3.1 A warm up and useful result – propagation in a region of constant depth

To begin with, we shall recover the shallow water equation over channels of constant depth $h$ . To do so, we consider the following expansions

(3.5a,b ) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D711}^{\mathit{3d}}=\unicode[STIX]{x1D711}^{0}(\boldsymbol{r},z_{\mathit{r}})+\unicode[STIX]{x1D700}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r},z_{\mathit{r}})+\cdots \,,\quad \boldsymbol{u}^{\mathit{3d}}=\boldsymbol{u}^{0}(\boldsymbol{r},z_{\mathit{r}})+\unicode[STIX]{x1D700}\boldsymbol{u}^{1}(\boldsymbol{r},z_{\mathit{r}})+\cdots \,, & & \displaystyle\end{eqnarray}$$

and the horizontal gradient is replaced by

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}+\frac{\boldsymbol{e}_{\mathit{z}}}{\unicode[STIX]{x1D700}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{\mathit{r}}},\quad \text{with }\unicode[STIX]{x1D735}=\boldsymbol{e}_{\mathit{x}}\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}+\boldsymbol{e}_{\mathit{y}}\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}, & & \displaystyle\end{eqnarray}$$

so that (3.3) becomes, for $n\geqslant 0$ ,

(3.7) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}z_{\mathit{r}}}=0,\quad \boldsymbol{u}^{n}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{n}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{n+1}}{\unicode[STIX]{x2202}z_{\mathit{r}}}\,\boldsymbol{e}_{\mathit{z}},\\[12.0pt] \displaystyle \frac{\unicode[STIX]{x2202}u_{z}^{0}}{\unicode[STIX]{x2202}z_{\mathit{r}}}=0,\quad \text{div}\,\boldsymbol{u}^{n}+\frac{\unicode[STIX]{x2202}u_{z}^{n+1}}{\unicode[STIX]{x2202}z_{\mathit{r}}}=0,\\[12.0pt] \displaystyle {u_{z}^{0}}_{|z_{\mathit{r}}=0}=0,\quad {u_{z}^{n+1}}_{|z_{\mathit{r}}=0}=-\unicode[STIX]{x2202}_{tt}{\unicode[STIX]{x1D711}^{n}}_{|z_{\mathit{r}}=0},\quad {u_{z}^{n}}_{|z_{\mathit{r}}=-h^{\ast }}=0,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $h^{\ast }=h/\ell$ is the channel depth in dimensionless form.

3.2 The first order

From (3.7), $\unicode[STIX]{x1D711}^{0}$ and $u_{z}^{0}$ do not depend on $z_{\mathit{r}}$ . Besides, since $u_{z}^{0}(\boldsymbol{r})$ vanishes at $z_{\mathit{r}}=0$ , we deduce that $u_{z}^{0}=0$ and eventually

(3.8) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{0}(\boldsymbol{r})=\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r}). & & \displaystyle\end{eqnarray}$$

Also, it follows from $0=u_{z}^{0}=\unicode[STIX]{x2202}_{z_{\mathit{r}}}\unicode[STIX]{x1D711}^{1}$ that $\unicode[STIX]{x1D711}^{1}(\boldsymbol{r})$ does not depend on $z_{\mathit{r}}$ , hence

(3.9) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{1}(\boldsymbol{r},z_{\mathit{r}})=\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r})+\frac{\unicode[STIX]{x2202}u_{z}^{2}}{\unicode[STIX]{x2202}z_{\mathit{r}}}(\boldsymbol{r},z_{\mathit{r}})\;\boldsymbol{e}_{z}, & & \displaystyle\end{eqnarray}$$

and $\text{div}\,\boldsymbol{u}^{1}=\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r})$ since the divergence is two-dimensional. Thus, integrating for $n=0,1$ the relation $\text{div}\,\boldsymbol{u}^{n}+\unicode[STIX]{x2202}_{z_{\mathit{r}}}u_{z}^{n+1}=0$ between $z_{\mathit{r}}=-h^{\ast }$ and $z_{\mathit{r}}=0$ , along with the boundary conditions ${u_{z}^{n+1}}_{|z_{\mathit{r}}=-h^{\ast }}=0$ and ${u_{z}^{n+1}}_{|z_{\mathit{r}}=0}=\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{n}(\boldsymbol{r})$ , we get that

(3.10) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle h^{\ast }\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})-\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})=0,\\[4.0pt] \displaystyle h^{\ast }\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r})-\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r})=0,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

which is the usual shallow water equation applying at the first and second orders.

3.3 Effective propagation in a region with structured sea bed

Over the structured ridge and far away from its end boundaries, we assume that the fields can be expanded as follows

(3.11a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{\mathit{3d}}=\unicode[STIX]{x1D711}^{0}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})+\unicode[STIX]{x1D700}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})+\cdots \,,\quad \boldsymbol{u}^{\mathit{3d}}=\boldsymbol{u}^{0}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})+\unicode[STIX]{x1D700}\boldsymbol{u}^{1}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})+\cdots \,, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $\boldsymbol{r}=(x,y)$ well suited to describe the variations of the propagating field at the wavelength scale, and $(x_{\mathit{r}},z_{\mathit{r}})$ well suited to describe the variations of the evanescent field in the vertical direction (as previously) and along $x$ because of the microstructure. In this region, the horizontal gradient is replaced by

