4.1 Closed-form solution far from the patch’s plane of symmetry

Here we derive the envelope equations for wave propagation over a bi-chromatic patch, i.e. with bottom corrugations of the form

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D701}=\frac{d_{1}}{2}(\text{e}^{\text{i}\boldsymbol{k}_{b_{1}}\boldsymbol{\cdot }\boldsymbol{ x}}+\text{e}^{-\text{i}\boldsymbol{k}_{b_{1}}\boldsymbol{\cdot }\boldsymbol{x}})+\frac{d_{2}}{2}(\text{e}^{\text{i}\boldsymbol{k}_{b_{2}}\boldsymbol{\cdot }\boldsymbol{ x}}+\text{e}^{-\text{i}\boldsymbol{k}_{b_{2}}\boldsymbol{\cdot }\boldsymbol{x}}),\end{eqnarray}$$

with $d_{1}$ and $d_{2}$ the (real) positive corrugation amplitudes. We assume that the incident wave number $\boldsymbol{k}_{1}$ satisfies both the class I and class II Bragg resonance conditions (which we refer to as class I $_{2}$ ), i.e. ${\mathcal{D}}(\unicode[STIX]{x1D714},|\boldsymbol{k}_{t}|)={\mathcal{D}}(\unicode[STIX]{x1D714},|\boldsymbol{k}_{2}|)=0$ (cf. (2.4) and (2.5)), such that $\boldsymbol{k}_{t}=\boldsymbol{k}_{1}+\boldsymbol{k}_{b_{1}}$ and $\boldsymbol{k}_{2}=\boldsymbol{k}_{t}+\boldsymbol{k}_{b_{2}}$ are free propagating waves. Since the wave $\boldsymbol{k}_{t}$ is the first wave resonated, it can be shown using a regular perturbation method that initially the $\boldsymbol{k}_{t}$ wave amplitude increases linearly in time while the $\boldsymbol{k}_{2}$ wave growth is quadratic.

To obtain the multiple-scale equations for the bi-modal corrugation problem under the assumption of small bottom steepness, we consider a leading-order velocity potential of the form

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D719}^{(1)}=\frac{g}{\unicode[STIX]{x1D714}}\frac{\cosh [k_{1}(z+h)]}{\cosh (k_{1}h)}\left(\frac{A_{1}}{2}\text{e}^{\text{i}\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{ x}}+\frac{A_{t}}{2}\text{e}^{\text{i}\boldsymbol{k}_{t}\boldsymbol{\cdot }\boldsymbol{ x}}+\frac{A_{2}}{2}\text{e}^{\text{i}\boldsymbol{k}_{2}\boldsymbol{\cdot }\boldsymbol{ x}}\right)\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}+\text{c.c.},\end{eqnarray}$$

where the envelope amplitudes $A_{1}$ , $A_{t}$ and $A_{2}$ are functions of the slow variables $(\bar{x},\bar{y},\bar{t})\sim \unicode[STIX]{x1D716}(x,y,t)$ . Similar to the monochromatic problem, we derive the evolutionary equations for the slowly varying amplitudes from the so-called compatibility condition, obtained upon inspection of the second-order problem in terms of the bottom steepness. The derivation is provided in Appendix I, and the final equations read

(4.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}A_{1}+\boldsymbol{C}_{g_{1}}\boldsymbol{\cdot }\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D735}}A_{1}+\frac{A_{1}}{2}\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{C}_{g_{1}}=\text{i}A_{t}\unicode[STIX]{x1D6FA}_{1}, & \displaystyle\end{eqnarray}$$

(4.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}A_{t}+\boldsymbol{C}_{g_{t}}\boldsymbol{\cdot }\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D735}}A_{t}+\frac{A_{t}}{2}\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{C}_{g_{t}}=\text{i}A_{1}\unicode[STIX]{x1D6FA}_{1}+\text{i}A_{2}\unicode[STIX]{x1D6FA}_{2}, & \displaystyle\end{eqnarray}$$

(4.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}A_{2}+\boldsymbol{C}_{g_{2}}\boldsymbol{\cdot }\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D735}}A_{2}+\frac{A_{2}}{2}\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{C}_{g_{2}}=\text{i}A_{t}\unicode[STIX]{x1D6FA}_{2}, & \displaystyle\end{eqnarray}$$

$\boldsymbol{C}_{g_{j}}$

$j=1,t,2$

(4.4a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{1}=\frac{\unicode[STIX]{x1D714}d_{1}\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{k}_{t}}{2k_{1}\sinh 2k_{1}h},\quad \unicode[STIX]{x1D6FA}_{2}=\frac{\unicode[STIX]{x1D714}d_{2}\boldsymbol{k}_{t}\boldsymbol{\cdot }\boldsymbol{k}_{2}}{2k_{1}\sinh 2k_{1}h}.\end{eqnarray}$$

The governing equations (4.3) are valid for slow mean bottom variations, but for mathematical expediency, we will now assume that $h$ is constant (i.e. such that $\bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{C}_{g_{1}}=\bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{C}_{g_{t}}=\bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{C}_{g_{2}}=0$ in (4.3)). The case of a slowly varying mean water depth, of particular interest near the coast where the beach is usually gently sloped (see e.g. Mei, Hara & Yu Reference Mei, Hara and Yu2001), does not change the qualitative behaviour of Bragg resonance and can be in fact estimated relatively simply. The wave–bottom interactions indeed remain resonant even when the mean depth is changing, as long as the Bragg condition, which changes as the depth increases or decreases, is satisfied everywhere locally. As a result, the Bragg resonance can be maintained by adjusting the corrugation wavelength to the local surface wavelength, which is what occurs naturally for sandbar formation on sloped beaches (see e.g. Dulou, Belzons & Rey Reference Dulou, Belzons and Rey2000). Doing this, it is expected that the efficiency of the Bragg reflection (or here deflection) of the total corrugated patch over the varying mean depth (say $h_{s}<h<h_{d}$ ) is less than the efficiency obtained in the constant shallow water case ( $h=h_{s}$ ), but more than in the constant deep water case ( $h=h_{d}$ ) (cf. e.g. Alam Reference Alam2012a ).

The conservation of the wave action equation

(4.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}\left(\frac{|A_{1}|^{2}}{2\unicode[STIX]{x1D714}}+\frac{|A_{t}|^{2}}{2\unicode[STIX]{x1D714}}+\frac{|A_{2}|^{2}}{2\unicode[STIX]{x1D714}}\right)+\bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\left(\boldsymbol{C}_{g_{1}}\frac{|A_{1}|^{2}}{2\unicode[STIX]{x1D714}}+\boldsymbol{C}_{g_{t}}\frac{|A_{t}|^{2}}{2\unicode[STIX]{x1D714}}+\boldsymbol{C}_{g_{2}}\frac{|A_{2}|^{2}}{2\unicode[STIX]{x1D714}}\right)=0,\end{eqnarray}$$

derived from (4.3), is the natural extension of (3.5) to the three-wave interaction problem. By defining the normalized transitional wave amplitude $a_{t}^{\ast }=|A_{t}|\sqrt{\cos \unicode[STIX]{x1D703}_{t}}/a_{0}$ , equation (4.5) reduces to $a_{1}^{\ast ^{2}}+a_{t}^{\ast ^{2}}+a_{2}^{\ast ^{2}}=1$ when $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{t}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{y}\equiv 0$ (cf. (3.6) for $a_{1,2}^{\ast }$ ).

