2 Mathematical formulation

Figure 1 shows the schematic of the H-shaped breakwater submerged in deep water. The impermeable H-shaped breakwater comprises two vertical thin plates connected by a horizontal thin plate. To avoid edge effects, the breakwater is assumed to be indefinitely long in the z-direction. In other dimensions, the breakwater has width $2b$ , depth of submersion c and the breakwater occupies up to d. Thus, the wave-scattering problem in this article is formulated as a two-dimensional problem rather than a three-dimensional problem. Since the H-shaped breakwater has a symmetrical configuration, the entire fluid region is split into three regions, as shown in Figure 1. Waves on the water surface shall impinge on the breakwater either from the positive x-direction or negative x-direction, and the wave scattering is identical by the symmetrical breakwater. The fluid motion is governed by the Laplace equation, which results from the equation of continuity under the assumptions of an incompressible and inviscid fluid with irrotational motion. Furthermore, waves are assumed to have small wave steepness. When a train of monochromatic waves, incoming from the positive x-direction, impinge on the fixed H-shaped breakwater obliquely at an angle $\alpha $ , they will be scattered in different directions. So, the problem involves determining the wave reflection and transmission amplitudes. Because the geometry of the breakwater is invariant in the z-direction, we seek a periodic solution in the $ z$ -direction. The z part can be separated from the solution in this case.

Figure 1

Sketch of an H-shaped breakwater submerged in deep water. Region 1: $\{(x,y): b<x<\infty , 0<y<\infty \}$ and $\{(x,y):-\infty <x<-b, 0<y<\infty \}$ ; Region 2: $\{(x,y): 0<x<b, 0<y<h\}$ and $\{(x,y):-b<x<0, 0<y<h\}$ ; Region 3: $\{(x,y): 0<x<b, h<y<\infty \}$ and $\{(x,y):-b<x<0, h<y<\infty \}$ . In each region, $-\infty <z<\infty $ is assumed.

Based on the fluid and wave assumptions, we write the velocity potential in the form , where $\mathit {i}=\sqrt {-1}$ . Here, $\upsilon =K\sin {\alpha }$ with wavenumber K, and $\omega $ is the angular frequency and t is time. In deep water, the wavenumber with gravitational constant . The expression $\mathit {e}^{\mathit {i}(\upsilon z-\omega t)}$ is suppressed throughout the analysis. Since the breakwater has an invariant configuration in the z-direction, matching of the solutions at the interface between regions is possible by Snell’s law [Reference Kirby and Dalrymple13].

The entire fluid motion is governed by the modified Helmholtz equation

(2.1)

where $\partial ^2_{x}$ and $\partial ^2_{y}$ are the second-order partial derivatives with respect to variables x and y, respectively. The free-surface boundary condition is

(2.2)

Conditions on each vertical side of the breakwater are

(2.3)

(2.4)

(2.5)

Conditions on the horizontal side of the breakwater are

(2.6)

(2.7)

The deep-water condition is

(2.8)

The edge conditions [Reference Mandal and Chakrabarti16] are

(2.9)

where $r=\sqrt {(x\mp b)^2+(y-c)^2}$ and $s=\sqrt {(x\pm b)^2+(y-d)^2}$ .

The far-field conditions are

(2.10)

where the incident wave with . The constants R and T are the coefficients of wave reflection and transmission to be determined.

3 Reduction to integral equations

When the breakwater geometry is symmetric with respect to some coordinate axis, the velocity potential is usually split into symmetric and antisymmetric functions (see [Reference Linton and McIver15, Ch. 2]). Here, we write the function as the sum of symmetric and antisymmetric parts and , respectively. That is,

where

(3.1)

Thus, the domain of the scattering problem is restricted to the region $x \geq 0$ only. Now, and satisfy (2.1)–(2.7). From (3.1),

(3.2)

(3.3)

Let the behaviour of be represented by

(3.4)

where is read as . The constants $R^s$ and $R^a$ are to be determined. From (2.10) and (3.4), we find that

Now, the Havelock expansions [Reference Havelock10] of satisfying (2.1)–(2.8), (3.2) and (3.3) (for $x \geq 0$ ) are the following. The expansions are usually deduced by means of separation of variables in each region.

Region 1  $(x> b,\; 0 < y < \infty )$ :

(3.5)

where and .