(3.12a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}+\frac{1}{\unicode[STIX]{x1D700}}\,\unicode[STIX]{x1D735}_{\mathit{r}},\quad \text{with }\unicode[STIX]{x1D735}_{\mathit{r}}=\boldsymbol{e}_{\mathit{x}}\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{\mathit{r}}}+\boldsymbol{e}_{\mathit{z}}\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{\mathit{r}}}, & & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D735}$ is defined in (3.6); it is useful to remark that

(3.13) $$\begin{eqnarray}\displaystyle \text{div}\,\unicode[STIX]{x1D735}_{\mathit{r}}=\text{div}_{\mathit{r}}\unicode[STIX]{x1D735}=\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x\unicode[STIX]{x2202}x_{\mathit{r}}}. & & \displaystyle\end{eqnarray}$$

In addition, we assume that $\unicode[STIX]{x1D711}^{n}$ and $\boldsymbol{u}^{n}$ are periodic with respect to $x_{\mathit{r}}$ , and we define $\unicode[STIX]{x1D6FA}$ the elementary cell in $(x_{\mathit{r}},z_{\mathit{r}})$ coordinates, with $x_{\mathit{r}}\in (-1/2,1/2)$ (figure 3). Now, equation (2.1) reads as

(3.14) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{0}=0,\quad \boldsymbol{u}^{n}=\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{n+1}+\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{n},\\[4.0pt] \displaystyle \text{div}_{\mathit{r}}\,\boldsymbol{u}^{0}=0,\quad \text{div}_{\mathit{r}}\,\boldsymbol{u}^{n+1}+\text{div}\,\boldsymbol{u}^{n}=0,\\[4.0pt] \displaystyle {u_{z}^{0}}_{|z_{\mathit{r}}=0}=0,\quad {u_{z}^{n+1}}_{|z_{\mathit{r}}=0}=-\unicode[STIX]{x2202}_{tt}{\unicode[STIX]{x1D711}^{n}}_{|z_{\mathit{r}}=0},\quad \boldsymbol{u}^{n}\boldsymbol{\cdot }\boldsymbol{n}_{|\unicode[STIX]{x1D6E4}}=0,\\[4.0pt] \unicode[STIX]{x1D711}^{n},\boldsymbol{u}^{n}\;\text{periodic with respect to}~x_{\mathit{r}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Figure 3.

Homogenization in the bulk. On the left, the ridge of finite extent in $(x,y,z)$ coordinates ( $\boldsymbol{r}=(x,y)$ ). On the right, the unit cell $\unicode[STIX]{x1D6FA}$ in rescaled in $(x_{\mathit{r}},z_{\mathit{r}})$ coordinates, with $h^{\ast }(x_{\mathit{r}})=h(x/\unicode[STIX]{x1D700})/\ell$ .

3.3.1 First order

At the leading order in $1/\unicode[STIX]{x1D700}$ , equation (3.14) gives $\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{0}=0$ , from which $\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})$ depends on the macroscopic coordinates $\boldsymbol{r}$ only. It follows that $\unicode[STIX]{x1D711}^{1}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})$ satisfies the following problem

(3.15) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D6FB}_{\mathit{r}}^{2}\unicode[STIX]{x1D711}^{1}=0,\\[4.0pt] \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{1}}{\unicode[STIX]{x2202}z_{\mathit{r}}}_{|z_{\mathit{r}}=0}=0,\quad \unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{1}\boldsymbol{\cdot }\boldsymbol{n}=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x}(\boldsymbol{r})\,{n_{x}}_{|\unicode[STIX]{x1D6E4}},\\[10.0pt] \unicode[STIX]{x1D711}^{1},\quad \unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{1}\text{ periodic with respect to}~x_{\mathit{r}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where we have used that $\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{0}\boldsymbol{\cdot }\boldsymbol{n}=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D711}^{0}\,n_{x}$ (since the problem is invariant along $y$ hence $n_{y}=0$ ). Next, the velocity at the leading order reads as

(3.16) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{0}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})=\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})+\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}}). & & \displaystyle\end{eqnarray}$$

The problem (3.15) is linear with respect to $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})$ so that we can set

(3.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{1}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x}(\boldsymbol{r})\,\unicode[STIX]{x1D6F7}(x_{\mathit{r}},z_{\mathit{r}})+\langle \unicode[STIX]{x1D711}^{1}\rangle (\boldsymbol{r}), & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D6F7}$ satisfying the two-dimensional cell problem defined in $\unicode[STIX]{x1D6FA}$

(3.18) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D6FB}_{\mathit{r}}^{2}\unicode[STIX]{x1D6F7}=0,\quad \text{in}\;\unicode[STIX]{x1D6FA}\\[4.0pt] \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}z_{\mathit{r}}}(x_{\mathit{r}},0)=0,\quad \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}n}_{|\unicode[STIX]{x1D6E4}}=-n_{x},\quad \unicode[STIX]{x1D6F7},\quad \unicode[STIX]{x2202}_{x_{\mathit{r}}}\unicode[STIX]{x1D6F7}\;\text{periodic with respect to}~x_{\mathit{r}}.\end{array}\right\}\quad & & \displaystyle\end{eqnarray}$$

Figure 4.