In the case of perfectly tuned waves far from the patch’s plane of symmetry, a closed-form solution can be derived at the steady state. Specifically, assuming $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{t}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{y}\equiv 0$ and enforcing the boundary conditions for the incident wave $A_{1}=a_{0}$ , and resonated waves $A_{t}=A_{2}=0$ (which all propagate in the $+x$ direction) at $x=0$ , we obtain the solution

(4.6) $$\begin{eqnarray}\left(\begin{array}{@{}c@{}}A_{1}\\ A_{t}\\ A_{2}\end{array}\right)=\frac{a_{0}K_{1}K_{2}}{K_{12}}\left(\begin{array}{@{}c@{}}K_{2}/K_{1}+K_{1}/K_{2}\cos K_{12}x\\ \text{i}K_{12}/K_{2}\sin K_{12}x\\ -1+\cos K_{12}x\end{array}\right),\end{eqnarray}$$

where

(4.7a-c ) $$\begin{eqnarray}K_{1}=\unicode[STIX]{x1D6FA}_{1}/(C_{g}\sqrt{\cos \unicode[STIX]{x1D703}_{t}}),\quad K_{2}=\unicode[STIX]{x1D6FA}_{2}/(C_{g}\sqrt{\cos \unicode[STIX]{x1D703}_{t}\cos \unicode[STIX]{x1D703}_{2}}),\quad K_{12}=\sqrt{K_{1}^{2}+K_{2}^{2}}.\end{eqnarray}$$

The envelope amplitudes (4.6) are oscillatory over the patch, but the initial $\boldsymbol{k}_{1}$ wave does not necessarily cede all its energy to the $\boldsymbol{k}_{t}$ or $\boldsymbol{k}_{2}$ waves. For $d_{2}\neq 0$ , the $\boldsymbol{k}_{1}$ wave amplitude reaches a minimum of $A_{1}=a_{0}(K_{2}^{2}-K_{1}^{2})/K_{12}$ at $x=\unicode[STIX]{x03C0}/K_{12}$ , which is zero if and only if

(4.8) $$\begin{eqnarray}K_{2}=K_{1}\quad \Leftrightarrow \quad d_{2}/d_{1}=\sqrt{\cos \unicode[STIX]{x1D703}_{2}}\frac{\cos \unicode[STIX]{x1D703}_{t}}{\cos (\unicode[STIX]{x1D703}_{2}-\unicode[STIX]{x1D703}_{t})},\end{eqnarray}$$

in which case $K_{12}$ reduces to

(4.9) $$\begin{eqnarray}K_{12}=\sqrt{2}K_{1}.\end{eqnarray}$$

Equation (4.8) describes the necessary balance between the efficiency of the $\boldsymbol{k}_{1}\leftrightarrow \boldsymbol{k}_{t}$ and $\boldsymbol{k}_{t}\leftrightarrow \boldsymbol{k}_{2}$ Bragg interactions to achieve a synchronous energy transfer from  $\boldsymbol{k}_{1}$ to  $\boldsymbol{k}_{2}$ : if one of the deflection angles increases relative to the other, which diminishes the corresponding interaction efficiency, then the associated ripple amplitude should also increase in order to balance the effect. The condition (4.8) is illustrated in figure 4(a) where we show the ratio $d_{2}/d_{1}$ as a function of $\unicode[STIX]{x1D703}_{2}$ for four different transitional angles. Clearly $d_{2}/d_{1}$ decreases when $\unicode[STIX]{x1D703}_{t}$ increases since the $\boldsymbol{k}_{t}\leftrightarrow \boldsymbol{k}_{2}$ Bragg interaction becomes more effective with decreasing $\unicode[STIX]{x1D703}_{2}-\unicode[STIX]{x1D703}_{t}$ . The ratio $d_{2}/d_{1}$ also decreases almost monotonically with increasing $\unicode[STIX]{x1D703}_{2}$ since the $\boldsymbol{k}_{2}$ wave’s time spent over the patch, or, in other words, the number of wave–seabed interactions, scales with $\sqrt{1+\tan ^{2}\unicode[STIX]{x1D703}_{2}}$ . For instance, when $\boldsymbol{k}_{1}$ waves travel one unit distance in the $x$ direction, $\boldsymbol{k}_{2}$ waves only move by $\cos \unicode[STIX]{x1D703}_{2}$ in  $x$ .

Figure 4.

(a) Ratio of the corrugation amplitudes $d_{2}/d_{1}$ as a function of $\unicode[STIX]{x1D703}_{2}$ , such that full energy transfer is achieved between the $\boldsymbol{k}_{1}$ wave and the $\boldsymbol{k}_{2}$ wave at the end of a patch of width $w=w_{cr_{2}}=\unicode[STIX]{x03C0}/K_{12}$ (cf. (4.8)). The four different curves correspond to different transitional angles $\unicode[STIX]{x1D703}_{t}$ . (b) The unique optimal transitional angle $\unicode[STIX]{x1D703}_{t}^{opt}$ as a function of $\unicode[STIX]{x1D703}_{2}$ , which minimizes the critical patch width $w_{cr_{2}}$ for fixed total ripple amplitude $(d_{1}+d_{2})/h=0.12$ and normalized water depth $k_{1}h=0.2$ , and provided that (4.8) is satisfied. (c) Plot of the critical patch width $w_{cr_{2}}$ for bi-modal corrugations (2-corr. patch curves; corr. short for corrugation) for various transitional angles (same line-style notation as in figure 4 a). We also plot the critical patch width for a monochromatic patch $w_{cr_{1}}=\unicode[STIX]{x03C0}/(2K_{c})$ (1-corr. patch) for which the corrugation amplitude is $d/h=0.12$ (cf. (3.8)). The critical monochromatic patch width becomes rapidly much larger than the critical bi-chromatic patch width when $\unicode[STIX]{x1D703}_{2}\rightarrow \unicode[STIX]{x03C0}/2$ .

When (4.8) is satisfied, the energy carried by the $\boldsymbol{k}_{1}$ waves transfers to $\boldsymbol{k}_{2}$ waves after travelling the distance $w_{cr_{2}}=\unicode[STIX]{x03C0}/K_{12}$ , which we call the critical bi-chromatic patch width. For fixed $\unicode[STIX]{x1D703}_{2}$ and normalized total ripple amplitude $(d_{1}+d_{2})/h$ , the critical bi-chromatic patch width has a unique minimum at $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{t}^{opt}$ , such that $\unicode[STIX]{x1D703}_{t}^{opt}$ maximizes $K_{12}$ as given by (4.9). Figure 4(b) shows the variations of $\unicode[STIX]{x1D703}_{t}^{opt}$ as a function of  $\unicode[STIX]{x1D703}_{2}$ .