Region 2  $(0 < x < b,\; 0 < y < h)$ :

(3.6)

where $\alpha _0,\pm \mathit {i}\alpha _n\; (n=1,2,3,\ldots ) $ satisfy $\alpha \tanh {\alpha h}-K=0$ , , , $n=1,2,3,\ldots .$

Region 3  $(0 < x < b,\; h < y < \infty )$ :

(3.7)

Let be the horizontal component of fluid velocity across the line $x=b$ , that is,

(3.8)

then

Also, since

(3.9)

we find that

From the edge condition (2.9), we find that

On using the expansion (3.5) for in (3.8) and by Havelock’s inversion formula,

(3.10)

In the same way, on using the expansion (3.6) for in (3.9) and by Havelock’s inversion formula, we obtain the integral relations

Substituting the expansion (3.7) for into (3.9) and using Fourier cosine transformation, we find the integral relations

Fluid pressure is continuous across the line $x=b$ for all $y\in (0,c)\cup (d,\infty )$ , that is,

This matching condition ultimately yields the first-kind Fredholm-type integral equations

(3.11)

where

with

It should be noted that the expression of $\mathcal {M}^{a}(y,u)$ is obtained by replacing “ $\cot $ ” by “ $-\tan $ ” and “ $\coth $ ” by “ $\tanh $ ” in $\mathcal {M}^{s}(y,u)$ . Next, the substitution of the solution into (3.10) gives rise to

(3.12)

where

$$ \begin{align*} \mathcal{P}^{s,a}=-\displaystyle\frac{\mathit{i}\cos{\alpha}}{2} \frac{1-R^{s,a}}{1+R^{s,a}}. \end{align*} $$

The real valued nature of the kernels $\mathcal {M}^{s,a}(y,u)$ implies that have a similar nature to the integral equation (3.11). As such, the relation (3.12) shows that $\mathcal {P}^{s,a}$ are also real valued. The integral expressions of $\mathcal {P}^{s,a}$ shall be evaluated once the solutions of the integral equation (3.11) are obtained. Subsequently, the required wave reflection and transmission coefficients can be estimated from the expressions

$$ \begin{align*} |R|=\frac{|1+(2 \sec{\alpha})^2\mathcal{P}^s \mathcal{P}^a|}{\sqrt{\{1+(2 \mathcal{P}^s \sec{ \alpha})^2\} \{(1+(2 \mathcal{P}^a \sec{\alpha})^2 \}}}, \end{align*} $$

$$ \begin{align*} |T|=\frac{2 |\mathcal{P}^s -\mathcal{P}^a| \sec{ \alpha}}{\sqrt{\{1+(2\mathcal{P}^s \sec{ \alpha})^2\} \{(1+(2 \mathcal{P}^a \sec{\alpha})^2 \}}}, \end{align*} $$

satisfying the energy conservation equation $|R|^2+|T|^2=1$ .

4 Approximate solution of the integral equations

In this section, the Galerkin approximation technique is illustrated to solve the integral equations (3.11) associated with wave scattering by the H-shaped breakwater. This approximation is constructed by using simple polynomials multiplied by a suitable weight function. We choose as

(4.1)

where $ \{1-{y}/{c}\}^{-{1}/{2}}$ and $ \{{y}/{d}-1\}^{-{1}/{2}}$ are the weight functions, N is any nonnegative integer, and $\alpha ^{s,a}_p$ and $\beta ^{s,a}_p$ are unknown constants. Here, $ ({y}/{c})^p \mathit {e}^{-K (p+1) y}$ and $ ({y}/{d})^p \mathit {e}^{-K (p+1) y}$ are the basis functions. The approximation is chosen based on the behaviour of at the edges $y=c \; \mbox {and}\; y=d$ . Note that the exponential term $\mathit {e}^{-K (p+1) y}$ in the expansion preserves the decay properties of Laguerre functions, which can be used alternatively. With this exponential term, rapid convergence of the series shall be achieved with a few terms in the expansion (4.1) which, in turn, shows the accuracy of the method.