Example of field $\unicode[STIX]{x1D6F7}(x_{\mathit{r}},z_{\mathit{r}})$ solution of (3.18) in the unit cell $\unicode[STIX]{x1D6FA}$ ( $h^{+}/\ell =1$ , $h^{-}/\ell =0.3$ and $\unicode[STIX]{x1D709}=0.4$ ).

It is worth noting that this problem is independent of the frequency, and $(\unicode[STIX]{x1D6F7}+x_{\mathit{r}})$ corresponds to the problem of potential flow in a corrugated rigid duct (figure 4). In (3.15), $\unicode[STIX]{x1D711}^{1}$ is defined up to a function of $\boldsymbol{r}$ ; in (3.17), we have fixed this function equal to

(3.19) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D711}^{1}\rangle (\boldsymbol{r})=\frac{1}{\langle h^{\ast }\rangle }\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D711}^{1}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})\,\text{d}x_{r}\,\text{d}z_{\mathit{r}}, & & \displaystyle\end{eqnarray}$$

being the mean value of $\unicode[STIX]{x1D711}^{1}$ , with $\langle h^{\ast }\rangle$ the surface of $\unicode[STIX]{x1D6FA}$ . Doing so, we have imposed the condition

(3.20) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6F7}\,\text{d}x_{r}\,\text{d}z_{\mathit{r}}=0, & & \displaystyle\end{eqnarray}$$

hence for symmetric layers, $\unicode[STIX]{x1D6F7}$ is an odd function of $x_{\mathit{r}}$ . It is now sufficient to integrate, from (3.14), the relation $\text{div}_{\mathit{r}}\,\boldsymbol{u}^{1}+\text{div}\,\boldsymbol{u}^{0}=0$ over the unit cell $\unicode[STIX]{x1D6FA}$ to get that

(3.21) $$\begin{eqnarray}\displaystyle h_{x}^{\ast }\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x^{2}}(\boldsymbol{r})+h_{y}^{\ast }\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(\boldsymbol{r})-\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})=0, & & \displaystyle\end{eqnarray}$$

where $(h_{x}^{\ast },h_{y}^{\ast })$ are given by

(3.22a,b ) $$\begin{eqnarray}\displaystyle h_{x}^{\ast }=\int _{\unicode[STIX]{x1D6FA}}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}x_{\mathit{r}}}+1\right)\text{d}x_{r}\,\text{d}z_{\mathit{r}},\quad h_{y}^{\ast }=\langle h^{\ast }\rangle . & & \displaystyle\end{eqnarray}$$

To derive the above relation, we have used that (i) $\int _{\unicode[STIX]{x1D6FA}}\text{div}_{\mathit{r}}\,\boldsymbol{u}^{1}=\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})$ from Green’s theorem with ${u_{z}^{1}}_{|z_{\mathit{r}}=0}=\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})$ , $\boldsymbol{u}^{1}\boldsymbol{\cdot }\boldsymbol{n}$ vanishing on $\unicode[STIX]{x1D6E4}$ and being periodic with respect to $x_{\mathit{r}}$ and (ii) $\text{div}\,\boldsymbol{u}^{0}=\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})+\unicode[STIX]{x2202}_{xx}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})\unicode[STIX]{x2202}_{x_{r}}\unicode[STIX]{x1D6F7}$ making use of (3.16).

3.3.2 Second order

To get the effective propagation at the second order, the problem on $\unicode[STIX]{x1D711}^{2}$ has to be defined. From (3.14), it is easy to see that $\unicode[STIX]{x1D711}^{2}$ satisfies the following problem

(3.23) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D6FB}_{\mathit{r}}^{2}\unicode[STIX]{x1D711}^{2}=-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x^{2}}(\boldsymbol{r})\left(2\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}x_{\mathit{r}}}+1\right)-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(\boldsymbol{r}),\quad \text{in}\;\unicode[STIX]{x1D6FA},\\[10.0pt] \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{2}}{\unicode[STIX]{x2202}z_{\mathit{r}}}_{|z_{\mathit{r}}=0}=-h_{x}^{\ast }\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x^{2}}(\boldsymbol{r})-h_{y}^{\ast }\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(\boldsymbol{r}),\\ \displaystyle {\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{2}\boldsymbol{\cdot }\boldsymbol{n}}_{|\unicode[STIX]{x1D6E4}}=-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x^{2}}(\boldsymbol{r})\,\unicode[STIX]{x1D6F7}_{|\unicode[STIX]{x1D6E4}}\;n_{x}-\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D711}^{1}\rangle }{\unicode[STIX]{x2202}x}(\boldsymbol{r})\,{n_{x}}_{|\unicode[STIX]{x1D6E4}},\\[10.0pt] \unicode[STIX]{x1D711}^{2},\quad \unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{1}\;\text{periodic with respect to}~x_{\mathit{r}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The above system has been obtained using that

(3.24) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{1}=\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D711}^{2}+\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{1}, & & \displaystyle\end{eqnarray}$$

and the boundary conditions on $\boldsymbol{u}^{1}$ in (2.1), along with (3.21). The problem (3.23) on $\unicode[STIX]{x1D711}^{2}$ in $(x_{\mathit{r}},z_{\mathit{r}})$ coordinates is linear with respect $\unicode[STIX]{x2202}_{xx}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})$ , $\unicode[STIX]{x2202}_{yy}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})$ and $\unicode[STIX]{x2202}_{x}\langle \unicode[STIX]{x1D711}^{1}\rangle (\boldsymbol{r})$ , thus we can set

