Kx}+R^{*}_{1}\mathrm{e}^{\mathrm{i}Kx})\mathrm{e}^{-Ky}+\int _{0}^{\infty }A^{*}(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k~\quad \text{ for }~x>0,~0<y<\infty ,\\ \\ T^{*}_{1}I_{0}(y)\mathrm{e}^{-\mathrm{i}k_{0}x}+\sum _{n=1}^{\infty }A^{*}_{n}I_{n}(y)\mathrm{e}^{k_{n}x}~~~\quad \quad \quad \quad \quad \quad \quad \quad \text{ for }~x<0,~0<y<h; \end{array} \right. \end{aligned}$$

(A.6)

$$\begin{aligned} \phi ^{*}_{2}(x,y)= & {} \left\{ \begin{array}{lr} (\mathrm{e}^{-\mathrm{i}Kx}+R^{*}_{2}\mathrm{e}^{\mathrm{i}Kx})\mathrm{e}^{-Ky}+\int _{0}^{\infty }B^{*}(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k~\quad \text{ for }~x>0,~0<y<\infty ,\\ \\ \left( C^{*}_{0}\cos k_{0}x+D^{*}_{0}\sin k_{0}x\right) I_{0}(y)+\sum _{n=1}^{\infty }\left( C^{*}_{n}\cosh k_{n}x+D^{*}_{n}\sinh k_{n}x\right) I_{n}(y) \\ ~\quad \quad \quad \quad \quad \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \text{ for }~-b<x<0,~0<y<h,\\ \\ T^{*}_{2}I_{0}(y)\mathrm{e}^{-\mathrm{i}k_{0}x}+\sum _{n=1}^{\infty }B^{*}_{n}I_{n}(y)\mathrm{e}^{k_{n}x}~~~\quad \quad \text{ for }~x<-b,~0<y<h ; \end{array} \right. \end{aligned}$$

(A.7)

$$\begin{aligned} \phi ^{*}_{3}(x,y)= & {} \left\{ \begin{array}{lr} (\mathrm{e}^{-\mathrm{i}Kx}+R^{*}_{3}\mathrm{e}^{\mathrm{i}Kx})\mathrm{e}^{-Ky}+\int _{0}^{\infty }E^{*}(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k\quad ~\text{ for }~x>b,~0<y<\infty ,\\ \\ \left( F^{*}_{0}\mathrm{e}^{\mathrm{i}Kx}x+G^{*}_{0}\mathrm{e}^{-\mathrm{i}Kx}\right) \mathrm{e}^{-Ky}+\int _{0}^{\infty }\left( F^{*}(k)\mathrm{e}^{-kx}x+G^{*}(k)\mathrm{e}^{kx}\right) S(k,y)\mathrm{e}^{-kx}\mathrm{d}k\\ ~\text{ for }~0<x<b,~0<y<\infty ,\\ \\ T^{*}_{3}I_{0}(y)\mathrm{e}^{-\mathrm{i}k_{0}x}+\sum _{n=1}^{\infty }E^{*}_{n}I_{n}(y)\mathrm{e}^{k_{n}x}~~~\quad \quad \quad \quad \quad \quad \quad \quad \text{ for }~x<0,~0<y<h; \end{array} \right. \end{aligned}$$

(A.8)

$$\begin{aligned} \phi ^{*}_{4}(x,y)= & {} \left\{ \begin{array}{lr} (\mathrm{e}^{-\mathrm{i}Kx}+R^{*}_{1}\mathrm{e}^{\mathrm{i}Kx})\mathrm{e}^{-Ky}+\int _{0}^{\infty }H^{*}(k)S(k,y)\mathrm{e}^{-kx}\mathrm{d}k~\quad \text{ for }~x>0,~0<y<\infty ,\\ \\ T^{*}_{4}I_{0}(y)\mathrm{e}^{-\mathrm{i}k_{0}x}+\sum _{n=1}^{\infty }H^{*}_{n}I_{n}(y)\mathrm{e}^{k_{n}x}~~~\quad \quad \quad \quad \quad \quad \quad \quad \text{ for }~x<0,~0<y<h . \end{array} \right. \end{aligned}$$

(A.9)

Here $A^{*}(k)$, $B^{*}(k)$, $E^{*}(k)$, $F^{*}(k)$, $G^{*}(k)$, $H^{*}(k)$ are unknown functions and $A^{*}_{n}$, $B^{*}_{n}$, $C^{*}_{n}$, $D^{*}_{n}$, $E^{*}_{n}$, $H^{*}_{n}~(n=1,2,\ldots )$, $C^{*}_{0}$, $D^{*}_{0}$, $F^{*}_{0}$, $G^{*}_{0}$ are unknown constants.