We rewrite as

(4.2)

where

(4.3) $$ \begin{align} \psi_{1p}(y)=\bigg\{1-\bigg(\frac{y}{c}\bigg)\bigg\}^{-{1}/{2}}\mathit{e}^{-K (p+1) y} \bigg(\frac{y}{c}\bigg)^p, \quad \psi_{2p}(y)=\bigg\{\bigg(\frac{y}{d}\bigg)-1\bigg\}^{-{1}/{2}}\mathit{e}^{-K (p+1) y} \bigg(\frac{y}{d}\bigg)^p. \end{align} $$

We substitute the expansion (4.2) in (3.11) and then multiply by $\psi _{1q}(y)$ and integrate from $0$ to c to obtain the linear system of equations

(4.4) $$ \begin{align} \sum_{p=0}^{N} \alpha^{s,a}_p \mathcal{A}^{s,a}_{pq}+\sum_{p=0}^{N} \beta^{s,a}_p \mathcal{B}^{s,a}_{pq}=\mathcal{H}_{1q}, \quad (q = 0, 1,\ldots, N), \end{align} $$

where

with

Similarly, we substitute the expansion (4.2) in (3.11) and then multiply by $\psi _{2q}(y)$ , and integrate from d to $\infty $ to obtain the linear system of equations

(4.5) $$ \begin{align} \sum_{p=0}^{N} \alpha^{s,a}_p \mathcal{C}^{s,a}_{pq}+\sum_{p=0}^{N} \beta^{s,a}_p \mathcal{D}^{s,a}_{pq}=\mathcal{H}_{2q}, \quad (q = 0, 1,\ldots, N), \end{align} $$

where

with

The expression for $\mathcal {A}^{a}_{pq}$ and $\mathcal {C}^{a}_{pq}$ shall be obtained by simply replacing “ $\coth $ ” by “ $\tanh $ ” and “ $\cot $ ” by “ $-\tan $ ” in $\mathcal {A}^{s}_{pq}$ and $\mathcal {C}^{s}_{pq}$ , respectively. The expression for $\mathcal {B}^{a}_{pq}$ and $\mathcal {D}^{a}_{pq}$ shall be obtained by simply replacing “ $\coth $ ” by “ $\tanh $ ” in $\mathcal {B}^{s}_{pq}$ and $\mathcal {D}^{s}_{pq}$ , respectively.

Next, the substitution of the expansion equation (4.2) in (3.12) yields

(4.6) $$ \begin{align} \mathcal{P}^{s,a} \approx \mathcal{P}_N^{s,a} = \sum_{p=0}^N \alpha_p^{s,a} \mathcal{H}_{1p}+\sum_{p=0}^N \beta_p^{s,a} \mathcal{H}_{2p}. \end{align} $$

Thus, after solving the system of equations (4.4) and (4.5) for the unknown constants $\alpha _p^{s,a}$ and $\beta _p^{s,a}$ ( $p = 0, 1,\ldots , N$ ), these solutions can be used to obtain $\mathcal {P}^{s,a}$ from (4.6). In addition, numerical estimates of the reflection and transmission coefficients $|R|$ and $|T|$ can be found subsequently.

5 Computational results

The solution method with two-term Galerkin approximation (that is, $N=1$ ) is considered as it appears to be sufficient here. The computation is carried out with MATHEMATICA programming. The wave reflection and transmission coefficients $(|R|\;\mathrm {and} \;|T|)$ are computed and illustrated graphically with various values of the nondimensional parameters: $Kd$ , $b/d$ , $c/d$ for H-shaped breakwater, where d is the total length of the breakwater; $Kd$ , $b/d$ , $h/d$ for submerged $\large \boldsymbol {\sqcap }$ -shaped breakwater, where d is the total length of breakwater; $Kh$ , $b/h$ , $c/h$ for submerged $\large \boldsymbol {\sqcup }$ -shaped breakwater having the total length h. The effect of these relative parameters on wave scattering is studied. We show the convergence of scattering coefficients with $N=1$ for different wavenumbers $Kd$ in Table 1. It may be noted that the values of the wavenumber $Kd$ are kept moderate here. For a very high value of the wavenumber (high frequency), the waves will be confined within a thin layer below the free surface (and well above the submerged breakwater) so that most of the incident wave energy will be transmitted and very little will be reflected back. The gravitational constant is fixed in the calculation. The integrals that arise in the approximate solution of integral equations are evaluated by applying the Gauss–Laguerre quadrature with 11 points. In addition, the horizontal wave force ( $F_x$ ), vertical wave force ( $F_y$ ) on the breakwater and the overturning moment (OTM) of the breakwater are calculated in magnitude by using the nondimensional formulae [Reference Naftzger and Chakrabarti22]

(5.1)

where A is the incident wave amplitude and

(5.2)

(5.3)

(5.4)

Table 1

Convergence in $|R|$ and $|T|$ of H-shaped breakwater when $h/d=0.5$ , $c/d=0.01$ , $b/d=1.0$ and $\alpha =30^\circ $ . Three decimal places of accuracy is achieved with $N=1$ .