(3.25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{2}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})=\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D711}^{1}\rangle }{\unicode[STIX]{x2202}x}(\boldsymbol{r})\,\unicode[STIX]{x1D6F7}(x_{\mathit{r}},z_{\mathit{r}})-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x^{2}}(\boldsymbol{r})\,\tilde{\unicode[STIX]{x1D6F7}}(x_{\mathit{r}},z_{\mathit{r}})-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(\boldsymbol{r})\,\hat{\unicode[STIX]{x1D6F7}}(x_{\mathit{r}},z_{\mathit{r}})+\langle \unicode[STIX]{x1D711}^{2}\rangle (\boldsymbol{r}), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $\tilde{\unicode[STIX]{x1D6F7}}$ and $\hat{\unicode[STIX]{x1D6F7}}$ satisfying the cell problems

(3.26a,b ) $$\begin{eqnarray}\displaystyle \begin{array}{@{}ll@{}}\!\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D6FB}_{\mathit{r}}^{2}\tilde{\unicode[STIX]{x1D6F7}}=2\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}x_{\mathit{r}}}+1,\quad \text{in}\;\unicode[STIX]{x1D6FA},\\[7.0pt] \displaystyle \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}z_{\mathit{r}}}_{|z_{\mathit{r}}=0}=h_{x}^{\ast },\quad \displaystyle \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}n}_{|\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6F7}\,{n_{x}}_{|\unicode[STIX]{x1D6E4}},\\[12.0pt] \displaystyle \tilde{\unicode[STIX]{x1D6F7}}\;\text{and }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}x_{\mathit{r}}}\;\text{periodic with respect to}~x_{\mathit{r}}.\end{array}\right. & \left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D6FB}_{\mathit{r}}^{2}\hat{\unicode[STIX]{x1D6F7}}=1,\quad \text{in}\;\unicode[STIX]{x1D6FA},\\[4.0pt] \displaystyle \frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}z_{\mathit{r}}}_{|z_{\mathit{r}}=0}=h_{y}^{\ast },\quad \displaystyle \frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}n}_{|\unicode[STIX]{x1D6E4}}=0,\\[10.0pt] \displaystyle \hat{\unicode[STIX]{x1D6F7}}\;\text{and }\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}x_{\mathit{r}}}\;\text{periodic with respect to}~x_{\mathit{r}}.\end{array}\right.\end{array} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In (3.25), we have imposed that $\tilde{\unicode[STIX]{x1D6F7}}$ and $\hat{\unicode[STIX]{x1D6F7}}$ have zero mean, hence for a symmetric sea bed shape in $\unicode[STIX]{x1D6FA}$ they are even functions of $x_{\mathit{r}}$ . It is now sufficient to integrate $\text{div}_{\mathit{r}}\,\boldsymbol{u}^{2}+\text{div}\,\boldsymbol{u}^{1}=0$ over $\unicode[STIX]{x1D6FA}$ to get that

(3.27) $$\begin{eqnarray}\displaystyle \displaystyle h_{x}^{\ast }\,\frac{\unicode[STIX]{x2202}^{2}\langle \unicode[STIX]{x1D711}^{1}\rangle }{\unicode[STIX]{x2202}x^{2}}+h_{y}^{\ast }\,\frac{\unicode[STIX]{x2202}^{2}\langle \unicode[STIX]{x1D711}^{1}\rangle }{\unicode[STIX]{x2202}y^{2}}-\unicode[STIX]{x2202}_{tt}\langle \unicode[STIX]{x1D711}^{1}\rangle =0. & & \displaystyle\end{eqnarray}$$

Here, we have used that $\int _{\unicode[STIX]{x1D6FA}}\text{div}_{\mathit{r}}\,\boldsymbol{u}^{2}=\unicode[STIX]{x2202}_{tt}\langle \unicode[STIX]{x1D711}^{1}\rangle (\boldsymbol{r})$ , using Green’s theorem along with the boundary conditions on $\boldsymbol{u}^{2}$ in (2.1) and $\unicode[STIX]{x1D711}^{1}$ in (3.17) along with (3.20). We also used $\boldsymbol{u}^{1}$ in (3.24) with $\unicode[STIX]{x1D711}^{1},\unicode[STIX]{x1D711}^{2}$ in (3.17) and (3.25), where $\int _{\unicode[STIX]{x1D6E4}}\,\text{d}s\,\tilde{\unicode[STIX]{x1D6F7}}\;n_{x}=\int _{\unicode[STIX]{x1D6E4}}\,\text{d}s\,\hat{\unicode[STIX]{x1D6F7}}\;n_{x}=0$ and (3.20).

Figure 5.

Asymptotic analysis at the end boundary of the structured ridge. On the left, we zoom in on the vicinity of the boundary at $x=0$ . On the right, the elementary cell $\unicode[STIX]{x1D654}$ in $(x_{\mathit{r}}^{\prime },z_{\mathit{r}})$ coordinates is infinite along $x_{\mathit{r}}^{\prime }$ , with $h^{\ast }(x_{\mathit{r}}^{\prime })=h(x/\ell )/\ell$ , $x_{\mathit{r}}^{\prime }\in (-\infty ,+\infty )$ .

