3.1 Eigenfunction expansion for $\phi ^{a,s}(x,y)$ in various regions

The eigen expansion of $\phi ^{a,s}(x,y)$ satisfying (2.1)–(2.7), (3.2), (3.3) for $x\geq 0$ are given below.

Region R1 $~(x>L,~0<y<H)$

(3.4) $$ \begin{align} \phi^{a,s}(x,y)&= \frac{\cosh{P_0(H-y)}}{\cosh{P_0 H}}\{e^{-iP_0(x-L)}+ \mathcal{R}^{a,s}e^{iP_0(x-L) }\}\nonumber \\ &\quad +\sum_{r=1}^{\infty}\mathcal{A}^{a,s}_{r}\cos P_{r}(H-y) e^{-P_r(x-L) }, \end{align} $$

where $k= P_{0}$ and $k= iP_{r}(r=1,2,\ldots )$ are the roots of the equation

$$ \begin{align*} k\tanh kH-K=0 , \end{align*} $$

and $\mathcal {A}^{a,s}_{r},~ r=1,\ldots, \infty $ are unknowns to be determined.

Region R2 $~(B<x<L,~0<y<h)$

(3.5) $$ \begin{align} \begin{pmatrix} \phi^{a}(x,y) \\ \phi^{s}(x,y) \end{pmatrix}=\begin{pmatrix} \mathcal{B}^{a}_{0}\cos p_{0}x \\ \mathcal{B}^{s}_{0}\sin p_{0}x \end{pmatrix} \frac{\cosh{p_0(h-y)}}{\cosh{p_0 h}}+\sum_{r=1}^{\infty}\begin{pmatrix} \mathcal{B}^{a}_{r}\cosh p_{r}x \\ \mathcal{B}^{s}_{r}\sinh p_{r}x \end{pmatrix}\cos p_{r}(h-y), \end{align} $$

where $k= p_{0}$ and $k= ip_{r}(r=1,2,\ldots )$ are the roots of the equation

$$ \begin{align*} k\tanh kh-K=0, \end{align*} $$

and $\mathcal {B}^{a,s}_{r},~ r=0,1,\ldots, \infty $ are unknowns to be determined.

Region R3 $~(0<x<B,~A<y<h)$

(3.6) $$ \begin{align} \begin{pmatrix} \phi^{a}(x,y) \\ \phi^{s}(x,y) \end{pmatrix}=\begin{pmatrix}\mathcal{ C}^{a}_{0} \\ \mathcal{ C}^{s}_{0}x \end{pmatrix} +\sum_{r=1}^{\infty}\begin{pmatrix} \mathcal{ C}^{a}_{r}\cosh \bigg(\dfrac{r\pi}{h-A}\bigg)x \\[.2cm] \mathcal{C}^{s}_{r}\sinh\bigg(\dfrac{r\pi}{h-A}\bigg)x \end{pmatrix}\cos\bigg(\dfrac{r\pi}{h-A}\bigg)(h-y), \end{align} $$

where $\mathcal {C}^{a,s}_{r},~ r=0,1,\ldots, \infty $ are unknowns to be determined.

3.2 Determination of unknowns in the eigenfunction expansion for $\phi ^{a,s}(x,y)$

Let us now define

(3.7) $$ \begin{align} f^{a,s}(y)=\phi^{a,s}_{x}(L+0,y) \quad \text{for~} 0<y<H,~ \end{align} $$

(3.8) $$ \begin{align}\kern-6pt F^{a,s}(y)=\phi^{a,s}_{x}(L-0,y) \quad \text{for~} 0<y<h.~ \end{align} $$

Then,

$$ \begin{align*} F^{a,s}(y)=\begin{cases} f^{a,s}(y) & \text{ for } 0<y<H, \\ 0 &\text{ for } H<y<h, \end{cases} \end{align*} $$

and

(3.9) $$ \begin{align}\kern-6pt G^{a,s}(y)=\phi^{a,s}_{x}(B+0,y) \quad \text{for } 0<y<h, \end{align} $$

(3.10) $$ \begin{align} g^{a,s}(y)=\phi^{a,s}_{x}(B-0,y) \quad \text{for } D<y<h. \end{align} $$

Then,

$$ \begin{align*} G^{a,s}(y)= \begin{cases} 0 \quad & \text{ for }0<y<D, \\ g^{a,s}(y) & \text{ for }D<y<h. \end{cases} \end{align*} $$

The edge conditions given by (2.8) and (2.9) assert that

$$ \begin{align*} F^{a,s}(y)&=O (|y-H|^{-{1}/{3}}) \quad \text{as~} y\rightarrow H, \\G^{a,s}(y)&=O (|y-D|^{-{1}/{2}}) \quad\text{as~} y\rightarrow D.\end{align*} $$

Using the expansion (3.4) in relation (3.7) and applying Havelock’s inversion formula, we obtain

(3.11) $$ \begin{align} 1-\mathcal{R}^{a,s}&=\frac{4 i \cosh{P_0H}}{\delta_{0}}\int_{0}^{H}f^{a,s}(y)\cosh{P_0(H-y)}~dy, \\\mathcal{A}_{r}^{a,s}&=-\frac{4}{\delta_{r}}\int_{0}^{H}f^{a,s}(y)\cos{P_r(H-y)}~dy ,\nonumber \end{align} $$

with $\delta _{0}=2P_0H+\sinh {2P_0H}$ and $\delta _{r}=2P_rH+\sin {2P_rH}$ for $r=1,2,3,\ldots .$

Using the expansion (3.5) in (3.8) and (3.9), and applying Fourier cosine transformation, we obtain

$$ \begin{align*} \mathcal{B}_{0}^{a,s}&=\frac{4\cosh{p_0h}}{\gamma_{0}}\bigg(\dfrac{-1}{\sin{p_0L}},\dfrac{1}{\cos{p_0L}}\bigg)\int_{0}^{h}F^{a,s}(y)\cosh{p_0(h-y)}~dy \\ &=\frac{4\cosh{p_0h}}{\gamma_{0}}\bigg(\dfrac{-1}{\sin{p_0B}},\dfrac{1}{\cos{p_0B}}\bigg)\int_{0}^{h}G^{a,s}(y)\cosh{p_0(h-y)}~dy, \\ \mathcal{B}_{r}^{a,s}&=\frac{4}{\gamma_{r}}\bigg(\dfrac{1}{\sinh{p_rL}},\dfrac{1}{\cosh{p_rL}}\bigg)\int_{0}^{h}F^{a,s}(y)\cos{p_r(h-y)}~dy \\ &=\frac{4}{\gamma_{r}}\bigg(\dfrac{1}{\sinh{p_rB}},\dfrac{1}{\cosh{p_rB}}\bigg)\int_{0}^{h}G^{a,s}(y)\cos{p_r(h-y)}~dy, \end{align*} $$

with $\gamma _{0}=2p_0h+\sinh {2p_0h}$ and $\gamma _{r}=2p_rh+\sin {2p_rh}$ for $r=1,2,3,\ldots .$