In practice, $\unicode[STIX]{x1D703}_{t}^{opt}$ can be approximated by $\unicode[STIX]{x1D703}_{2}/2$ because the critical patch widths for $\unicode[STIX]{x1D703}_{t}^{opt}$ and $\unicode[STIX]{x1D703}_{2}/2$ are nearly the same. This can be seen from the overlap of the $w_{cr_{2}}$ curve obtained for $\unicode[STIX]{x1D703}_{t}^{opt}$ with the dashed curve obtained for $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{2}/2$ in figure 4(c). Figure 4(c) also shows the critical patch width $w_{cr_{1}}$ for a monochromatic patch. Interestingly, $w_{cr_{2}}$ is greater than $w_{cr_{1}}$ for relatively small deflection angles because we have assumed $d_{2}\neq 0$ in deriving (4.8). By setting $d_{2}=0$ , we get $w_{cr_{2}}=w_{cr_{1}}$ since both patches involve only one Bragg interaction. Since $w_{cr_{1}}=w_{cr_{2}}$ at $\unicode[STIX]{x1D703}_{2}\sim 11\unicode[STIX]{x03C0}/25$ for $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{2}/2$ , as depicted in figure 4(c) (intersection of the dashed line with the 1-corr. curve, i.e. the curve obtained for a monochromatic patch; corr. short for corrugation), the bi-chromatic patch becomes significantly more efficient for shore protection when deflection angles satisfy $\unicode[STIX]{x1D703}_{2}>11\unicode[STIX]{x03C0}/25$ .

While the values predicted by the method of multiple scales for $d_{2}/d_{1}$ (see figure 4 a), $\unicode[STIX]{x1D703}_{t}^{opt}$ (figure 4 b), and $w_{cr_{2}}$ (figure 4 c) are valid for most $\unicode[STIX]{x1D703}_{2}$ , the trends in the limit of $\unicode[STIX]{x1D703}_{2}\rightarrow 0$ or $\unicode[STIX]{x1D703}_{2}\rightarrow \unicode[STIX]{x03C0}/2$ may be inaccurate. Indeed, $\unicode[STIX]{x1D703}_{2}\sim 0$ implies seabed bar crests almost parallel to $\boldsymbol{k}_{1}$ , which invalidates the multiple-scale assumption of a rapidly varying water depth in $x$ . The configuration $\unicode[STIX]{x1D703}_{t}\sim \unicode[STIX]{x1D703}_{2}$ also violates the multiple-scale assumption because some of the seabed bar crests become almost parallel to $\boldsymbol{k}_{t}$ . The case of $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ also requires special care, mainly because the incident wave energy cannot be transferred to the target $\boldsymbol{k}_{2}$ waves in a steady state by assuming $y$ invariance. Shore protection by perpendicular deflection, for which $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ , will be discussed in details in § 5 using direct simulations.

4.2 Effect of the patch’s plane of symmetry

We now take into account the patch’s plane of symmetry, that is, we consider bottom corrugations given by $\unicode[STIX]{x1D701}(x,y)$ in (4.1) for $y>0$ , and by $\unicode[STIX]{x1D701}(x,-y)$ for $y<0$ . We solve the system of (4.3) with the same finite-difference scheme used for the case of a monochromatic patch in § 3, and we enforce the synchronization condition (4.8) such that full energy transfer from the $\boldsymbol{k}_{1}$ waves to the $\boldsymbol{k}_{2}$ waves is achieved far from the plane of symmetry. The incident wave has wavenumber $\boldsymbol{k}_{1}=k_{1}\hat{x}$ and the boundary conditions are $A_{1}=a_{0}$ and $A_{t}=A_{2}=0$ at $x=0$ . Similar to the case of a monochromatic patch, the symmetry of the problem with respect to the $y$ axis for normally incident and perfectly tuned waves results in the boundary condition $A_{t}=A_{2}=0$ at $y=0$ , which implies $A_{1}=a_{0}$ at $y=0$ . These boundary conditions provide us with a well-posed problem, as discussed in § 3; diffraction effects are yet also neglected which is a limitation of our model, in particular near the edges of the corrugated patch. We shall emphasize the fact that the solution obtained for a semi-infinite patch (i.e. with seabed bars in $x,y\geqslant 0$ ) is the most general and informative solution (and therefore it is the problem we investigate here), since any smaller finite-size patch would result in envelope amplitudes equal to the semi-infinite solution over the patch, and constant envelope amplitudes outside of it. This is in contrast with the case of Bragg reflection, and due to the fact that the resonated waves are forward scattered, rather than backward scattered.

Figure 5.

Normalized envelope amplitudes (a) $a_{1}^{\ast }$ , (b) $a_{t}^{\ast }$ and (c) $a_{2}^{\ast }$ , over a bi-chromatic patch, as obtained from (4.3) with $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{t}=0$ . The patch is semi-infinite, starting from $x=0$ onward, and the bottom corrugations are given by $\unicode[STIX]{x1D701}(x,y)$ for $y>0$ and by $\unicode[STIX]{x1D701}(x,-y)$ for $y<0$ (cf. (4.1)). The amplitude modulations far from the plane of symmetry (i.e. highlighted in the rectangles) are clearly similar to the $y$ -invariant solutions (shown in the top figures). We only show the amplitudes for $y>0$ since the problem is symmetric with respect to the $y=0$ plane. The small vertical arrows show the critical patch width $w_{cr_{2}}$ . The physical parameters are $\unicode[STIX]{x1D703}_{2}=7\unicode[STIX]{x03C0}/15$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{2}/2$ , $k_{1}h=0.2$ , $(d_{1}+d_{2})/h=0.12$ , with (4.8) satisfied. The simulation parameters are $\unicode[STIX]{x1D6FF}x/\unicode[STIX]{x1D706}_{1}=0.05$ and $\unicode[STIX]{x1D6FF}y/\unicode[STIX]{x1D706}_{1}=0.2$ .

Figure 6.

Normalized envelope amplitudes $a_{1}^{\ast }$ , $a_{t}^{\ast }$ , $a_{2}^{\ast }$ at the end of a patch truncated at the critical patch width $x=w_{cr_{2}}$ (indicated by a small vertical arrow in figure 5) for $\unicode[STIX]{x1D703}_{2}=11\unicode[STIX]{x03C0}/25$ (plots a–c) and $\unicode[STIX]{x1D703}_{2}=7\unicode[STIX]{x03C0}/15$ (plots d–f). The $x$ -axis is on a logarithmic scale so as to show details near the plane of symmetry. Solid, dashed, and dotted lines represent results obtained for $\unicode[STIX]{x1D703}_{t}=0.2\unicode[STIX]{x1D703}_{2}$ , $0.5\unicode[STIX]{x1D703}_{2}$ , $0.8\unicode[STIX]{x1D703}_{2}$ respectively. The dash-dotted lines show the $\boldsymbol{k}_{1}$ -wave and $\boldsymbol{k}_{2}$ -wave amplitudes at the end of a monochromatic patch of width $w_{cr_{1}}$ . Physical and simulation parameters are the same as those in figure 5.