4.1 The first order

From (4.8), we have $\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D713}^{0}=0$ , thus $\unicode[STIX]{x1D713}^{0}(y)$ does not depend on the rapid coordinate. Hence, from (4.5), we get

(4.9a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}^{0}(y)=\unicode[STIX]{x1D711}^{0}(0^{\pm },y),\quad \unicode[STIX]{x27E6}\unicode[STIX]{x1D711}^{0}\unicode[STIX]{x27E7}=0. & & \displaystyle\end{eqnarray}$$

Now, we also have $\text{div}_{\mathit{r}}\,\boldsymbol{v}^{0}=0$ that we integrate over

(4.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D654}_{m}=\{x_{\mathit{r}}^{\prime }\in (-x_{m}^{\prime },x_{m}^{\prime }),z_{\mathit{r}}\in (-h(x_{\mathit{r}}^{\prime }),0)\}, & & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D654}_{m}\rightarrow \unicode[STIX]{x1D654}$ when $x_{m}^{\prime }\rightarrow +\infty$ , see figure 6. Using the boundary conditions $\boldsymbol{v}^{0}\boldsymbol{\cdot }\boldsymbol{n}=0$ at $z_{\mathit{r}}=0$ and on $\unicode[STIX]{x1D6E4}$ , we get that

(4.11) $$\begin{eqnarray}\displaystyle \lim _{x_{\mathit{m}}^{\prime }\rightarrow +\infty }\left[\int _{-h^{\ast }(x_{\mathit{m}}^{\prime })}^{0}v_{x}^{0}(y,x_{\mathit{m}}^{\prime },z_{\mathit{r}})\,\text{d}z_{\mathit{r}}-\int _{-h^{\ast }}^{0}v_{x}^{0}(y,-x_{\mathit{m}}^{\prime },z_{\mathit{r}})\,\text{d}z_{\mathit{r}}\right]=0. & & \displaystyle\end{eqnarray}$$

It is now sufficient to use the matching conditions (4.5), with $u_{x}^{0}(\boldsymbol{r})=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})$ which applies at $x=0^{-}$ and with $u_{x}^{0}(\boldsymbol{r},x_{\mathit{r}},z_{\mathit{r}})=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D711}^{0}(\boldsymbol{r})(\unicode[STIX]{x2202}_{x_{\mathit{r}}}\unicode[STIX]{x1D6F7}(x_{\mathit{r}},z_{\mathit{r}})+1)$ from (3.16) to (3.17) which applies at $x=0^{+}$ , and we get

(4.12) $$\begin{eqnarray}\displaystyle D^{0}(y)=h^{\ast }\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x}(0^{-},y)=h_{x}^{\ast }\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x}(0^{+},y), & & \displaystyle\end{eqnarray}$$

where we have used that

(4.13) $$\begin{eqnarray}\displaystyle \int _{-h^{\ast }(x_{\mathit{r}})}^{0}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}x_{\mathit{r}}}+1\right)\text{d}z_{\mathit{r}}=h_{x}^{\ast }, & & \displaystyle\end{eqnarray}$$

is independent of $x_{\mathit{r}}$ (and thus, independent of $x_{\mathit{r}}^{\prime }$ ), see § A.1. At this leading order, we find that the potential and the flow rate are continuous. We shall see that this is not the case anymore at the next order.

4.2 The second order

First, we shall consider the problem on $\unicode[STIX]{x1D713}^{1}$ , which from (4.8) reads as

(4.14) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FB}_{\mathit{r}}^{2}\unicode[STIX]{x1D713}^{1}=0,\\[4.0pt] \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}^{1}}{\unicode[STIX]{x2202}z_{\mathit{r}}}_{|z_{\mathit{r}}=0}=0,\quad \unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D713}^{1}\boldsymbol{\cdot }\boldsymbol{n}_{|\unicode[STIX]{x1D6E4}}=0,\\[9.0pt] \displaystyle \lim _{x_{\mathit{r}}^{\prime }\rightarrow -\infty }\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D713}^{1}=\frac{D^{0}(y)}{h^{\ast }}\,\boldsymbol{e}_{\mathit{x}},\quad \displaystyle \lim _{x_{\mathit{r}}^{\prime }\rightarrow +\infty }\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D713}^{1}=\frac{D^{0}(y)}{h_{x}^{\ast }}\,(\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}(x_{\mathit{r}},z_{\mathit{r}})+\boldsymbol{e}_{\mathit{x}}),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with

(4.15) $$\begin{eqnarray}\displaystyle \boldsymbol{v}^{0}(y,x_{\mathit{r}}^{\prime },z_{\mathit{r}})=\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D713}^{1}(y,x_{\mathit{r}}^{\prime },z_{\mathit{r}})+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y}(0,y)\,\boldsymbol{e}_{\mathit{y}}. & & \displaystyle\end{eqnarray}$$

To determine the limits when $x_{\mathit{r}}^{\prime }\rightarrow \pm \infty$ , we have used the matching condition (4.5) on $\boldsymbol{v}^{0}$ with $\boldsymbol{u}^{0}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}^{0}$ which applies at $x=0^{-}$ and with $\boldsymbol{u}^{0}$ in (3.16) which applies at $x=0^{+}$ , along with (3.17). The problem on $\unicode[STIX]{x1D6F9}^{1}$ in (4.14) is linear with respect to $D^{0}(y)$ and we can set