Again, using (3.6) in (3.10) and applying Fourier cosine transformation, we get

$$ \begin{align*} \mathcal{ C}_{0}^{s}&=\frac{1}{h-A }\int_{D}^{h}g^{s}(y)~dy, \\ \mathcal{ C}_{r}^{a,s}&=\frac{2}{r\pi}\bigg(\dfrac{1}{\sinh(\tfrac{r\pi}{h-A})B}, \dfrac{1}{\cosh(\tfrac{r\pi}{h-A})B}\bigg)\int_{D}^{h}g^{a,s}(y)\cos{\bigg(\dfrac{r\pi}{h-A}\bigg)(h-y)}~dy. \end{align*} $$

Thus, all the unknowns $\mathcal {R}^{a,s},~\mathcal { A}_{r}^{a,s},~ \mathcal { B}_{r}^{a,s}~ \text {and}~\mathcal { C}_{r}^{a,s}$ are expressed in terms of unknown functions $F^{a,s}(y)$ and $G^{a,s}(y)$ . Next, we shall derive integral equations for the functions $F^{a,s}(y)$ and $G^{a,s}(y)$ .

3.3 Integral equations for $F^{a,s}(y)$ and $G^{a,s}(y)$

We consider the continuity of the functions $\phi ^{a,s}(x,y)$ across the gaps in the boundaries of the different regions.

At the interface of Regions $R1$ and $R2$ , that is, at $x=L$ ,

$$ \begin{align*} \phi^{a,s}(L+0,y)=\phi^{a,s}(L-0,y), \quad 0<y<H. \end{align*} $$

In the gap below the barrier at the interface of Regions $R2$ and $R3$ , that is, at $x=B$ ,

$$ \begin{align*} \phi^{a,s}(B+0,y)=\phi^{a,s}(B-0,y),\quad D<y<h. \end{align*} $$

These matching conditions provide the following system of integral equations for $q_{j}^{a,s}(u),~j=1,2$ :

(3.12a) $$ \begin{align} \int_{0}^{H}q_{1}^{a,s}(u)M_{1}^{a,s}(y,u)\,du+\int_{D}^{h}q_{2}^{a,s}(u)N_{1}^{a,s}(y,u)\,du=X_{1}^{a,s},\quad 0<y<H,~ \end{align} $$

(3.12b) $$ \begin{align} \int_{0}^{H}q_{1}^{a,s}(u)M_{2}^{a,s}(y,u)\,du+\int_{D}^{h}q_{2}^{a,s}(u)N_{2}^{a,s}(y,u)\,du=X_{2}^{a,s},\quad D<y<h,~ \end{align} $$

where

(3.13) $$ \begin{align} q_{1}^{a,s}(u)&=\dfrac{f^{a,s}(u)}{1+\mathcal{ R}^{a,s}}, \nonumber \\ q_{2}^{a,s}(u)&=\dfrac{g^{a,s}(u)}{1+\mathcal{ R}^{a,s}}, \\ X_{1}^{a,s}&=\frac{\cosh{P_0(H-y)}}{\cosh{P_0 H}}, \nonumber \\ X_{2}^{a,s}&=0, \nonumber \end{align} $$

and the kernels

$$ \begin{align*} M_{1}^{a,s}(y,u)&=\sum_{r=1}^{\infty}\dfrac{4}{\delta_{r}}\cos P_{r}(H-u)\cos P_{r}(H-y), \\M_{2}^{a}(y,u)&=\dfrac{4\cos p_{0}B}{\gamma_{0}\sin p_{0}L}\cosh p_{0}(h-u)\bigg\{\cosh p_{0}(h-y)-\dfrac{\sinh p_{0}(h-D)}{p_{0}(h-D)}\bigg\} \nonumber \\&\quad -\sum_{r=1}^{\infty}\dfrac{4\cosh p_{r}B}{\gamma_{r}\sinh p_{r}L}\cos p_{r}(h-u)\bigg\{\cos p_{r}(h-y)-\dfrac{\sin p_{r}(h-D)}{p_{r}(h-D)}\bigg\}, \\M_{2}^{s}(y,u)&=\dfrac{4\sin p_{0}B}{\gamma_{0}\cos p_{0}L}\cosh p_{0}(h-u)\cosh p_{0}(h-y) \nonumber \\&\quad +\sum_{r=1}^{\infty}\dfrac{4\sinh p_{r}B}{\gamma_{r}\cosh p_{r}L}\cos p_{r}(h-u)\cos p_{r}(h-y), \\N_{1}^{a}(y,u)&=-\dfrac{4\cos p_{0}L}{\gamma_{0}\sin p_{0}B}\cosh p_{0}(h-u)\cosh p_{0}(h-y) \nonumber \\&\quad + \sum_{r=1}^{\infty}\dfrac{4\cosh p_{r}L}{\gamma_{r}\sinh p_{r}B}\cos p_{r}(h-u)\cos p_{r}(h-y), \\N_{1}^{s}(y,u)&=\dfrac{4\sin p_{0}L}{\gamma_{0}\cos p_{0}B}\cosh p_{0}(h-u)\cosh p_{0}(h-y) \nonumber \\&\quad +\sum_{r=1}^{\infty}\dfrac{4\sinh p_{r}L}{\gamma_{r}\cosh p_{r}B}\cos p_{r}(h-u)\cos p_{r}(h-y),\\N_{2}^{a}(y,u)&=\sum_{r=1}^{\infty}\dfrac{2}{r\pi}\coth \bigg(\dfrac{r\pi}{h-A}\bigg)B\cos\bigg(\dfrac{r\pi}{h-A}\bigg)(h-u)\bigg\{\cos\bigg(\dfrac{r\pi}{h-A}\bigg)(h-y) \nonumber \\&\quad -\frac{1}{r\pi}\bigg(\frac{h-A}{h-D}\bigg)\sin\bigg(\dfrac{r\pi}{h-A}\bigg)(h-D)\bigg\}, \\N_{2}^{s}(y,u)&=\dfrac{-B}{h-A}-\sum_{r=1}^{\infty}\dfrac{2}{r\pi}\tanh \bigg(\dfrac{r\pi}{h-A}\bigg)B\cos\bigg(\dfrac{r\pi}{h-A}\bigg)(h-u)\cos\bigg(\dfrac{r\pi}{h-A}\bigg)(h-y). \end{align*} $$

The substitution of $q_{1}^{a,s}(y)$ from (3.13) into (3.11) gives rise to

(3.14) $$ \begin{align} \frac{4\cosh{P_0H}}{\delta_{0}}\int_{0}^{H}q_{1}^{a,s}(y)\cosh P_{0}(H-y)~dy=C^{a,s}, \end{align} $$

where

(3.15) $$ \begin{align} C^{a,s}=-i\dfrac{1-\mathcal{R}^{a,s}}{1+\mathcal{R}^{a,s}}. \end{align} $$

Using (3.15), the reflection and transmission coefficient are obtained as

(3.16) $$ \begin{align} |R|=\dfrac{|1+C^{a}C^{s}|}{\Delta},\quad |T|=\dfrac{|C^{a}-C^{s}|}{\Delta}, \end{align} $$

where $\Delta =\{1+(C^{a})^{2}+(C^{s})^{2}+(C^{a}C^{s})^{2}\}^{{1}/{2}}.$ Thus, solving the system of integral equations (3.12a), (3.12b), the functions $q_j,~j=1,2$ are determined which are then used in (3.14)–(3.16) to evaluate $|R|$ and $|T|$ .