We show the normalized envelope amplitudes $a_{1}^{\ast }$ , $a_{t}^{\ast }$ and $a_{2}^{\ast }$ obtained for a semi-infinite patch in $x$ direction with $\unicode[STIX]{x1D703}_{2}=7\unicode[STIX]{x03C0}/15$ and $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{2}/2$ in figure 5. We have marked in figure 5 the directions of the $\boldsymbol{k}_{t}$ and $\boldsymbol{k}_{2}$ waves by oblique arrows starting from the origin of each axis. Far from $y=0$ , the problem becomes one-dimensional and the separation wavelength between each successive crest and trough of the $\boldsymbol{k}_{1}$ or $\boldsymbol{k}_{2}$ wave envelope is $w_{cr_{2}}=\unicode[STIX]{x03C0}/K_{12}$ as predicted by (4.6) (see the figures on top of each frame that show the $y$ -invariant solutions). As for the case of the monochromatic patch, the far field is a function of $x$ , i.e. the one-dimensional behaviour is obtained further away from the patch’s plane of symmetry as we look further down in $x$ (i.e. considering a longer patch). This comes from the fact that the far-field solution should only be obtained in the region where all three waves co-exist from $x=0$ , which is the region given by $y>x\cos \unicode[STIX]{x1D703}_{2}$ (i.e. above the vector $\boldsymbol{k}_{2}$ in figure 5).

The wave patterns near $y=0$ show the remarkable complexity of the three-wave interaction process near the $y=0$ plane of symmetry. Generally, wave amplitudes near $y=0$ are smaller further down the patch due to the deflection. The qualitative behaviour of waves in each one of the three distinguishable regions (i.e. $y/x>\tan \unicode[STIX]{x1D703}_{2}$ , $\tan \unicode[STIX]{x1D703}_{2}>y/x>\tan \unicode[STIX]{x1D703}_{t}$ , and $y/x<\tan \unicode[STIX]{x1D703}_{t}$ ), changes with a change in $\unicode[STIX]{x1D703}_{t}$ or $\unicode[STIX]{x1D703}_{2}$ . It is to be noted that once we introduce a plane of symmetry at $y=0$ , neither $a_{1}^{\ast }$ nor $a_{t}^{\ast }$ become uniformly zero in $y$ at $x=w_{cr_{2}}$ (cf. the near field $y\sim 0$ and $x=w_{cr_{2}}$ in figure 6 a). Therefore, even though the critical patch width minimizes wave energy in the protected wake, there will always be leaked waves emanating near $y=0$ at the end of the patch and propagating to the downstream.

Figure 6 shows the variations of the normalized wave amplitudes as functions of $y$ at the end of a bi-chromatic patch truncated at $x=w_{cr_{2}}$ for $\unicode[STIX]{x1D703}_{2}=11\unicode[STIX]{x03C0}/25$ (a–c) and $\unicode[STIX]{x1D703}_{2}=7\unicode[STIX]{x03C0}/15$ (d–f). For comparison, we also plot $a_{1}^{\ast }$ and $a_{2}^{\ast }$ for a monochromatic patch ending at $x=w_{cr_{1}}$ (dash-dotted line). Clearly, the bi-chromatic patch achieves a much better reduction in the incident wave amplitude everywhere (cf. figure 6 a,d). The $\boldsymbol{k}_{1}$ wave amplitude drops much more quickly in $y$ for the bi-chromatic patch, because some of its energy is stored in the hump of $\boldsymbol{k}_{t}$ wave energy seen in figure 6(b,e) (note that the horizontal axis is on a logarithmic scale truncated at $10^{-1}$ ). Using two corrugation modes instead of one thus not only decreases the critical patch width for large $\unicode[STIX]{x1D703}_{2}$ , but also spreads out a fraction of the energy leaked near $y=0$ further to the sides in the form of $\boldsymbol{k}_{t}$ waves. Interestingly, the location of the second hump of $\boldsymbol{k}_{1}$ wave energy (figure 6 a,d) scales with $1/\sqrt{\cos \unicode[STIX]{x1D703}_{t}}$ , which suggests that the $\boldsymbol{k}_{1}\leftrightarrow \boldsymbol{k}_{t}$ wave energy exchange is partially oscillatory in $y$ with a period similar to the period of oscillation in the $x$ direction (cf. $K_{12}$ in (4.9)).

Because the wave amplitudes are shown at the end of the mono- and bi-chromatic patches with width $w=w_{cr_{1}}$ and $w=w_{cr_{2}}$ , we obtain $a_{1}^{\ast }\sim 0$ and $a_{2}^{\ast }\sim 1$ for $y/\unicode[STIX]{x1D706}_{1}\gg 1$ in figure 6(c,f). In both cases we also observe that the $\boldsymbol{k}_{2}$ wave gains energy more rapidly for the smaller $\unicode[STIX]{x1D703}_{2}$ angle: for instance, considering the case of $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{2}/2$ (solid lines), we find that $a_{2}^{\ast }=0.9$ at $y/\unicode[STIX]{x1D706}_{1}=80\,134$ for $\unicode[STIX]{x1D703}_{2}=11\unicode[STIX]{x03C0}/25$ , $7\unicode[STIX]{x03C0}/15$ . This suggests that there might be a compromise to be found between deflecting at a very large angle and minimizing the energy content of the transmitted wave beams. The effect of the transitional angle on the amount of energy captured by the $\boldsymbol{k}_{2}$ wave is relatively minor, as can be seen from the overlap of the $a_{2}^{\ast }$ curves obtained for $\unicode[STIX]{x1D703}_{t}=0.2\unicode[STIX]{x1D703}_{2}$ , $0.5\unicode[STIX]{x1D703}_{2}$ , $0.8\unicode[STIX]{x1D703}_{2}$ in figure 6(c,f).

As a note of caution, we would like to note that the variations of the $\boldsymbol{k}_{1}$ wave amplitude are relatively fast close to $y=0$ , where $a_{1}^{\ast }$ drops from 1 to about 0 in just five wavelengths in figure 5(a) (see also figure 3 a). Similarly, $a_{t}^{\ast }$ shows relatively rapid variations in figure 5(b) along the line $y=x\cos \unicode[STIX]{x1D703}_{t}$ , as does $a_{2}^{\ast }$ in figure 3(b) along the line $y=x\cos \unicode[STIX]{x1D703}_{2}$ . The multiple-scale assumption becomes violated when the modulations of the wave envelopes are fast, but here we expect the analytical predictions to remain reasonably valid because the most rapid variations are still five times slower than the carrier wave oscillations. The comparison between theory and direct numerical simulation in the next section will further support this claim as we will demonstrate a very good agreement between the two methods.