Note that the symmetric potential function has no effect on the horizontal wave force and the OTM, whereas the vertical wave force depends only on the symmetric potential function.

First, we validate the developed method of solution by producing a known benchmark reflection coefficient $(|R|)$ of a fully submerged breakwater of a rectangular cross section whose length is infinite, established by Mei and Black [Reference Mei and Black19]. This case is considered with the H-shaped breakwater taking into account $h/c=1.0$ and a large value of $d(=150\;\mbox {m})$ . This limiting case is reasonable with the H-shaped configuration. The reflection coefficient $|R|$ , depicted in Figure 2, almost matches that of [Reference Mei and Black19, Figure 2] concerning a bottom-standing breakwater of infinite length. Thus, the accuracy of the proposed Galerkin approximation is assured with a single term and the correctness of the computation is justified.

Figure 2

Reflection coefficient of Mei and Black [Reference Mei and Black19] for a fully submerged breakwater of rectangular cross section versus wavenumber $Kc$ when $h/c=1.0$ , $\alpha =0^{\circ }$ , $d=150\;\mbox {m}$ .

Next, we validate our solution method again in a different limiting case that illustrates wave scattering by a thin vertical barrier of finite length fully submerged below the free surface of deep water. This case is concerned with Evans [Reference Evans6], who demonstrated water-wave scattering by a thin submerged vertical plate for normal incident waves ( $\alpha =0^\circ $ ). We used his wave reflection and transmission coefficients ( $|R|$ , $|T|$ ) to validate our method, illustrated in Figure 3(a). The H-type breakwater shrinks to a thin barrier when $b/d=10^{-6}$ , which is sufficient to show Evans’ [Reference Evans6] scattering coefficients for $c/d=0.01,0.05, 0.25$ , where $d=10\;\mbox {m}$ . In Figure 3(a), the present method is proved to show the reflection and transmission coefficients of Evans exactly. Thus, the correctness of the computation is confirmed from Figure 3(a) in the case of a thin submerged barrier as well.

Figure 3

(a) Reflection and transmission coefficients as in [Reference Evans6] when $b/d=10^{-6}$ . (b) Reflection coefficients when $c/d=0.05$ versus wavenumber $Kd$ with $h/d=0.5$ and $\alpha =0^\circ $ .

In addition, Figure 3(b) shows how a variation in the reflection coefficient $|R|$ occurs when the H-shaped breakwater gradually shrinks to a thin vertical barrier ( ${b/d=10^{-6}}$ ). This demonstrates that $|R|$ starts from one for the H-shaped breakwater having a certain width, and zero or minimum in $|R|$ occurs for some incident wavelengths. It is to be noticed here that so long as the gap $(b/d)$ remains moderate but small, the H-shaped barrier consists of a pair of thin vertical barriers for which the reflection coefficient $|R|$ is very near to unity for very long waves ( $ Kd\rightarrow 0$ ) as is well known for two submerged thin vertical barriers [Reference De, Mandal and Chakrabarti3]. The incident wavelengths of zero reflection gradually vanish as the H-shaped breakwater becomes a thin vertical barrier, and $|R|$ ultimately tends to Evans’ reflection curve for $b/d=10^{-6}$ , which starts from zero value. Somewhat analogous behaviour of $|R|$ in the very low wavenumber ( $ Kd\rightarrow 0$ ) range was observed by Tuck [Reference Tuck35] as the depth of the water changes from finite to infinite. Thus, from these observations, we point out that the H-shaped breakwater results in higher wave reflection for the most incident wavelengths.

In the following, we show the computational results of the H-, $\large \boldsymbol {\sqcap }$ - and $\large \boldsymbol {\sqcup }$ -shaped breakwaters. The parameter $d=10\;\mbox {m}$ is fixed for all the results presented here.