Also, the horizontal force and vertical force on the $\large \boldsymbol \sqcap $ -shaped breakwater can be determined by using Bernoulli’s equation [Reference Bai2, Reference Bhattacharjee and Soares4, Reference Kaligatla, Singh and Mandal10] and is given by

$$ \begin{align*} W_{h1}&=2 i\rho \omega \, \bigg(\!\int_{0}^{D}{\phi^{s}(B+0,y)}\,dy-\int_{A}^{D}{\phi^{s}(B-0,y)}\, dy\bigg),\\ W_{v1}&=2 i\rho \omega\int_{0}^{B}{\phi^{a}(x,A)}\,dx, \end{align*} $$

where $W_{h1}$ is the horizontal force and $W_{v1}$ is the vertical force.

The horizontal force and vertical force per unit width can be expressed in the nondimensional form as

$$ \begin{align*} W_{h}=\dfrac{|W_{h1}|}{\rho g \xi B}, \quad W_{v}=\dfrac{|W_{v1}|}{\rho g \xi B}. \end{align*} $$

Here, $\xi $ is the incident wave amplitude and $\rho $ is the density of the water.

3.4 Solution of integral equations by the Galerkin method

Now, we solve the integral equations (3.12a), (3.12b) using the multi-term Galerkin approximation method involving orthogonal polynomials multiplied by appropriate weight functions as basis functions. Here, it is important to note that the choice of basis functions depends on the types of singularities at the corner of the trench and at the submerged sharp edge of the breakwater as given by the edge conditions (2.8), (2.9).

Noting the edge condition (2.9), we express the unknown function $q_{1}^{a,s}(y)$ in terms of Gegenbauer polynomials and noting the edge condition (2.8), we express the unknown function $q_{2}^{a,s}(y)$ in terms of Chebychev’s polynomial as

(3.17) $$ \begin{align} q_{1}^{a,s}(y)&=\sum_{n=0}^{N}a_{n}^{a,s}U_{n}^{a,s}(y),\quad 0<y<H, \end{align} $$

(3.18) $$ \begin{align} q_{2}^{a,s}(y)&=\sum_{n=0}^{N}b_{n}^{a,s}V_{n}^{a,s}(y),\quad D<y<h. \end{align} $$

Here,

$$ \begin{align*} U_{n}^{a,s}(y)=U_{n}(y)=-\frac{d}{dy}\bigg[e^{-Ky}\int_{y}^{H}e^{Kz}\tilde{U_{n}}(z)\,dz\bigg],\quad 0<y<H, \end{align*} $$

with

$$ \begin{align*} \tilde{U_{n}}(z)=\dfrac{2^{{7}/{6}}\Gamma({1}/{6})(2n)!}{\pi \Gamma(2n+{1}/{3})H^{{1}/{3}}(H^{2}-z^{2})^{{1}/{3}}}C_{2n}^{{1}/{6}}\bigg(\dfrac{z}{H}\bigg),\quad n=0,1,2,\ldots, \end{align*} $$

where $C_{2n}^{{1}/{6}}({z}/{H})$ are Gegenbauer polynomials of order $({1}/{6})$ , and

$$ \begin{align*} V_{n}^{a,s}(y)=V_{n}(y)=\dfrac{2(-1)^{n}}{\pi \{(h-D)^{2}-(h-y)^{2}\}^{{1}/{2}}}T_{2n}\bigg(\dfrac{h-y}{h-D}\bigg),\quad D<y<H, \end{align*} $$

where $T_{2n}(y)$ are Chebychev’s polynomial of first kind of order $2n$ .

We substitute the expansion (3.17) and (3.18) in (3.12a), then multiplying by $U_{m}^{a,s}(y)$ and integrating y from $0$ to H, we get

(3.19) $$ \begin{align} &\sum_{n=0}^{N}a_{n}^{a,s}\int_{0}^{H}\int_{0}^{H}M_{1}^{a,s}(y,u)U_{n}^{a,s}(u)U_{m}^{a,s}(y)\,du\,dy \nonumber \\ &\quad +\sum_{n=0}^{N}b_{n}^{a,s}\int_{0}^{H}\int_{D}^{h}N_{1}^{a,s}(y,u)V_{n}^{a,s}(u)U_{m}^{a,s}(y)\,du\,dy =\int_{0}^{H}\frac{\cosh{P_0(H-y)}}{\cosh{P_0 H}}U_{m}^{a,s}(y)\,dy,\nonumber \\ &\sum_{n=0}^{N}a_{n}^{a,s}K_{1mn}^{a,s}+\sum_{n=0}^{N}b_{n}^{a,s}J_{1mn}^{a,s}=d_{1m,}^{a,s},\quad m=0,1,\ldots ,N. \end{align} $$

Again substituting the expansion (3.17) and (3.18) in (3.12b), multiplying by $V_{m}^{a,s}(y)$ and integrating y from D to h, we get

(3.20) $$ \begin{align} &\sum_{n=0}^{N}a_{n}^{a,s}\int_{D}^{h}\int_{0}^{H}M_{2}^{a,s}(y,u)U_{n}^{a,s}(u)V_{m}^{a,s}(y)\,du\,dy\nonumber \\ &\quad +\sum_{n=0}^{N}b_{n}^{a,s}\int_{D}^{h}\int_{D}^{h}N_{2}^{a,s}(y,u)V_{n}^{a,s}(u)V_{m}^{a,s}(y)\,du\,dy =0,\nonumber \\ &\sum_{n=0}^{N}a_{n}^{a,s}K_{2mn}^{a,s}+\sum_{n=0}^{N}b_{n}^{a,s}J_{2mn}^{a,s}=d_{2m,}^{a,s},\quad m=0,1,\ldots ,N, \end{align} $$