5.1 Cross-validation

We compare the direct simulation and multiple-scale results of a normal-to-shore monochromatic wavetrain impinging on a bi-chromatic patch with a $y=0$ plane of symmetry. The numerical domain is a square of size $L_{0}\times L_{0}$ and we select the dimensionless wavenumber $k_{1}=110\times (2\unicode[STIX]{x03C0})/L_{0}$ , such that $110$ waves of wavelength $\unicode[STIX]{x1D706}_{1}$ can fit in the computational space in both $x$ and $y$ (note in practice that we set $L_{0}=2\unicode[STIX]{x03C0}$ for the simplicity of calculations in Fourier space). In the numerical simulations we offset the centreline of the patch (i.e. the $y=0$ axis of symmetry) from the centreline of the numerical domain such that we can simulate a patch of width greater than $55\unicode[STIX]{x1D706}_{1}$ (the half-width of the numerical domain) in the upper half-plane. Specifically, with this offset we are able to obtain the numerical simulation results in the upper half-plane for a patch of width $85\unicode[STIX]{x1D706}_{1}$ ; this greatly reduces the computational cost. The solution for the lower half-plane is obtained by symmetry. We use the same parameters as in figures 5 and 6(d–f) such that $\unicode[STIX]{x1D703}_{2}=7\unicode[STIX]{x03C0}/15$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{2}/2$ , $k_{1}h=0.2$ and $(d_{1}+d_{2})/h=0.12$ with (4.8) satisfied. The seabed bars are laid out from $x/\unicode[STIX]{x1D706}_{1}=0$ to $x/\unicode[STIX]{x1D706}_{1}=w_{cr_{2}}/\unicode[STIX]{x1D706}_{1}=18.3$ such that the patch width is critical. The incident wave amplitude is taken as $a_{0}/h=10^{-3}$ , i.e. two orders of magnitude smaller than the combined seabed bars amplitudes $(d_{1}+d_{2})/h=0.12$ , in order to minimize nonlinear wave–wave effects. It is to be noted that we still obtained very similar results for larger wave amplitudes (i.e. such as $a_{0}/h=0.02$ ), which is expected because the theory is applicable for wave steepness smaller or of same order as the bottom slope.

The generation of the initial waves coming from $x<0$ is done using a numerical wavemaker at the beginning of the numerical domain, upstream of the seabed bars. We enforce a radiation condition on the incident and forward-scattered waves at the outgoing boundary, which is several wavelength downstream of the end of the corrugated patch, with the help of a numerical damping layer. The damping layer works efficiently and minimizes spurious reflections because the (right) outgoing boundary is downstream of the corrugated patch (by more than 40 wavelengths $\unicode[STIX]{x1D706}_{1}$ ), i.e. such that there are no seabed bars there. We also enforce radiation conditions at the side boundaries of the numerical domain (i.e. in $y$ ) with damping layers. This implies that the patch has a finite width in $y$ , since the radiation condition assumes no further energy exchanges between the different wave modes. It should be noted that the damping layers might not work perfectly on the sides of the numerical domain since the water depth is not constant. Spurious reflections are yet minimal as can be seen in the upper half of figure 5. The use of damping layers on the sides of the numerical domain implies that the numerical results are obtained assuming that there are no waves outside of the simulation domain. This is not a problem for waves leaving the domain. However, incident waves unaffected by the seabed bars but propagating in $x$ just adjacent to the patch could spread over into the numerical domain by diffraction due to the fact that the neighbouring waves over the patch have non-uniform amplitude. Diffraction effects due to areas of energy dissipation (which can be representative of the situation here where the incident waves cede their energy to other waves) have been addressed by Dalrymple, Kirby & Hwang (Reference Dalrymple, Kirby and Hwang1984), and shown to be relatively minimal such that we do not study them here. Some effects of a patch’s finite transverse length (i.e. here in $y$ ) have also been reported in the experimental work of Magne, Rey & Ardhuin (Reference Magne, Rey and Ardhuin2005).

Figure 7(a) shows the normalized wave elevation $\unicode[STIX]{x1D702}/a_{0}$ when steady state is reached. The plot is split into two parts, with the lower part ( $y<0$ ) showing the semi-analytical solution (cf. § 4), and with the upper part ( $y>0$ ) showing direct simulation results. As can be seen, analytical and computational results endorse each other, with the same features already described in § 4 standing out: (i) the incident waves propagate unaffected upstream of the patch, (ii) the complex pattern due to the three-wave interactions over the patch becomes $y$ -invariant away from the centreline (i.e. $|y|\gg \unicode[STIX]{x1D706}_{1}$ ), (iii) two beams of $\boldsymbol{k}_{1}$ wave energy propagate near $y=0$ downstream of the patch (one near the centre at $y\sim 0$ and the other at $y\sim 15\unicode[STIX]{x1D706}_{1}$ ), (iv) a beam of $\boldsymbol{k}_{t}$ wave energy emanates from the end of the patch at $y\sim \pm 19\unicode[STIX]{x1D706}_{1}$ .

Figure 7.

(a) Free-surface elevation $\unicode[STIX]{x1D702}/a_{0}$ and (b) time-averaged energy (cf. (5.1)) at the steady state obtained from direct simulation (upper plots $y>0$ ) and multiple-scale analysis (lower plots $y<0$ ; cf. (4.3)). A monochromatic wave-train incident from $x<0$ impinges on a bi-chromatic patch with parameters $\unicode[STIX]{x1D703}_{2}=7\unicode[STIX]{x03C0}/15$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x1D703}_{2}/2$ , $k_{1}=110$ (dimensionless), $k_{1}h=0.2$ , $(d_{1}+d_{2})/h=0.12$ (with $d_{2}/d_{1}$ given by (4.8)),  $a_{0}/h=10^{-3}$ (cf. figure 2 b). The seabed bars occupy the space between $x=0$ and $x/\unicode[STIX]{x1D706}_{1}=w_{cr_{2}}/\unicode[STIX]{x1D706}_{1}=18.3$ such that all the incident $\boldsymbol{k}_{1}$ wave energy transfers to the target $\boldsymbol{k}_{2}$ wave away from the patch’s plane of symmetry $y=0$ . The direct simulation parameters are $\unicode[STIX]{x1D6FF}x/\unicode[STIX]{x1D706}_{1}=\unicode[STIX]{x1D6FF}y/\unicode[STIX]{x1D706}_{1}=1/9$ and $\unicode[STIX]{x1D6FF}t/T_{1}=1/32$ , and the resolution of the multiple-scale solution is the same as in figure 5. The vertical dashed lines show the transects where we plot the maximum free-surface elevation in figure 8.

The protection provided by the bi-chromatic patch can be further highlighted by looking at the time-averaged energy (cf. figure 7 b), i.e.