Considering the gap between the H-shaped breakwater and the free surface is very small, the reflection and transmission coefficients $(|R|\;\mathrm {and}\;|T|)$ are shown in Figure 4 as functions of wavenumber $Kd$ to study the effect of width $b/d$ . Here, we compared our results with that of De et al. [Reference De, Mandal and Chakrabarti4], who reconsidered the problem of wave scattering by a breakwater consisting of two thin vertical barriers piercing the free surface of deep water without a horizontal plate between them. For numerical results, we set $h/c=0.5$ , $\alpha =0^\circ $ and $b/d=1,3,5$ as in [Reference De, Mandal and Chakrabarti4]. According to De et al. [Reference De, Mandal and Chakrabarti4], increased space between vertical barriers leads to increased oscillation behaviour in the wave reflection coefficient. The breakwater was shown to reflect short waves considerably, but not long waves in deep water. However, the surface-piercing H-shaped breakwater considerably reflects both long and short waves. The horizontal plate between the two vertical plates accounts for this higher reflection. Since more oscillations arise in $|T|$ as the width $b/d$ widens, and cause zero wave reflection or full wave transmission at some wavelengths, the H-shaped breakwater with smaller values of $b/d$ may be employed to achieve low wave transmission.

Figure 4

Reflection and transmission coefficients of the surface piercing H-shaped breakwater versus wavenumber $Kd$ for $c/d=10^{-4}$ , $h/d=0.5$ and $\alpha =0^\circ $ .

In Figure 5, we plot the reflection and transmission coefficients $(|R|\;\mathrm {and}\;|T|)$ versus wavenumber $Kd$ for the effect of the gap ( $c/d$ ) between the H-shaped breakwater and the free surface when $b/d = 1.0$ and $h/d = 0.5$ . As expected, wider gaps allow more waves to be transmitted, and, as a result, they yield less wave reflection. Furthermore, it reveals that a few incoming long waves fully transmit over the H-type breakwater. In spite of this, incoming waves of almost all wavelengths are greatly attenuated by this breakwater.

Figure 5

Reflection and transmission coefficients of the submerged H-shaped breakwater versus wavenumber $Kd$ for $b/d=1.0$ , $h/d=0.5$ and $\alpha =30^\circ $ .

Figure 6 depicts the reflection and transmission coefficients $(|R|\;\mathrm {and}\;|T|)$ versus the wavenumber $Kd$ for different incident wave angles $\alpha $ when $b/d = 1.0$ , $c/d = 0.25$ and $h/d = 0.5$ . It illustrates that incident waves of higher angles are reflected greatly, except at a few wavelengths.

Figure 6

Reflection and transmission coefficients of the submerged H-shaped breakwater versus wavenumber $Kd$ for $b/d=1.0$ , $c/d=0.25$ and $h/d=0.5$ .

Figures 7 and 8 demonstrate the reflection and transmission coefficients of submerged $\large \boldsymbol {\sqcap }$ -shaped breakwater versus wavenumber $Kd$ when the width $b/d$ varies, and the gap $h/d$ between the free surface and the breakwater varies, respectively. Like the H-shaped breakwater, the $\large \boldsymbol {\sqcap }$ -shaped breakwater shows higher wave reflection and lower wave transmission in an oscillatory manner with varying width $b/d$ . However, the $\large \boldsymbol {\sqcap }$ -shaped one shows the scattering coefficients in an oscillatory manner when $h/d$ also changes. This is not the case with the H-shaped breakwater. Furthermore, a comparison of Figure 7 with Figure 8 reveals that the change in $h/d$ is apparently more effective than the change in width $b/d$ of this type of breakwater for wave attenuation.

Figure 7

Reflection and transmission coefficients of submerged $\large \boldsymbol {\sqcap }$ -shaped breakwater versus wavenumber $Kd$ for $h/d=0.25$ , $\alpha =30^\circ $ .

Figure 8

Reflection and transmission coefficients of submerged $\large \boldsymbol {\sqcap }$ -shaped breakwater versus wavenumber $Kd$ for $b/d=1.0$ , $\alpha =30^\circ $ .

The reflection and transmission coefficients of $\large \boldsymbol {\sqcup }$ -shaped breakwater are depicted with wavenumber $Kh$ for the varying width $b/h$ in Figure 9 and for the varying gap $c/h$ in Figure 10. The scattering performance of $\large \boldsymbol {\sqcup }$ -shaped breakwater is similar to that of H-shaped breakwater. However, a comparison of Figure 10 with Figure 5 reveals that the $\large \boldsymbol {\sqcup }$ -shaped breakwater reflects short waves considerably more than the H-shaped breakwater. This results in even more reduction in wave transmission.