where

$$ \begin{align*} K_{1mn}^{a,s}&=\int_{0}^{H}\int_{0}^{H}M_{1}^{a,s}(y,u)U_{n}^{a,s}(u)U_{m}^{a,s}(y)\,du\,dy \\ &=\sum_{r=1}^{\infty}(-1)^{m+n}\dfrac{16}{\delta_{r}}\dfrac{\cos^{2}P_{r}H}{(P_{r}H)^{{1}/{3}}}J_{2m+{1}/{6}}(P_{r}H)J_{2n+{1}/{6}}(P_{r}H), \\ J_{1mn}^{a}&=\int_{D}^{h}\int_{0}^{H}N_{1}^{a}(y,u)V_{n}^{a}(u)U_{m}^{a}(y)\,du\,dy \\ &=(-1)^{n+1}\dfrac{8}{\gamma_{0}}\dfrac{\cos p_{0}L}{\sin p_{0}B}\dfrac{\cosh p_{0}h}{(p_{0}H)^{{1}/{6}}}I_{2n}(p_{0}(h-D))I_{2m+{1}/{6}}(p_{0}H) \\ &\quad +\sum_{r=1}^{\infty}(-1)^{m}\dfrac{8}{\gamma_{r}}\dfrac{\cosh p_{r}L}{\sinh p_{r}B}\dfrac{\cos p_{r}h}{(p_{r}H)^{{1}/{6}}}J_{2n}(p_{r}(h-D))J_{2m+{1}/{6}}(p_{r}H), \end{align*} $$

$$ \begin{align*} J_{1mn}^{s}&=\int_{D}^{h}\int_{0}^{H}N_{1}^{s}(y,u)V_{n}^{s}(u)U_{m}^{s}(y)\,du\,dy \nonumber\\&=(-1)^{n}\dfrac{8}{\gamma_{0}}\dfrac{\sin p_{0}L}{\cos p_{0}B}\dfrac{\cosh p_{0}h}{(p_{0}H)^{{1}/{6}}}I_{2n}(p_{0}(h-D))I_{2m+{1}/{6}}(p_{0}H)\nonumber\\&\quad +\sum_{r=1}^{\infty}(-1)^{m}\dfrac{8}{\gamma_{r}}\dfrac{\sinh p_{r}L}{\cosh p_{r}b}\dfrac{\cos p_{r}h}{(p_{r}H)^{{1}/{6}}}J_{2n}(p_{r}(h-D))J_{2m+{1}/{6}}(p_{r}H), \\K_{2mn}^{a}&=\int_{D}^{h}\int_{0}^{H}M_{2}^{a}(y,u)U_{n}^{a}(u)V_{m}^{a}(y)\,du\,dy \nonumber\\[3pt]&=(-1)^{m}\dfrac{8}{\gamma_{0}}\dfrac{\cos p_{0}B}{\sin p_{0}L}\dfrac{\cosh p_{0}h}{(p_{0}H)^{{1}/{6}}}I_{2n+{1}/{6}}(p_{0}H)I_{2m}(p_{0}(h-D))\nonumber\\[3pt]&\quad -\sum_{r=1}^{\infty}(-1)^{n}\dfrac{8}{\gamma_{r}}\dfrac{\cosh p_{r}B}{\sinh p_{r}L}\dfrac{\cos p_{r}h}{(p_{r}H)^{{1}/{6}}}J_{2n+{1}/{6}}(p_{r}H)J_{2m}(p_{r}(h-D)) , \\[3pt]K_{2mn}^{s}&=\int_{D}^{h}\int_{0}^{H}M_{2}^{s}(y,u)U_{n}^{s}(u)V_{m}^{s}(y) \,du\,dy \nonumber\\[3pt]&=(-1)^{m}\dfrac{8}{\gamma_{0}}\dfrac{\sin p_{0}B}{\cos p_{0}L}\dfrac{\cosh p_{0}h}{(p_{0}H)^{{1}/{6}}}I_{2n+{1}/{6}}(p_{0}H)I_{2m}(p_{0}(h-D))\nonumber\\[3pt]&\quad -\sum_{r=1}^{\infty}(-1)^{n+1}\dfrac{8}{\gamma_{r}}\dfrac{\sinh p_{r}B}{\cosh p_{r}L}\dfrac{\cos p_{r}h}{(p_{r}H)^{{1}/{6}}}J_{2n+{1}/{6}}(p_{r}H)J_{2m}(p_{r}(h-D)), \\[3pt]J_{2mn}^{a}&=\int_{D}^{h}\int_{D}^{h}N_{2}^{a}(y,u)V_{n}^{a}(u)V_{m}^{a}(y)\,du\,dy \nonumber\\[3pt]&=\sum_{r=1}^{\infty}\dfrac{2}{r\pi}\coth\bigg(\dfrac{r\pi}{h-A}\bigg)BJ_{2n}\bigg(\dfrac{r\pi}{h-A}(h-D)\bigg)J_{2m}\bigg(\dfrac{r\pi}{h-A}(h-D)\bigg), \end{align*} $$

(3.21) $$ \begin{align} J_{2mn}^{s}&=\int_{D}^{h}\int_{D}^{h}N_{2}^{s}(y,u)V_{n}^{s}(u)V_{m}^{s}(y)\,du\,dy \nonumber\\[3pt]&=\sum_{r=1}^{\infty}\dfrac{-2}{r\pi}\tanh\bigg(\dfrac{r\pi}{h-A}\bigg)BJ_{2n}\bigg(\dfrac{r\pi}{h-A}(h-D)\bigg)J_{2m}\bigg(\dfrac{r\pi}{h-A}(h-D)\bigg), \end{align} $$

(3.22) $$ \begin{align}\kern11pt d_{1m}^{a,s}&=\int_{0}^{H}\frac{\cosh{P_0(H-y)}}{\cosh{P_0 H}}U_{m}^{a,s}(y)\,dy=\dfrac{2}{(P_{0}H)^{{1}/{6}}}I_{2m+{1}/{6}}(P_{0}H)\quad \text{and} \end{align} $$

(3.23) $$ \begin{align}\kern6pt d_{2m}^{a,s}&=0. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\end{align} $$

Here, $J_\nu (x)$ is a Bessel function of first kind of order $\nu $ and $I_\nu (x)$ is a modified Bessel function of first kind of order $\nu $ .

The constants $a_{n}^{a,s},b_{n}^{a,s}(n=0,1,\ldots ,N)$ are now obtained by solving the linear systems (3.19), (3.20). Knowing $a_{n}^{a,s},b_{n}^{a,s}(n=0,1,\ldots ,N)$ , the functions $q_{1}^{a,s}(y),~ q_{2}^{a,s}(y)$ can be evaluated from (3.17) and (3.18). Hence, using the function $q_{1}^{a,s}(y)$ , the relation (3.14) produces

$$ \begin{align*} C^{a,s}=\dfrac{8\cosh^{2}P_{0}H}{\delta_{0}(P_{0}H)^{{1}/{6}}}\sum_{n=0}^{N}a_{n}^{a,s}I_{2n+{1}/{6}}(P_{0}H). \end{align*} $$

Once $C^{a,s}$ are determined, $|R|,|T| $ can be evaluated from (3.16).