(5.1) $$\begin{eqnarray}\bar{E}(x,y)=\frac{1}{a_{0}^{2}/2}\int _{t_{0}}^{t_{f}}\unicode[STIX]{x1D702}^{2}(x,y,t)\frac{\text{d}t}{t_{f}-t_{0}},\end{eqnarray}$$

where the pre-factor in (5.1) is inversely proportional to the averaged wave energy that would be obtained without the seabed bars. It should be noted that (5.1) is the first-order normalized wave energy, and that it includes contributions from both the kinetic and potential wave energy, which are equal. The time integration is performed when the system reaches steady state with $t_{0}/T_{1}=140$ and $t_{f}/T_{1}=170$ . A good agreement is obtained for $\bar{E}$ in figure 7(b) between the direct simulation (lower half) and the multiple-scale results (upper half). The features described for figure 7(a) also appear in figure 7(b) and are in fact strongly enhanced (e.g. $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{t}$ wave beams). The rapid variations of wave energy are due to the formation of a standing-wave pattern downstream of the patch (e.g. along the $\boldsymbol{k}_{t}$ wave beam direction) resulting from the superposition of $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{t}$ waves. The strongest energy reduction is about 99 % (for both direct simulation and multiple scales) and is achieved within the stripe of width $2\unicode[STIX]{x1D706}_{1}$ centred on $y=\pm 5\unicode[STIX]{x1D706}_{1}$ .

5.2 Effect of detuning and protection efficiency

To quantify the effect of frequency detuning on the Bragg deflection mechanism, we calculate the maximum free-surface elevation $\unicode[STIX]{x1D702}^{\ast }=\max |\unicode[STIX]{x1D702}|/a_{0}$ (in time) with and without detuning along the $y$ axis at a virtual shoreline (no reflection considered) that is assumed $\unicode[STIX]{x0394}x=26\unicode[STIX]{x1D706}_{1}$ downstream of the patch (i.e. along the dashed lines in figure 7). The incident wave frequency is $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D71B}$ where $\unicode[STIX]{x1D714}_{1}$ is the perfectly tuned wave frequency and $\unicode[STIX]{x1D71B}\ll \unicode[STIX]{x1D714}_{1}$ is the detuning parameter. The corresponding detuning wavenumber $\unicode[STIX]{x1D705}$ , such that $k=k_{1}+\unicode[STIX]{x1D705}$ , is obtained from $\unicode[STIX]{x1D71B}=C_{g}\unicode[STIX]{x1D705}$ . In general, detuning effects result in having to solve the time-dependent multiple-scale equations, which can be used also to investigate the transient boundary-value problem of wave group scattering (see e.g. Yu & Mei Reference Yu and Mei2000). For small detuning, however, the frequency detuning $\unicode[STIX]{x1D71B}$ can be combined with the slowly varying amplitudes $A_{1}$ , $A_{t}$ and $A_{2}$ in (4.2), i.e., for instance, $A_{1}$ is rewritten as $A_{1}\exp (-\text{i}\unicode[STIX]{x1D71B}t)$ , such that following the method of Mei (Reference Mei1985) we simply have to replace $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{t}$ by $-\text{i}\unicode[STIX]{x1D71B}$ in the envelope equations (4.3). Similarly to the case of class I Bragg reflection, we normalize the detuning frequency by the characteristic long time scale of the system $\unicode[STIX]{x1D6FA}_{12}=C_{g}K_{12}$ (cf. (4.7)). The normalized detuning $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D6FA}_{12}$ thus represents the strength of detuning, which suggests that shorter waves are more strongly affected by detuning than longer waves (a detailed analysis of detuning for a bi-chromatic patch infinitely long in $y$ direction is provided in appendix B).

Figure 8.

Effect of detuning on the normalized maximum free-surface elevation $\unicode[STIX]{x1D702}^{\ast }=\max |\unicode[STIX]{x1D702}|/a_{0}$ at a distance $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D706}_{1}=26$ downstream of the patch (i.e. along the dashed lines in figure 7). The detuning parameter is $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D714}_{1}=0$ for the lower two curves and $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D714}_{1}=0.1$ (or $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D6FA}_{12}=3.7$ ) for the upper two curves. Other physical parameters are as in figure 7. The direct simulation (HOS) parameters are $\unicode[STIX]{x1D6FF}x/\unicode[STIX]{x1D706}_{1}=\unicode[STIX]{x1D6FF}y/\unicode[STIX]{x1D706}_{1}=1/7.3$ and $\unicode[STIX]{x1D6FF}t/T_{1}=1/32$ , and the resolution of the multiple-scale solution (MS) is the same as in figure 5. Detuning clearly results in a larger maximum wave amplitude downstream of the patch.

The result of our numerical experiments on the effect of detuning is presented in figure 8. In the absence of detuning, i.e. $\unicode[STIX]{x1D71B}=0$ , a global minimum of $\unicode[STIX]{x1D702}^{\ast }$ for both direct simulation and multiple scales is observed: this minimum in $\unicode[STIX]{x1D702}^{\ast }$ corresponds to the minimum of incident wave amplitude $a_{1}^{\ast }$ at the end of the patch as seen in figure 6(d) (dotted line at $y/\unicode[STIX]{x1D706}_{1}\sim 6$ ). The large standing-wave oscillations seen for $y/\unicode[STIX]{x1D706}_{1}>25$ are, as mentioned previously, due to the superposition of the leaked $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{t}$ waves and predicted by both methods. Of course, the superposition would not occur if we were to observe $\unicode[STIX]{x1D702}^{\ast }$ further downstream of the patch since the $\boldsymbol{k}_{t}$ beams of energy propagate progressively away from the $y=0$ plane of symmetry to which the $\boldsymbol{k}_{1}$ beams of energy are confined.

The effect of detuning on the maximum free-surface elevation is strongest away from the $y=0$ plane of symmetry. There is still a global minimum near $y=0$ for $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D6FA}_{12}=3.7$ but $\unicode[STIX]{x1D702}^{\ast }$ reaches a unitary value (which corresponds to no wave amplitude reduction) near $y=15\unicode[STIX]{x1D706}_{1}$ , with oscillations farther out to the sides again due to the superposition of $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{t}$ waves. The wave field away from $y=0$ changes radically because the second interaction of the two-step Bragg resonance mechanism, which is key to reducing the incident wave mode energy far from the plane of symmetry, is more strongly impacted than the first Bragg interaction. The fact that detuning affects the successive Bragg interactions cumulatively is further discussed in appendix B for the case of an infinitely long patch (see figures 12 and 13).