(4.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}^{1}(y,x_{\mathit{r}}^{\prime },z_{\mathit{r}})=D^{0}(y)\,\unicode[STIX]{x1D6F9}(x_{\mathit{r}}^{\prime },z_{\mathit{r}})+\unicode[STIX]{x1D711}^{1}(0^{-},y). & & \displaystyle\end{eqnarray}$$

Since $\unicode[STIX]{x1D713}_{1}$ in (4.14) is defined up to a function of $y$ , we set this function equal to $\unicode[STIX]{x1D711}^{1}(0^{-},y)$ ; next $\unicode[STIX]{x1D6F9}$ satisfies the elementary problem

(4.17) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D6FB}_{\mathit{r}}^{2}\unicode[STIX]{x1D6F9}=0,\quad \text{in }\unicode[STIX]{x1D654},\\[4.0pt] \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}z_{\mathit{r}}}_{|z_{\mathit{r}}=0}=0,\quad \unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D6F9}\boldsymbol{\cdot }\boldsymbol{n}_{|\unicode[STIX]{x1D6E4}}=0,\\[10.0pt] \displaystyle \lim _{x_{\mathit{r}}^{\prime }\rightarrow -\infty }\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D6F9}=\frac{\boldsymbol{e}_{\mathit{x}}}{h^{\ast }},\quad \displaystyle \lim _{x_{\mathit{r}}^{\prime }\rightarrow +\infty }\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D6F9}=\frac{\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}+\boldsymbol{e}_{\mathit{x}}}{h_{x}^{\ast }},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Figure 7.

(a) Example of field $\unicode[STIX]{x1D6F9}(x_{\mathit{r}}^{\prime },z_{\mathit{r}})$ satisfying the elementary problem (4.17). (b) Illustration of the extension of transition effects due to evanescent field, with a resulting coefficient ${\mathcal{B}}=1.08$ in (4.18).

(an example of the field $\unicode[STIX]{x1D6F9}$ is given in figure 7). From the limits on $\unicode[STIX]{x1D735}_{\mathit{r}}\unicode[STIX]{x1D6F9}$ , we have that

(4.18) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}ll@{}}\displaystyle \unicode[STIX]{x1D6F9}(x_{\mathit{r}}^{\prime },z_{\mathit{r}})\sim \frac{x_{\mathit{r}}^{\prime }}{h^{\ast }}, & \text{when }x_{\mathit{r}}^{\prime }\rightarrow -\infty ,\\[10.0pt] \displaystyle \unicode[STIX]{x1D6F9}(x_{\mathit{r}}^{\prime },z_{\mathit{r}})\sim \frac{\unicode[STIX]{x1D6F7}(x_{\mathit{r}},z_{\mathit{r}})+x_{\mathit{r}}^{\prime }}{h_{x}^{\ast }}+{\mathcal{B}}, & \text{when }x_{\mathit{r}}^{\prime }\rightarrow +\infty ,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with ${\mathcal{B}}$ a constant. Using these limits in (4.16) and the matching conditions for $\unicode[STIX]{x1D713}^{1}$ in (4.6), we get that

(4.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{1}(0^{+},y,x_{\mathit{r}},z_{\mathit{r}})=D^{0}(y)\left(\frac{\unicode[STIX]{x1D6F7}(x_{\mathit{r}},z_{\mathit{r}})}{h_{x}^{\ast }}+{\mathcal{B}}\right)+\unicode[STIX]{x1D711}^{1}(0^{-},y). & & \displaystyle\end{eqnarray}$$

Expectedly, we recover that $\unicode[STIX]{x1D711}_{|x=0^{+}}^{1}$ is periodic with respect $x_{\mathit{r}}$ . Eventually, with $\langle \unicode[STIX]{x1D6F7}\rangle =0$ , we get the jump in $\langle \unicode[STIX]{x1D711}^{1}\rangle$

(4.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x27E6}\langle \unicode[STIX]{x1D711}^{1}\rangle \unicode[STIX]{x27E7}=\langle \unicode[STIX]{x1D711}^{1}\rangle (0^{+},y)-\unicode[STIX]{x1D711}^{1}(0^{-},y)={\mathcal{B}}\,D^{0}(y), & & \displaystyle\end{eqnarray}$$

(with $\langle \unicode[STIX]{x1D711}^{1}\rangle =\unicode[STIX]{x1D711}^{1}$ for $x<0$ ). At this order, the potential is not continuous anymore and we shall see that the mass flux is not continuous either. To do so, we integrate over $\unicode[STIX]{x1D654}_{m}$ the incompressibility relation $(\text{div}_{\mathit{r}}\,\boldsymbol{v}^{1}+\unicode[STIX]{x2202}_{y}v_{y}^{0})=0$ from (4.8). With $\unicode[STIX]{x2202}_{y}v_{y}^{0}=\unicode[STIX]{x2202}_{yy}\unicode[STIX]{x1D711}^{0}(0,y)$ applying in $\unicode[STIX]{x1D654}_{m}$ from (4.15), we get that