3.5 Expression for fluid velocities in various regions

1. (I) Region R1 $~(x>L,~0<y<H)$

   $$ \begin{align*} \phi_{x}(x,y)&= -4P_0 \frac{\cosh{P_0(H-y)}}{\cosh{P_0 H}}\bigg[\sin{P_0(x-L)}\cos \omega t-\dfrac{4 i}{\delta_{0}}\dfrac{(\cosh P_{0}H)^2}{(P_{0}H)^{{1}/{6}}} \nonumber\\ &\quad \times \sin(P_0(x-L)-\omega t) \sum_{n=0}^{N}\bigg\{\dfrac{a_{n}^{a}}{(1+iC^{a})} +\dfrac{a_{n}^{s}}{(1+iC^{s})}\bigg\}I_{2n+{1}/{6}}(P_{0}H)\bigg] \nonumber\\ &\quad +\sum_{r=1}^{\infty}\dfrac{16P_r}{\delta_{r}}\dfrac{\cos P_{r}H}{(P_{r}H)^{{1}/{6}}}\cos P_{r}(H-y)e^{-P_r(x-L)}\nonumber \\ &\quad\times\cos \omega t\sum_{n=0}^{N}\bigg\{\dfrac{a_{n}^{a}}{(1+iC^{a})} +\dfrac{a_{n}^{s}}{(1+iC^{s})}\bigg\}(-1)^nJ_{2n+{1}/{6}}(P_{r}H), \nonumber \end{align*} $$
   $$ \begin{align*} \phi_{y}(x,y)&= 4P_0 \frac{\sinh{P_0(H-y)}}{\cosh{P_0 H}}\bigg[\cos{P_0(x-L)}\cos \omega t-\dfrac{4 i}{\delta_{0}}\dfrac{(\cosh P_{0}H)^2}{(P_{0}H)^{{1}/{6}}} \nonumber\\ &\quad \times \cos(P_0(x-L)-\omega t) \sum_{n=0}^{N}\bigg\{\dfrac{a_{n}^{a}}{(1+iC^{a})} +\dfrac{a_{n}^{s}}{(1+iC^{s})}\bigg\}I_{2n+{1}/{6}}(P_{0}H)\bigg] \nonumber\\ &\quad +\sum_{r=1}^{\infty}\dfrac{16P_r}{\delta_{r}}\dfrac{\cos P_{r}H}{(P_{r}H)^{{1}/{6}}}\sin P_{r}(H-y)e^{-P_r(x-L)}\nonumber \\ &\quad\times \cos \omega t\sum_{n=0}^{N}\bigg\{\dfrac{a_{n}^{a}}{(1+iC^{a})} +\dfrac{a_{n}^{s}}{(1+iC^{s})}\bigg\}(-1)^nJ_{2n+{1}/{6}}(P_{r}H). \nonumber \end{align*} $$

2. (II) Region R2 $~(B<x<L,~0<y<h)$

   $$ \begin{align*} \phi_{x}(x,y) & = \dfrac{16p_0}{\gamma_{0}}\dfrac{\cosh p_{0}h}{(p_{0}H)^{{1}/{6}}}\cosh p_{0}(h-y)\cos \omega t \sum_{n=0}^{N}\bigg\{\dfrac{a_{n}^{a}}{(1+iC^{a})}\dfrac{\sin{p_{0}x}}{\sin{p_{0}L}} \nonumber\\ &\quad +\dfrac{a_{n}^{s}} {(1+iC^{s})}\dfrac{\cos{p_{0}x}}{\cos{p_{0}L}}\bigg\} I_{2n+{1}/{6}}(p_{0}H)\nonumber\\ &\quad +\sum_{r=1}^{\infty}\dfrac{16p_r}{\gamma_{r}}\dfrac{\cos p_{r}h}{(p_{r}H)^{{1}/{6}}}\cos p_{r}(h-y)\cos \omega t \nonumber\\ &\quad \times \sum_{n=0}^{N}(-1)^n \bigg\{\dfrac{a_{n}^{a}}{(1+iC^{a})}\dfrac{\sinh{p_{r}x}}{\sinh{p_{r}L}}+\dfrac{a_{n}^{s}}{(1+iC^{s})}\dfrac{\cosh{p_{r}x}}{\cosh{p_{r}L}}\bigg\}J_{2n+{1}/{6}}(P_{r}H), \nonumber \end{align*} $$
   $$ \begin{align*} \phi_{y}(x,y)&= \dfrac{16p_0}{\gamma_{0}}\dfrac{\cosh p_{0}h}{(p_{0}H)^{{1}/{6}}}\sinh p_{0}(h-y)\cos \omega t\sum_{n=0}^{N}\bigg\{-\dfrac{a_{n}^{a}}{(1+iC^{a})}\dfrac{\cos{p_{0}x}}{\sin{p_{0}L}} \nonumber\\ &\quad +\dfrac{a_{n}^{s}}{(1+iC^{s})}\dfrac{\sin{p_{0}x}}{\cos{p_{0}L}}\bigg\} I_{2n+{1}/{6}}(p_{0}H) \nonumber\\ &\quad -\sum_{r=1}^{\infty}\dfrac{16p_r}{\gamma_{r}}\dfrac{\cos p_{r}h}{(p_{r}H)^{{1}/{6}}}\sin p_{r}(h-y)\cos \omega t \nonumber\\ &\quad \times \sum_{n=0}^{N}(-1)^n \bigg\{\dfrac{a_{n}^{a}}{(1+iC^{a})}\dfrac{\cosh{p_{r}x}}{\sinh{p_{r}L}}+\dfrac{a_{n}^{s}}{(1+iC^{s})}\dfrac{\sinh{p_{r}x}}{\cosh{p_{r}L}}\bigg\}J_{2n+{1}/{6}}(P_{r}H). \nonumber \end{align*} $$

3. (III) Region R3 $~(0<x<B,~A<y<h)$

   $$ \begin{align*} \phi_{x}(x,y)&=\dfrac{4}{h-A}\sum_{r=1}^{\infty}\cos{\bigg(\dfrac{r\pi}{h-A}\bigg)(h-y)}\cos \omega t \nonumber \\ &\quad \times\sum_{n=0}^{N}\bigg\{\dfrac{b_{n}^{a}}{(1+iC^{a})}\dfrac{\sinh{(\tfrac{r\pi}{h-A})x}}{\sinh{(\tfrac{r\pi}{h-A})B}} +\dfrac{b_{n}^{s}}{(1+iC^{s})}\dfrac{\cosh{(\tfrac{r\pi}{h-A})x}}{\cosh{(\tfrac{r\pi}{h-A})B}}\bigg\} \nonumber \\ &\quad\times J_{2n}\bigg(\dfrac{r\pi}{h-A}(h-d)\bigg), \nonumber \end{align*} $$
   $$ \begin{align*} \phi_{y}(x,y)&=\dfrac{-4}{h-A}\sum_{r=1}^{\infty}\sin{(\dfrac{r\pi}{h-A})(h-y)}\cos \omega t\nonumber \\ &\quad\times\sum_{n=0}^{N}\bigg\{\dfrac{b_{n}^{a}}{(1+iC^{a})}\dfrac{\cosh{(\tfrac{r\pi}{h-A})x}}{\sinh{(\tfrac{r\pi}{h-A})B}} +\dfrac{b_{n}^{s}}{(1+iC^{s})}\dfrac{\sinh{(\tfrac{r\pi}{h-A})x}}{\cosh{(\tfrac{r\pi}{h-A})B}}\bigg\} \nonumber \\ &\quad\times J_{2n}\bigg(\dfrac{r\pi}{h-A}(h-d)\bigg). \nonumber \end{align*} $$