Quantitative differences between direct simulation and multiple-scale results are in general expected and obtained here (as is seen in figure 8). The discrepancies are due to the fact that first-order multiple scales neglects wave–wave nonlinearities, higher-order wave–bottom interactions and the effect of discontinuity in the seabed topography at the beginning and end of the patch. This latter item is known to result in, among other effects, evanescent modes near boundaries. In the absence of detuning (cf. lower two curves in figure 8), the maximum free-surface elevation obtained from the direct simulation matches well with the multiple-scale solution except for a small local peak near $y=0$ and for the small oscillations from $y=0$ to $y=25\unicode[STIX]{x1D706}_{1}$ . The discrepancy at $y=0$ can be expected, even though the bottom variations are continuous there, because as discussed before the variations of the wave amplitudes near the patch’s plane of symmetry predicted by the theory are relatively rapid ( $\unicode[STIX]{x1D702}^{\ast }$ drops to 0 in approximately 5 wavelengths, cf. figure 8, and discussion at the end of § 4): weak diffraction effects may be obtained here that are considered in the numerical solution but not in the theoretical solution. These relatively fast variations, which are not as rapid in direct simulation, potentially contribute to the small phase shift observed between theory and experiment for the location of the superposed $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{t}$ waves. Similar discrepancies between the two methods are observed for the detuned case (upper two curves in figure 8), and are higher (in particular far from the patch’s plane of symmetry), which is consistent with the fact that multiple-scale performs best at perfect resonance, i.e. in the absence of detuning. Despite the simplifying assumptions of multiple-scale analysis, the comparison in figure 8 is however still good almost everywhere in the domain, with or without detuning.

Figure 9.

(a) Protection efficiency ${\mathcal{P}}$  (5.2) with and without detuning ( $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D714}_{1}=0$ , 0.01, 0.03, 0.05, 0.1 which corresponds to $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D6FA}_{12}=0$ , 0.37, 1.1, 1.8, 3.7) at a distance $\unicode[STIX]{x0394}x=26\unicode[STIX]{x1D706}_{1}$ downstream of the patch (i.e. at $x=w_{cr_{2}}+26\unicode[STIX]{x1D706}_{1}$ , which is along the dashed line in figure 7). The results are obtained from multiple-scale analysis. (b) Close-up with direct simulation results added for $\unicode[STIX]{x1D71B}/\unicode[STIX]{x1D714}_{1}=0,0.1$ (dashed lines). The physical and simulation parameters other than $\unicode[STIX]{x1D71B}$ are the same as those in figure 8.

To further assess the efficiency of the Bragg deflection mechanism, as well as how it is affected by detuning, we now define a protection efficiency variable ${\mathcal{P}}$ , where

(5.2) $$\begin{eqnarray}{\mathcal{P}}(x,y)=1-\frac{1}{y}\int _{0}^{y}\bar{E}(x,\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}.\end{eqnarray}$$

The protection efficiency ${\mathcal{P}}$ measures the $y$ -average wave energy reduction (with $\bar{E}$ obtained at steady state and given in (5.1)) on a line parallel and downstream of the patch’s trailing edge. Figure 9(a) shows ${\mathcal{P}}$ over a relatively large domain, i.e. from $y=0$ to $y=350\unicode[STIX]{x1D706}_{1}$ , with $\bar{E}$ evaluated at the distance $\unicode[STIX]{x0394}x=26\unicode[STIX]{x1D706}_{1}$ downstream of the patch. Figure 9(b) is a close-up with the addition of direct simulation results (dashed lines), to show that the multiple-scale predictions are reasonably accurate, even with 10 % detuning.

Figure 10.

Time-averaged energy $\bar{E}$ (cf. (5.1),  $t_{0}/T_{1}=140$ , $t_{f}/T_{1}=170$ ) obtained from direct simulation for a patch with a plane of symmetry at $y=0$ . The physical and simulation parameters are identical to those in figure 7 except for the deflection angle which is now set to $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ . The energy distribution is symmetric with respect to the $y=0$ axis which is why we only show the upper half-plane $y>0$ . The solid vertical lines show the beginning and end of the corrugated patch in $x$ direction.

The protection efficiency ${\mathcal{P}}$ for perfectly tuned waves (i.e. $\unicode[STIX]{x1D71B}=0$ ) is as high as 88 % at $y=270\unicode[STIX]{x1D706}_{1}$ , meaning that only 12 % of the incident wave energy makes it inside the protected wake behind the patch. From $y\sim 270\unicode[STIX]{x1D706}_{1}$ farther out to the sides, ${\mathcal{P}}$ starts decreasing because the deflected $\boldsymbol{k}_{2}$ waves, which captured most of the incident wave energy, propagate within the semi-infinite space $y>x/\cos \unicode[STIX]{x1D703}_{2}$ outside of the protected wake, which includes the half-line $y>270\unicode[STIX]{x1D706}_{1}>26\unicode[STIX]{x1D706}_{1}/\cos (7\unicode[STIX]{x03C0}/15)$ . As already discussed, the protection efficiency decreases rapidly with an increase in $\unicode[STIX]{x1D71B}$ away from $y=0$ since the two-step Bragg deflection mechanism is relatively sensitive to detuning. Detuning affects transmitted waves (as in Bragg deflection) more strongly than reflected waves, because transmitted waves have an oscillatory behaviour over the corrugations: detuning not only diminishes wave–bottom interactions but also changes the critical patch width where the incident wave amplitude reaches a minimum. Nonetheless, we can see that significant wave amplitude reduction is still achieved over a large domain with some small detuning.

5.3 Perpendicular deflection

In this section we investigate with direct numerical simulation the special case of shore-normal incident waves (with wavenumber $\boldsymbol{k}_{1}=k_{1}\hat{x}$ ) that are deflected by $90^{\circ }$ (i.e. perpendicularly) into shore-parallel resonated waves ( $\boldsymbol{k}_{2}=\pm k_{2}{\hat{y}}$ , cf. figure 2 b). Assuming perfectly tuned incident waves, we first show in figure 10 that the averaged wave energy (5.1) is again significantly decreased in the wake of the patch for $y<x\cos \unicode[STIX]{x1D703}_{t}$ . A bi-chromatic patch can therefore deflect incident waves to the shore-parallel direction, and offer shore protection near the patch’s plane of symmetry. It is to be noted that the physical parameters in figure 10 are the same as those in figure 7 (except for $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ in figure 10), such that the patch width in figure 10 is not critical. Wave reduction is thus obtained in the wake near the patch’s plane of symmetry regardless of its width.