(4.21) $$\begin{eqnarray}\displaystyle \displaystyle \int _{\unicode[STIX]{x1D654}_{m}}\frac{\unicode[STIX]{x2202}v_{y}^{0}}{\unicode[STIX]{x2202}y}=\left(h^{\ast }\,x_{m}^{\prime }+\int _{0}^{x_{m}^{\prime }}h^{\ast }(x_{\mathit{r}}^{\prime })\,\text{d}x_{r}^{\prime }\right)\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(0,y). & & \displaystyle\end{eqnarray}$$

Next, we use Green’s theorem and we account for the boundary conditions in (4.8) and for the matching condition on $\boldsymbol{v}^{1}$ in (4.6) to get that

(4.22) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D654}_{m}}\text{div}_{\mathit{r}}\,\boldsymbol{v}^{1}=I^{-}(x_{m}^{\prime })+I^{+}(x_{m}^{\prime }), & & \displaystyle\end{eqnarray}$$

with

(4.23) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle I^{-}(x_{m}^{\prime })=-h^{\ast }\left[u_{x}^{1}-x_{m}^{\prime }\frac{\unicode[STIX]{x2202}u_{x}^{0}}{\unicode[STIX]{x2202}x}\right]_{|x=0^{-}}-x_{m}^{\prime }\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{0}(0,y),\\[12.0pt] \displaystyle I^{+}(x_{m}^{\prime })=\int _{-h^{\ast }(x_{m}^{\prime })}^{0}\left[u_{x}^{1}+x_{m}^{\prime }\frac{\unicode[STIX]{x2202}u_{x}^{0}}{\unicode[STIX]{x2202}x}\right]_{|x=0^{+}}\text{d}z_{\mathit{r}}-x_{m}^{\prime }\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D711}^{0}(0,y).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

At $x=0^{-}$ , we simply have $u_{x}^{n}=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D711}^{n}$ , $n=0,1$ ; next, making use of (3.10) leaves us with

(4.24) $$\begin{eqnarray}\displaystyle \displaystyle I^{-}(x_{m}^{\prime })=-h^{\ast }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{1}}{\unicode[STIX]{x2202}x}(0^{-},y)-x_{m}^{\prime }h^{\ast }\;\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(0,y). & & \displaystyle\end{eqnarray}$$

The term $I^{+}$ is evaluated using that $u_{x}^{0}=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D711}^{0}+\unicode[STIX]{x2202}_{x_{\mathit{r}}}\unicode[STIX]{x1D711}^{1}$ from (3.16) and $u_{x}^{1}=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D711}^{1}+\unicode[STIX]{x2202}_{x_{\mathit{r}}}\unicode[STIX]{x1D711}^{2}$ from (3.24), with $\unicode[STIX]{x1D711}^{1}$ in (3.17) and $\unicode[STIX]{x1D711}^{2}$ in (3.25), namely

(4.25) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle u_{x}^{0}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x}(\boldsymbol{r})\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}x_{\mathit{r}}}+1\right),\\ \displaystyle u_{x}^{1}=\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D711}^{1}\rangle }{\unicode[STIX]{x2202}x}(\boldsymbol{r})\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}x_{\mathit{r}}}+1\right)-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x^{2}}(\boldsymbol{r})\left(\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}x_{\mathit{r}}}-\unicode[STIX]{x1D6F7}\right)-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(\boldsymbol{r})\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}x_{\mathit{r}}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

It follows that

(4.26) $$\begin{eqnarray}\displaystyle \displaystyle I^{+}(x_{m}^{\prime }) & = & \displaystyle h_{x}^{\ast }\,\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D711}^{1}\rangle }{\unicode[STIX]{x2202}x}(0^{+},y)-\tilde{g}(x_{m}^{\prime })\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}x^{2}}(0^{+},y)\nonumber\\ \displaystyle & & \displaystyle -\,{\hat{g}}(x_{m}^{\prime })\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(0,y)-x_{m}^{\prime }\,\langle h^{\ast }\rangle \,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(0,y),\end{eqnarray}$$

with the first term obtained using (4.13) and the last term obtained using (3.21). In the above expression, we have defined

(4.27a,b ) $$\begin{eqnarray}\displaystyle \tilde{g}(x_{m}^{\prime })=\int _{-h^{\ast }(x_{m}^{\prime })}^{0}\left(\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}x_{\mathit{r}}}-\unicode[STIX]{x1D6F7}\right)\text{d}z_{\mathit{r}},\quad {\hat{g}}(x_{m}^{\prime })=\int _{-h^{\ast }(x_{m}^{\prime })}^{0}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}x_{\mathit{r}}}\,\text{d}z_{\mathit{r}}. & & \displaystyle\end{eqnarray}$$

It is shown in appendix A that $\tilde{g}(x_{m}^{\prime })=0$ and ${\hat{g}}(x_{m}^{\prime })=\int _{x_{0m}}^{x_{m}^{\prime }}(h^{\ast }(x_{\mathit{r}}^{\prime })-\langle h^{\ast }\rangle )\,\text{d}x_{r}^{\prime }$ , with $x_{0m}$ any $x$ -abscissa where $\unicode[STIX]{x2202}_{x_{\mathit{r}}}\tilde{\unicode[STIX]{x1D6F7}}$ and $\unicode[STIX]{x2202}_{x_{\mathit{r}}}\hat{\unicode[STIX]{x1D6F7}}$ vanish, hence where $\unicode[STIX]{x1D6F7}$ vanishes. With the symmetry of our layers, the zeros of $\unicode[STIX]{x1D6F7}$ take place for $x_{\mathit{r}}=-1/2,0,1/2$ and for $x_{\mathit{r}}^{\prime }\in (x_{0m}-1/2,x_{0m}+1/2)$ , we have $x_{\mathit{r}}^{\prime }=x_{0m}+x_{\mathit{r}}$ ; it follows that $x_{0m}=\unicode[STIX]{x1D6FF}+(m+1)/2$ with $m=-1,0,\ldots$ integer and $\unicode[STIX]{x1D6FF}$ the distance from the origin to the first half-layer, see figure 6. Eventually, gathering (4.21) and (4.22) along with (4.24) and (4.26) leaves us with