1.2 B: Expressions for the reflection and transmission coefficients

1.2.1 Waves incident from the finite depth water region

For the partially immersed barrier occupying the first position shown in figure 1a, let $f_{1}(y)$ represent the horizontal component of velocity at $x=0$, and then

$$\begin{aligned} \phi _{1x}(0_{-},y)=\phi _{1x}(0_{+},y)=f_{1}(y),~~~y\in (0,h), \end{aligned}$$

(B.1)

and $f_{1}(y)=0$ for $0<y<c$. Also due to the edge condition (2.6),

$$\begin{aligned} f_{1}(y)=\left\{ \begin{array}{lr} O((y-c)^{-1/2})~~\text{ as }~~y\rightarrow c+0,\\ \\ O((h-y)^{-1/3})~~\text{ as }~~y\rightarrow h-0. \end{array}\right. \end{aligned}$$

Use of Havelock inversion formulae (cf. Mandal and Chakrabarti [28]) produces

$$\begin{aligned}&1-R_{1}=\frac{-4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{c}^{h}f_{1}(u)\cosh k_{0}(h-u)\mathrm{d}u, \end{aligned}$$

(B.2)

$$\begin{aligned}&A_{n}=\frac{4\cos k_{n}h}{2k_{n}h+\sin 2k_{n}h}\int _{c}^{h}f_{1}(u)\cos k_{n}(h-u)\mathrm{d}u, \end{aligned}$$

(B.3)

$$\begin{aligned}&T_{1}=-2\mathrm{i}\int _{c}^{h}f_{1}(u)\mathrm{e}^{-Ku}\mathrm{d}u, \end{aligned}$$

(B.4)

$$\begin{aligned}&A(k)=-\frac{2}{\pi }\frac{1}{k(k^{2}+K^{2})}\int _{c}^{h}f_{1}(u)S(k,u)\mathrm{d}u. \end{aligned}$$

(B.5)

For the partially immersed barrier occupying the positions at $x=-b$ (Fig. 1b), let $f_{2}(y)$ represent the horizontal component of velocity at $x=-b$, and then

$$\begin{aligned} \phi _{2x}(-b_{-},y)=\phi _{2x}(-b_{+},y)=f_{2}(y),~~~y\in (0,h), \end{aligned}$$

(B.6)

and due to the edge condition (2.6),

$$\begin{aligned} f_{2}(y)=O((y-c)^{-1/2})~~\text{ as }~~y\rightarrow c+0 \end{aligned}$$

and $f_{2}(y)=0$ for $0<y<c$ due to the condition (2.3),

$$\begin{aligned} \phi _{2x}(0_{-},y)=\phi _{2x}(0_{+},y)=f_{3}(y),~~~y\in (0,h), \end{aligned}$$

(B.7)

so that $f_{3}(y)=O((h-y)^{-1/3})~~\text{ as }~~y\rightarrow h-0$.

For the barrier in Fig. 1c, let

$$\begin{aligned} \phi _{3x}(0_{-},y)=\phi _{3x}(0_{+},y)=f_{4}(y),~~~y\in (0,h), \end{aligned}$$

(B.8)

so that $f_{4}(y)=O((h-y)^{-1/3})~~\text{ as }~~y\rightarrow h-0$ and let

$$\begin{aligned} \phi _{3x}(b_{-},y)=\phi _{3x}(b_{+},y)=f_{5}(y),~~~y\in (0,\infty ), \end{aligned}$$

(B.9)

so that $f_{5}(y)=0$ for $0<y<c$ and $f_{5}(y)=O((y-c)^{-1/3})~~\text{ as }~~y\rightarrow c+0$.

Finally, for the submerged plate (Fig. 1d), let

$$\begin{aligned} \phi _{4x}(0_{-},y)=\phi _{4x}(0_{+},y)=f_{6}(y),~~~y>0, \end{aligned}$$

(B.10)

so that $f_{6}(y)=0$ for $a<y<c$ due to the condition (2.3) and

$$\begin{aligned} f_{6}(y)=\left\{ \begin{array}{lr} O((a-y)^{-1/2})~~\text{ as }~~y\rightarrow a-0,\\ \\ O((y-c)^{-1/2})~~\text{ as }~~y\rightarrow c+0. \end{array}\right. \end{aligned}$$

Applying Havelock’s inversion formulae appropriately, the unknown functions B(k) and unknown constants $B_{n}$, $C_{n}$, $D_{n}(n=1,2,\ldots )$, $C_{0}$, $D_{0}$, $R_{2}$, $T_{2}$ involved in the expression for $\phi _{2}(x,y)$ in (A.2) can be expressed in terms of integrals involving $f_{2}(y)$ and $f_{3}(y)$. In particular, $R_{2}$ and $T_{2}$ are given by

$$\begin{aligned}&1-R_{2}&=\frac{-4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{c}^{h}f_{2}(u)\cosh k_{0}(h-u)\mathrm{d}u , \end{aligned}$$

(B.11)

$$\begin{aligned}&T_{2}&=-2\mathrm{i}\int _{0}^{h}f_{3}(u)\mathrm{e}^{-Ku}\mathrm{d}u . \end{aligned}$$

(B.12)