Far from the plane of symmetry where the solution is expected to be $y$ -invariant, the synchronization condition (4.8) suggests that the bi-chromatic patch should be turned into a monochromatic patch since we get $d_{2}/d_{1}\rightarrow 0$ when $\unicode[STIX]{x1D703}_{2}\rightarrow 0$ . However, as was mentioned earlier, the degeneracy of the Bragg resonance condition prevents a monochromatic patch from deflecting incident waves by a $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ angle. Therefore the solution (4.4) is not valid for perpendicular deflection and we should still consider a bi-chromatic patch (with $d_{1}$ , $d_{2}\neq 0$ ). Let us now suppose that the envelope amplitude $A_{2}$ need not be continuous at the patch boundaries, i.e. at $x=0$ or $x=w$ . Then, the steady-state ( $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{t}\equiv 0$ ) $y$ -invariant solution ( $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\bar{y}\equiv 0$ ) of (4.3) with $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ is obtained as

(5.3a-c ) $$\begin{eqnarray}A_{1}=a_{0},\quad A_{t}=0,\quad A_{2}=\frac{d_{1}\cos \unicode[STIX]{x1D703}_{t}}{d_{2}\cos (\unicode[STIX]{x1D703}_{2}-\unicode[STIX]{x1D703}_{t})}a_{0}.\end{eqnarray}$$

The incident wave amplitude in (5.3) is found to remain unchanged over the patch, which is in agreement with the conservation of wave action (4.5). If (5.3) is correct, then, a bi-chromatic patch set to deflect waves by a $90^{\circ }$ angle cannot be expected to protect the shore far from the patch’s plane of symmetry. This result is now validated in figure 11 as we show the evolution in time of the envelope amplitudes over a patch infinitely long in $y$ direction (and therefore the solution is independent of $y$ ) and for which $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ . We provide three different cases of ripple amplitudes, i.e. $d_{1}/d_{2}=0.5,1,2$ (see figure 11 a–c), in order to verify the linear relationship predicted in (5.3) between the amplitude of the deflected waves $|A_{2}|$ and the ratio $d_{1}/d_{2}$ . The incident and resonated envelope amplitudes are obtained from direct simulation and are extracted from the overall free-surface solution by fast Fourier transform in $y$ direction. We then also average in time over the carrier wave period (i.e. $T_{1}$ ) in order to remove the fast oscillations of the carrier wave phases. It should be noted that the wave amplitude is zero for all three modes for $t<k_{1}(x-x_{w})/\unicode[STIX]{x1D714}$ in figure 11 since the incident waves start propagating into the domain through the wavemaker on the left-hand side (at $x=x_{w}$ ) only at $t=0$ .

Figure 11.

Evolution of the envelope amplitudes $|A_{1}|/a_{0}$ , $|A_{t}|/a_{0}$ and $|A_{2}|/a_{0}$ in $x$ and $t$ for waves incident from $x<0$ and propagating over an infinitely long patch in $y$ direction with $\unicode[STIX]{x1D703}_{2}=2\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/2$ . We consider three different distributions of ripple amplitudes: (a) $d_{1}/d_{2}=0.5$ , (b) $d_{1}/d_{2}=1$ , (c) $d_{1}/d_{2}=2$ and keep $(d_{1}+d_{2})/h=0.12$ fixed. The results in the right column demonstrate that $|A_{2}|$ increases approximately linearly with $d_{1}/d_{2}$ . The patch width is $w/\unicode[STIX]{x1D706}_{1}=17.5$ and the other physical parameters are $k_{1}h=0.2$ , $a_{0}/h=10^{-3}$ . The direct simulation parameters are $\unicode[STIX]{x1D6FF}x/\unicode[STIX]{x1D706}_{1}=\unicode[STIX]{x1D6FF}y/\unicode[STIX]{x1D706}_{1}=1/7.31$ and $\unicode[STIX]{x1D6FF}t/T_{1}=1/32$ .

In all cases, we see that the $\boldsymbol{k}_{1}\leftrightarrow \boldsymbol{k}_{t}$ Bragg interaction is strongest at the initial time: $|A_{t}|$ increases very rapidly to the detriment of $|A_{1}|$ (figure 11 a,b in the left and middle columns). The duration and significance of energy transfer from the $\boldsymbol{k}_{1}$ to the $\boldsymbol{k}_{t}$ waves increases with increasing $d_{1}/d_{2}$ (i.e. from figure 11 a to 11 c), which is expected because the strength of the $\boldsymbol{k}_{1}\leftrightarrow \boldsymbol{k}_{t}$ Bragg interaction is proportional to $d_{1}\cos \unicode[STIX]{x1D703}_{t}$ . Energy exchanges still occur at later times, as seen from the small oscillations of $|A_{t}|$ , but the incident wave energy eventually becomes fully transmitted such that $|A_{1}|$ is as large downstream and upstream of the patch (i.e. $|A_{1}|/a_{0}\sim 1$ at the end of the patch). The $\boldsymbol{k}_{2}$ wave amplitude $|A_{2}|$ also only increases early on, with some energy redistribution clearly seen across the patch at later times for the smaller $d_{1}/d_{2}$ (figure 11 a–c in the right column). In particular, we find that the plateau reached by the $x$ -averaged $|A_{2}|$ amplitude grows approximately linearly with $d_{1}/d_{2}$ , which means that (5.3) is a good first-order approximation to the $y$ -invariant problem. That the initial net energy flow from $|A_{1}|$ to $|A_{2}|$ increases linearly with $d_{1}/d_{2}$ (note that the colour map scales change between figures) is somewhat surprising since larger $d_{1}$ imply smaller $d_{2}$ , hence weaker $\boldsymbol{k}_{t}\leftrightarrow \boldsymbol{k}_{2}$ Bragg interactions. However, since $\boldsymbol{k}_{2}$ waves are trapped by the patch for $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ , they experience an infinite number of Bragg interactions for an infinitely long patch in $y$ direction. The second Bragg interaction is therefore intrinsically much more efficient than the first $\boldsymbol{k}_{1}\leftrightarrow \boldsymbol{k}_{t}$ Bragg interaction, which becomes the limiting mechanism in the overall transfer of $\boldsymbol{k}_{1}$ wave energy to the $\boldsymbol{k}_{2}$ waves.

To summarize our results on perpendicular deflection, we found that the case of $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ offers significant wave energy reduction in the downstream near the patch’s plane of symmetry. However, no protection can be obtained when $\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x03C0}/2$ far from the patch’s plane of symmetry, because the conservation of wave action requires waves in the downstream propagating with a non-zero velocity in $x$ direction. How much far away from $y=0$ we get the $y$ -invariant behaviour is therefore of significance importance and can be estimated, to a good first-order approximation, by inspection of the steady-state solution (5.3). Indeed, the far field is expected to start where the deflected wave amplitude $|A_{2}|$ obtained for a patch with a plane of symmetry becomes as large as the $y$ -invariant solution (5.3), i.e. at a distance $|y|>0$ such that $|A_{2}(y)|/a_{0}=(d_{1}\cos \unicode[STIX]{x1D703}_{t})/[d_{2}\cos (\unicode[STIX]{x1D703}_{2}-\unicode[STIX]{x1D703}_{t})]$ . For the results of figure 10, we find that $|A_{2}|$ increases approximately linearly with $|y|$ . We estimated the growth slope to be ${\sim}0.02$ per wavelength $\unicode[STIX]{x1D706}_{1}$ , such that the maximum value predicted by (5.3), i.e. $|A_{2}|/a_{0}=3.4$ , is reached at $y/\unicode[STIX]{x1D706}_{1}=160$ in this case.