3.6 Energy balance relation

The energy balance is obtained by applying Green’s integral theorem to the functions $\phi (x,y)$ and $\bar {\phi (x,y)}$ in the region bounded by contour $\Gamma $ shown in Figure 3. This gives

$$ \begin{align*} \int_\Gamma \bigg(\phi~ \frac{\partial \bar{\phi}}{\partial n}- \bar{\phi}~ \frac{\partial \phi}{\partial n}\bigg)\,d\Gamma=0. \end{align*} $$

Computing the integrals on different boundaries using the boundary conditions (2.2)–(2.10), we obtain the energy identity as

(3.24) $$ \begin{align} |R|^2+ |T|^2= 1. \end{align} $$

Figure 3

Schematic diagram.

4.1 Convergence of the method

For numerical computation, the value of N in (3.19), (3.20) is chosen as $N=6$ . Table 1 depicts the convergence of $|R|$ . Here, some representative values of $|R|$ are presented for various $KH$ , taking $N=2,~4,~6$ and $A/H = 0.25,~ B/H=0.7,~ D/H= 0.45,~ h/H=1.1,~ L/H=0.8$ . It is observed from Table 1 that the values of $|R|$ coincide up to 3 to 4 decimal places for $N=4,~ 6$ .

Table 1

Reflection coefficient for $B/H =0.7,~ A/H = 0.25,~ D/H=0.45,~ L/H=0.8,~ h/H=1.1$ .

4.2 Energy identity

In Table 2, the energy identity (3.24) is verified and presented for various values of $KH$ and for $A/H = 0.25,~ D/H=0.45,~ h/H=1.2,~ L/H= 0.8, 2.5,~ B/H= 0.7,~ 1.5.$ Table 2 establishes the validity of energy balance relation.

Table 2

Energy identity table for $A/H = 0.25,~ D/H=0.45,~ h/H=1.2$ .

4.3 Comparison of the results obtained with known results

In this section, the results obtained in the present study are authenticated by comparing the values of $|R|$ with the known results in the literature.

In Figure 4, $|R|$ obtained in the present study is compared with the results of Kanoria et al. [Reference Kanoria, Dolai and Mandal11] considering ${h/H = 1,~L/H = 0.0000001,~ A/H = 0.2,~B/H = 1,~D/H=0.2.}$ With these values of parameters $h/H,~L/H,~ A/H ,~B/H,~D/H$ , the $\sqcap $ -shaped breakwater reduces to a rectangular box present in water with a uniform bottom. The reflection coefficient $|R|$ thus obtained agrees very well with the results of Kanoria et al. [Reference Kanoria, Dolai and Mandal11].

Figure 4

Comparison between the present method and [Reference Kanoria, Dolai and Mandal11, Figure 2(a)] for $B/H = 1,A/H = 0.2, D/H=0.2$ .

In Figure 5, $|T|$ obtained from the present analysis is compared with the results of Singh et al. [Reference Singh, Kaligatla and Mandal20] for $h/H = 1,~L/H = 0.005 ,~A/H = 0.25,~B/H = 0.5,~D/H=0.45$ . A good matching of the result is observed in the figure.

Figure 5

Comparison between the present method and [Reference Singh, Kaligatla and Mandal20, Figure 6] for $B/H = 0.5,~A/H = 0.25, D/H=0.45$ .

Porter and Evans [Reference Porter and Evans18] considered the problem of water wave diffraction by a thin rigid barrier partially immersed in water of uniform finite depth using a multiterm Galerkin approximation. In Figure 6, $|R|$ obtained from the present analysis is compared with the results of Porter and Evans [Reference Porter and Evans18] for $h/H = 1,~~A/H = 0.5,~B/H = 0.05, D/H=0.5$ . A good matching of $|R|$ is observed in the figure.

Figure 6

Comparison between the present method and [Reference Porter and Evans18, Figure 2] for $B/H = 0.05,~A/H = 0.5, D/H=0.5$ .

Thus, Figures 4, 5, 6 establish the authenticity of the results obtained in the present study.

4.4 Effect of length of side wall of the $\large \boldsymbol \sqcap $ -shaped breakwater on the waves

Figures 7 and 8 illustrate the effect of a submerged length of a rectangular box of the $\large \boldsymbol \sqcap $ -shaped breakwater on the reflection coefficient and transmission coefficients. Here, $|R|,~|T|$ are plotted against the nondimensional wavenumber $KH$ for two different draughts of rectangular $\large \boldsymbol \sqcap $ -shaped breakwater $(A/H = 0.3,~0.6)$ and fixed values of $B/H=0.3, ~D/H=0.8,~L/H=1.3, ~h/H=1.3$ .

Figure 7

$|R|$ against $KH$ for $B/H=0.3, ~D/H=0.8,~L/H=1.3, ~h/H=1.3$ , for various $A/H$ .

Figure 8

$|T|$ against $KH$ for $B/H=0.3,~D/H=0.8,~L/H=1.3, ~h/H=1.3$ , for various $A/H$ .

From these figures, it is observed that $|R|,~|T|$ exhibit oscillatory behaviour for various values of $A/H$ and, for certain values of wavenumber $KH$ , $|R|$ vanishes and $|T|$ becomes unity as is evident from Figure 7. Also, Figure 8 shows that $|T|$ vanishes and $|R|$ becomes unity for certain values of $KH$ . It is also observed that as $A/H$ increases, a phase shift occurs in oscillations of $|R|$ and $|T|$ . The zeros of $|R|$ shift to the left and the zeros of $|T|$ shift to the right as $A/H$ increases.

4.5 Effect of width of the trench and the width of the breakwater

Figures 9(a)–12(b) illustrate the effect of the width of the trench and that of the breakwater on $|R|$ and $|T|$ .

Figure 9

$|R|$ against $KH$ for $A/H=0.25, ~D/H=0.45,~L/H=0.8, ~h/H=1.2$ , for various $B/H$ .

Figures 9(a)–10(b) depict $|R|$ and $|T|$ against the wavenumber $KH$ for different widths of the breakwater $B/H= 0.5,\, 0.8,\, 1.5$ when the width of the trench is ${L/H=0.8}$ and $A/H=0.25,~ D/H=0.45,~ h/H=1.2$ .