Similarly, applying Havelock’s inversion formulae appropriately the unknown functions E(k) , F(k) , G(k) and unknown constants $E_{n}~(n=1,2,\ldots )$, $F_{0}$, $G_{0}$, $R_{3}$, $T_{3}$ involved in the expression for $\phi _{3}(x,y)$ in (A.3) can be expressed in terms of integrals involving $f_{4}(y)$ and $f_{5}(y)$. In particular, $R_{3}$ and $T_{3}$ are given by

$$\begin{aligned}&1-R_{3}&=\frac{-4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{0}^{h}f_{4}(u)\cosh k_{0}(h-u)\mathrm{d}u , \end{aligned}$$

(B.13)

$$\begin{aligned}&T_{3}&=-2\mathrm{i}\int _{c}^{\infty }f_{5}(u)\mathrm{e}^{-Ku}\mathrm{d}u . \end{aligned}$$

(B.14)

The unknown function H(k) and unknown constant $H_{n}~(n=1,2,\ldots )$, $R_{4}$, $T_{4}$ involved in the expression for $\phi _{4}(x,y)$ in (A.4) can be expressed in terms of integrals involving $f_{6}(y)$. In particular, $R_{4}$ and $T_{4}$ are given by

$$\begin{aligned}&1-R_{4}&=\frac{-4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{(0,a)\cup (c,h)}f_{6}(u)\cosh k_{0}(h-u)\mathrm{d}u , \end{aligned}$$

(B.15)

$$\begin{aligned}&T_{4}&=-2\mathrm{i}\int _{(0,a)\cup (c,h)}f_{6}(u)\mathrm{e}^{-Ku}\mathrm{d}u. \end{aligned}$$

(B.16)

1.2.2 Waves incident from the deep water region

In this case, for the partially immersed barrier at $x=0$ (Fig. 2a) the reflection and transmission coefficients can be expressed in terms of $f^{*}_{1}(y)(=\phi ^{*}_{1x}(0,y),c<y<h)$ as given by

$$\begin{aligned}&1-R^{*}_{1}&=2\mathrm{i}\int _{c}^{h}f^{*}_{1}(u)\mathrm{e}^{-Ku}\mathrm{d}u , \end{aligned}$$

(B.17)

$$\begin{aligned}&T^{*}_{1}&=\frac{4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{c}^{h}f^{*}_{1}(u)\cosh k_{0}(h-u)\mathrm{d}u . \end{aligned}$$

(B.18)

For the partially immersed barrier at $x=-b$ (Fig. 2b), the reflection coefficients $R^{*}_{2}$ can be expressed in terms of $f^{*}_{2}(y)(=\phi ^{*}_{2x}(0,y),0<y<h)$ as given by

$$\begin{aligned} 1-R^{*}_{2}=2\mathrm{i}\int _{0}^{h}f^{*}_{2}(u)\mathrm{e}^{-Ku}\mathrm{d}u. \end{aligned}$$

(B.19)

and transmission coefficients $T^{*}_{2}$ can be expressed in terms of $f^{*}_{3}(y)(=\phi ^{*}_{2x}(-b,y),c<y<h)$ as given by

$$\begin{aligned} T^{*}_{2}=\frac{4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{c}^{h}f^{*}_{3}(u)\cosh k_{0}(h-u)\mathrm{d}u. \end{aligned}$$

(B.20)

For the partially immersed barrier at $x=-b$ (Fig. 2c), the reflection coefficients $R^{*}_{3}$ can be expressed in terms of $f^{*}_{4}(y)(=\phi ^{*}_{3x}(b,y),c<y<\infty )$ as given by

$$\begin{aligned} 1-R^{*}_{3}=2\mathrm{i}\int _{c}^{\infty }f^{*}_{4}(u)\mathrm{e}^{-Ku}\mathrm{d}u \end{aligned}$$

(B.21)

and transmission coefficients $T^{*}_{3}$ can be expressed in terms of $f^{*}_{5}(y)(=\phi ^{*}_{3x}(0,y),0<y<h)$ as given by

$$\begin{aligned} T^{*}_{3}=\frac{4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{0}^{h}f^{*}_{5}(u)\cosh k_{0}(h-u)\mathrm{d}u. \end{aligned}$$

(B.22)

For the submerged plate at $x=0$ (Fig. 2d), the reflection and transmission coefficients can be expressed in terms of $f^{*}_{6}(y)(=\phi ^{*}_{4x}(0,y),y\in (0,a)\cup (c,h))$ as given by

$$\begin{aligned}&1-R^{*}_{4}&=2\mathrm{i}\int _{(0,a)\cup (c,h)}f^{*}_{6}(u)\mathrm{e}^{-Ku}\mathrm{d}u , \end{aligned}$$

(B.23)

$$\begin{aligned}&T^{*}_{4}&=\frac{4\mathrm{i}\cosh k_{0}h}{2k_{0}h+\sinh 2k_{0}h}\int _{(0,a)\cup (c,h)}f^{*}_{6}(u)\cosh k_{0}(h-u)\mathrm{d}u. \end{aligned}$$

(B.24)

It may be noted that $f^{*}_\mathrm{j}(y)$ satisfies the same conditions as $f_\mathrm{j}(y)(j=1,2,\ldots 6)$.