Figure 9

Reflection and transmission coefficients of submerged $\large \boldsymbol {\sqcup }$ -shaped breakwater versus wavenumber $Kh$ for $c/h=0.25$ , $\alpha =30^\circ $ .

Figure 10

Reflection and transmission coefficients of submerged $\large \boldsymbol {\sqcup }$ -shaped breakwater versus wavenumber $Kh$ for $b/h=1.0$ , $\alpha =30^\circ $ .

Figure 11 shows the wave forces and the overturning moment when the width of the H-shaped piercing surface breakwater ( $b/d$ ) changes. These physical quantities correspond to the wave scattering coefficients described in Figure 4. It reveals that the wider H-breakwater undergoes a lesser horizontal force ( $F_x$ ) and vertical force ( $F_y$ ) due to the oscillatory behaviour of the scattering coefficients shown in Figure 4. The incident wavelengths of a minimum wave reflection account for zero horizontal force. Furthermore, only smaller frequency waves cause a higher vertical force. The consequent OTM of the H-breakwater reduces with the increase in its width. Hence, a wider H-breakwater has lesser OTM, which is evident from the fact that a larger-size object needs much effort to spin. Generally, high-frequency waves cause a higher OTM, and low-frequency waves cause a lower OTM. Figure 11 is consistent with this fact. Ultimately, it turns out that the overall OTM of the free-surface H-breakwater with larger widths is much less.

Figure 11

Horizontal force ( $F_x$ ), vertical force ( $F_y$ ) and OTM of the surface piercing H-shaped breakwater associated with Figure 4.

Figure 12 shows the wave forces and OTM when the submerged gap of H-shaped breakwater ( $c/d$ ) changes. These physical quantities correspond to the wave scattering coefficients described in Figure 5. An increase in the submergence gap ( $c/d$ ) leads to a reduction in the horizontal force ( $F_x$ ) and an increment in the vertical force ( $F_y$ ), which is due to the reduction in wave reflection or increased transmission in waves as is evident from Figure 5. Furthermore, the breakwater does not undergo horizontal force where the wave reflection coefficient is zero or minimum. In other words, the breakwater undergoes a higher vertical force where the wave transmission coefficient tends to a maximum. This situation is consistent with wave scattering and wave forces. The resulting OTM reduces with the submergence gap ( $c/d$ ) since the horizontal force ( $F_x$ ) reduces. Higher horizontal wave force leads to higher OTM.

Figure 12

Horizontal force ( $F_x$ ), vertical force ( $F_y$ ) and overturning moment (OTM) of the submerged H-shaped breakwater associated with Figure 5.

Figures 13 and 14 present the effect of the submerged gap lengths of $\large \boldsymbol {\sqcap }$ - and $\large \boldsymbol {\sqcup }$ -shaped breakwaters on wave forces and OTM in the wave scattering situation illustrated in Figures 8 and 10, respectively. The variations in the physical quantities of $\large \boldsymbol {\sqcup }$ -breakwater are analogous to those of the H-breakwater as in Figure 12. However, these physical quantities possess an oscillatory nature in the case of $\large \boldsymbol {\sqcap }$ -breakwater because of the highly oscillatory reflection coefficient, depicted in Figure 8. In addition, these quantities are higher at low-frequency incident waves and lower at high-frequency incident waves, which account for the reversed breakwater configuration.

Figure 13

Horizontal force ( $F_x$ ), vertical force ( $F_y$ ) and overturning moment (OTM) of the submerged $\large \boldsymbol {\sqcap }$ -shaped breakwater associated with Figure 8.

Figure 14

Horizontal force ( $F_x$ ), vertical force ( $F_y$ ) and overturning moment (OTM) of the submerged $\large \boldsymbol {\sqcup }$ -shaped breakwater associated with Figure 10.

Finally, from the above results, one can determine the effect of wave incident angles on wave forces and OTM. Since the increase in wave angle results in higher wave reflection or lower wave transmission, as shown in Figure 6, higher wave angles account for a higher horizontal force and a lower vertical force, and therefore a higher OTM.