Figure 10

$|T|$ against $KH$ for $A/H=0.25, ~D/H=0.45,~L/H=0.8, ~h/H=1.2$ , for various $B/H$ .

Figures 11(a) and 12(b) depict $|R|$ and $|T|$ against the wavenumber $KH$ for various widths of the breakwater $B/H= 1.5,~ 2.5,~ 3.0$ when the width of the trench is ${L/H= 2.5}$ and $A/H=0.25,~ D/H=0.45,~ h/H=1.2$ .

Figure 11

$|R|$ against $KH$ for $A/H=0.25, ~D/H=0.45,~L/H=2.5, ~h/H=1.2$ , for various $B/H$ .

Figure 12

$|T|$ against $KH$ for $A/H=0.25, ~D/H=0.45,~L/H=2.5, ~h/H=1.2$ , for various $B/H$ .

We may mention here that Figures 9(b), 10(b), 11(b) and 12(b) depict $|R|$ and $|T|$ when the width of the breakwater is equal to the width of the trench. It may be noted here that when the width of the trench is equal to the width of the breakwater, that is, when $B=L$ , then Regions $R2$ and $R3$ coincide and in the boundary value problem (2.1)–(2.10), B is replaced by L in (2.1)–(2.3), (2.6)–(2.10) and the conditions (2.4)–(2.5) modify to

$$ \begin{align*} \phi_x=0 \quad \text{on~} x=\pm L,~0<y<D\quad \text{in Region R1, L1,} \end{align*} $$

$$ \begin{align*} \phi_x=0 \quad \text{on~} x=\pm L,~A<y<D\quad \text{in Region R2, L2,} \end{align*} $$

conditions on the horizontal side of the breakwater to

$$ \begin{align*} \phi_y=0 \quad \text{on~} y=A,~-L<x<L \quad \text{in Region R2, L2.} \end{align*} $$

A schematic diagram for when $B=L$ is given in Figure 13.

Figure 13

Subregions of the fluid domain when $B=L$ .

From Figures 9(a)–12(b), the following observations are made.

(i) When the width of the breakwater is equal to the width of the trench ${(B/H=L/H)}$ , $|R|$ and $|T|$ show uniform smooth oscillations as is evident from Figures 9(b), 10(b), 11(b) and 12(b). The frequency of oscillation is more for a wide trench $L/H= 2.5$ than a short trench $L/H= 0.8$ .

(ii) For a short trench, when the width of the breakwater is less than the width of the trench, $(B/H=0.5,~L/H=0.8)$ , $|R|$ and $|T|$ show nonuniform oscillations with $|R|$ vanishing and $|T|$ becoming unity at $KH \simeq 1.8$ , and $|T|$ vanishing and $|R|$ becoming unity at $KH \simeq 1.0$ . For $KH>1.8,$ the amplitude of oscillation of $|R|$ decreases as $KH$ increases.

However, when the width of the breakwater is more than the width of the trench $(B/H=1.5,~L/H=0.8)$ , $|R|$ and $|T|$ show nonuniform oscillations and a resonating behaviour in $|R|$ is observed for $KH \simeq 1.8$ .

(iii) For a wide trench, when the width of the breakwater is less than the width of the trench $(B/H=1.5,~L/H=2.5)$ , $|R|$ and $|T|$ show nonuniform oscillations with $|R|$ vanishing sharply at $KH \simeq 0.8$ and a resonating behaviour in $|R|$ is observed for $KH \simeq 2.0$ . However, when the width of the breakwater is more than the width of the trench $(B/H=3.0,~L/H=2.5)$ , $|R|$ and $|T|$ show nonuniform oscillations and, in this case, the frequency of oscillations is more than in the case when the width of the breakwater is less than the width of the trench.

Thus, when the width of the trench is equal to the width of the breakwater, the waves interact uniformly with the trench and the breakwater producing smooth uniform oscillations in the reflected and transmitted waves. However, when the width of the trench and the width of the breakwater are unequal, nonuniform, nonsmooth oscillations in the reflected and transmitted waves are observed and, in some cases, a resonance in the reflected wave is observed. These phenomena occur due to the interaction of waves with the trench and the breakwater.

(iv) For a trench with a particular width, it is observed that the frequency of oscillations increases with the increase of width of the breakwater. This is due to the fact that as the width of the breakwater increases, more interaction of waves with the breakwater and the nonuniform bottom occurs which results in increasing the frequency of oscillations in the reflected and transmitted waves.

4.6 Effect of depth of the trench

Figures 14 and 15 illustrate the effect of the depth of the trench on the reflection and transmission coefficients. Here, $|R|,~|T|$ are depicted against the nondimensional wavenumber $KH$ for various values of $h/H$ , that is, $(h/H = 1.3,~2.3)$ and for fixed values of $A/H=0.4, ~B/H=0.4,~D/H=0.6, L/H=1.1$ . From the figures, it is observed that when the width of the breakwater is less than the width of the trench, $|R|$ and $|T|$ show nonuniform oscillatory behaviour. However, when the width of the breakwater is equal to the width of the trench $(L/H=B/H=1.1)$ , $|R|$ and $|T|$ show uniform oscillatory behaviour as is evident from Figures 16 and 17. Thus, it is seen that the impact of the breakwater and the trench, when the width of the breakwater is equal to the width of the trench, produces uniform oscillations on the reflected and transmitted waves rather than when the width of the breakwater is not equal to the width of the trench.

Figure 14

$|R|$ against $KH$ for $A/H=0.4, ~B/H=0.4,~D/H=0.6, ~L/H=1.1$ , for various $h/H$ .

Figure 15

$|T|$ against $KH$ for $A/H=0.4, ~B/H=0.4,~D/H=0.6, ~L/H=1.1$ , for various $h/H$ .

Figure 16

$|R|$ against $KH$ for $A/H=0.4, ~B/H=1.1,~D/H=0.6, ~L/H=1.1$ , for various $h/H$ .

Figure 17

$|T|$ against $KH$ for $A/H=0.4, ~B/H=1.1,~D/H=0.6, ~L/H=1.1$ , for various $h/H$ .

4.7 Effect of trench on the reflection and transmission coefficients

If we compare the behaviour of $|R|$ and $|T|$ due to $\large \boldsymbol \sqcap $ -shaped breakwater present in water of uniform finite depth as studied by Singh et al. [Reference Singh, Kaligatla and Mandal20] with that due to the presence of a trench as analysed in the present study, it is found that the presence of a trench produces oscillatory behaviour in the reflection and transmission coefficients, and for certain values of wavenumber, the phenomena of total reflection, zero transmission or total transmission and zero reflection occur. Thus, we can infer that the phenomenon of water wave scattering by a $\large \boldsymbol \sqcap $ -shaped breakwater in the presence of a trench produces oscillations in the reflection and transmission coefficients which is in contrast with the behaviour of reflection and transmission coefficients in the absence of a trench as observed in [Reference Singh, Kaligatla and Mandal20].

4.8 Horizontal and vertical forces on the breakwater

Figures 18 and 19 illustrate the effect of the length of the side wall ( $D/H$ ) of the $\large \boldsymbol \sqcap $ -shaped breakwater on horizontal force $W_{h}$ and vertical force $W_{v}$ . Here, $W_{h}, ~W_{v}$ are plotted against the nondimensional wavenumber $KH$ for two different lengths of the side wall ( $D/H = 0.5,~ 0.7$ ) of the $\large \boldsymbol \sqcap $ -shaped breakwater for $(A/H = 0.3)$ and fixed values of $L/H=1.3, ~B/H=0.8,~h/H=1.3$ .

Figure 18

$W_{h}$ against $KH$ for $L/H=1.3, ~B/H=0.8,~A/H=0.3, ~h/H=1.3$ , for various $D/H$ .

Figure 19

$W_{v}$ against $KH$ for $L/H=1.3, ~B/H=0.8,~A/H=0.3, ~h/H=1.3$ , for various $D/H$ .

From Figures 18 and 19, it is observed that for a fixed width of the breakwater, as the length of side wall ( $D/H$ ) of the $\large \boldsymbol \sqcap $ -shaped breakwater increases, the horizontal force increases and the vertical force decreases. This is expected because as the length of side wall ( $D/H$ ) of the $\large \boldsymbol \sqcap $ -shaped breakwater increases, the impact of incident waves on the side walls of the breakwater increases substantially and so the horizontal force on the breakwater increases.

Kaligatla et al. [Reference Kaligatla, Singh and Mandal10] observed that for obliquely incident waves with angle of incidence as $30^\circ $ on a $\large \boldsymbol \sqcap $ -shaped breakwater present in deep water, the vertical force on the breakwater is more than the horizontal force. Also, they showed that the forces decrease with an increase of the width of the breakwater.

In the present study, in Figures 20 and 21, the horizontal and vertical forces $W_{h},W_{v}$ are plotted against the nondimensional wavenumber $KA$ for two different widths of the breakwater $B/A (=0.5,1)$ and for a normally incident wave train on the $\large \boldsymbol \sqcap $ -shaped breakwater present in deep water $H/A=h/A=4$ in the absence of a trench ( ${L/A=0.001}$ ).

Figure 20

$W_{h}$ against $KA$ for $L/A=0.001, ~D/A=2,~H/A=4, ~h/A=4$ , for various $B/A$ .

Figure 21

$W_{v}$ against $KH$ for $L/A=0.001, ~D/A=2,~H/A=4, ~h/A=4$ , for various $B/A$ .

Here, in the absence of a trench, when the water region is deep ( $H/A=h/A=4$ ), it is observed that for a fixed width of the breakwater, the vertical force is more than the horizontal force and this is significant for higher wavenumber. This phenomenon is similar to that observed by Kaligatla et al. [Reference Kaligatla, Singh and Mandal10, Figure 9] when the angle of incidence is $30^\circ $ . However, this behaviour of the forces depicted in Figures 20 and 21 is in contrast with the behaviour of the forces exerted on $\large \boldsymbol \sqcap $ -shaped breakwater present in the water region with a trench (see Figures 18 and 19) where the horizontal force exerted on the breakwater is more than the vertical force for any length of the side wall of the breakwater. This change in the behaviour of the forces illustrates the effect of the depth of the water region. For a uniformly deep water region, the large water column below the breakwater exerts a pressure force below the breakwater vertically. Due to this reason, the vertical force exceeds the horizontal force when the water is deep.

Thus, we can conclude from Figures 18–21 that due to presence of a trench: (1) the horizontal force increases and vertical force decreases with the increase in length of the side wall of the $\large \boldsymbol \sqcap $ -shaped breakwater; and (2) for a fixed dimension of the breakwater, when the bottom of the water region has nonuniform bathymetry (that is, the presence of a trench), the horizontal force is more than the vertical force. However, in the absence of a trench, when a $\large \boldsymbol \sqcap $ -shaped breakwater is present in uniformly deep water, the vertical force exerted on the breakwater is more than the horizontal force.

Figures 22 and 23 illustrate the horizontal and vertical forces exerted on the breakwater without the legs, that is, the $\large \boldsymbol \sqcap $ -shaped breakwater takes the shape of a rectangular box. From these figures, it is seen that for a rectangular breakwater present in uniformly deep water (without any trench), the horizontal force is less than the vertical force. This is also due to the reason that for deep water, the long column of water below the breakwater exerts a pressure force normally on it. This phenomenon was also observed by Kaligatla [Reference Kaligatla, Singh and Mandal10] when the angle of incidence is $30^\circ $ .

Figure 22

$W_{h}$ against $KA$ for $L/A=0.001, ~D/A=1,~H/A=4, ~h/A=4$ , for various $B/A$ .

Figure 23

$W_{v}$ against $KH$ for $L/A=0.001, ~D/A=1,~H/A=4, ~h/A=4$ , for various $B/A$ .

4.9 Velocity diagram as time elapses

Figures 24(a), 24(b), 24(c) capture the fluid velocity in the $(X=x/H,~ Y=y/H)$ plane, just after the motion started, at nondimensional time $\omega t=10^{-4}$ , while Figures 25(a), 25(b), 25(c) depict fluid velocity at nondimensional time $\omega t=3$ in the Regions R3, R2, R1, respectively, for ${A}/{H}=0.3, {D}/{H}=0.8,~ {B}/{H}=0.75,~ {L}/{H}=1.5,~ {h}/{H}=1.5.$

Figure 24

Fluid velocity at time $\omega t=10^{-4}$ in (a) Region R3, (b) Region R2, (c) Region R1 when ${A}/{H}=0.3,~{D/}{H}=0.8,~{B}/{H}=0.75,~{L}/{H}=1.5,~{h}/{H}=1.5$ .

Figure 25

Fluid velocity at time $\omega t=3$ in (a) Region R3, (b) Region R2, (c) Region R1, when ${A}/{H}=0.3,~{D}/{H}=0.8,~{B}/{H}=0.75,~{L}/{H}=1.5,~{h}/{H}=1.5$ .

Figures 24(a) and 25(a) represent the fluid velocity at time $\omega t=10^{-4},~3$ , respectively, in the Region R3 where R3: $0\leq X \leq {B}/{H}=0.75,~ {A}/{H}=0.3\leq Y \leq {h}/{H}=1.5.$

Figures 24(b) and 25(b) represent the fluid velocity at time $\omega t=10^{-4},~3$ , respectively, in the Region R2 where R2: $0.75\leq X \leq {L}/{H}=1.5,~ 0\leq Y \leq {h}/{H}=1.5.$

Figures 24(c) and 25(c) represent the fluid velocity at time $\omega t=10^{-4},~3$ , respectively, in the Region R1 where R1: ${L}/{H}\leq X \leq \infty ,~ 0\leq Y \leq {H}/{H}=1.$