(4.28) $$\begin{eqnarray}\displaystyle h_{x}^{\ast }\,\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D711}^{1}\rangle }{\unicode[STIX]{x2202}x}(0^{+},y)-h^{\ast }\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{1}}{\unicode[STIX]{x2202}x}(0^{-},y)={\mathcal{C}}\,\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}^{0}}{\unicode[STIX]{x2202}y^{2}}(0,y), & & \displaystyle\end{eqnarray}$$

with

(4.29) $$\begin{eqnarray}\displaystyle {\mathcal{C}}\equiv \int _{0}^{\unicode[STIX]{x1D6FF}}(\langle h^{\ast }\rangle -h^{\ast }(x_{\mathit{r}}^{\prime }))\,\text{d}x_{r}^{\prime }. & & \displaystyle\end{eqnarray}$$

4.3 Final homogenized problem

The final homogenized problem (2.2) along with (2.5) is constructed by considering the potential $(\unicode[STIX]{x1D711}^{0}+\unicode[STIX]{x1D700}\,\langle \unicode[STIX]{x1D711}^{1}\rangle )$ . Once coming back to the physical variables using (3.2) along with $h^{\ast }=h/\ell$ , equation (3.10) applying for $x<0$ , and (3.21), (3.27) applying for $x>0$ , gives simply (2.2). Then, from the transmission conditions (4.9), (4.20) and (4.12), (4.28), the potential $\unicode[STIX]{x1D711}$ satisfying (2.2) and (2.5) admits the same expansions as $(\unicode[STIX]{x1D711}^{0}+\unicode[STIX]{x1D700}\,\langle \unicode[STIX]{x1D711}^{1}\rangle )$ up to $O(\unicode[STIX]{x1D700}^{2})$ , hence the same expansion as the potential in the actual problem up to $O(\unicode[STIX]{x1D700}^{2})$ . For instance, equations (4.9) and (4.20) provide the relation

(4.30) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x27E6}\unicode[STIX]{x1D711}^{0}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D711}^{1}\unicode[STIX]{x27E7}=\unicode[STIX]{x1D700}\,{\mathcal{B}}\,D^{0}=\unicode[STIX]{x1D700}\,{\mathcal{B}}\,\overline{D}+O(\unicode[STIX]{x1D700}^{2}), & & \displaystyle\end{eqnarray}$$

and once coming back to the field with dimensions, $\unicode[STIX]{x27E6}\unicode[STIX]{x1D711}\unicode[STIX]{x27E7}={\mathcal{B}}\overline{D}$ . Eventually, we shall refer to the model at first order (that of $\unicode[STIX]{x1D711}^{0}$ ) the model obtained by setting ${\mathcal{B}}={\mathcal{C}}=0$ in (2.5).

5.1 The effective water depth $h_{x}$

To begin with, it is shown in § B.1 that $h_{x}\leqslant \langle h^{-1}\rangle ^{-1}\leqslant \langle h\rangle =h_{y}$ ; this means that the two-step homogenization procedure starting with the use of the shallow water equation overestimates the actual wavelength; this will be illustrated in § 6. Next, the effective depth $h_{x}$ depends on the geometry of the unit cell, see figure 4, and as such it depends on the spacing $\ell$ which is unusual in effective medium theories; this is a specificity in the context of water waves which do not appear in acoustics and in electromagnetism at leading order. In general, $h_{x}$ has to be determined numerically which is not too demanding. However, recalling that $h_{x}$ is related to a potential flow problem of a perfect fluid flowing over the ridge with Neumann boundary condition at the free surface, the following trends can be identified:

(i) When $\ell$ increases, $h_{x}$ tends to $\langle h^{-1}\rangle ^{-1}$ since all the dimensions of the periodic cell along $z_{\mathit{r}}$ decrease (see figure 4) and the shallow water approximation becomes increasingly valid. Conversely, when $\ell$ decreases, $h_{x}$ tends to $h^{-}$ since the flow cannot enter the dense region of the vertical array, which becomes similar to a flat bottom at depth $h^{-}$ (and this remains true for any filling $\unicode[STIX]{x1D709}$ ).

(ii) The limit $\unicode[STIX]{x1D709}=1$ simply corresponds to a ridge of constant water depth $h^{-}$ , hence $h_{x}=h^{-}$ . The limit $\unicode[STIX]{x1D709}\rightarrow 0$ is less trivial since it corresponds to vertical plates (layers of vanishing thickness) for which $h_{\mathit{x}0}={h_{x}}_{|\unicode[STIX]{x1D709}=0}$ is unknown. However, once $h^{-}\leqslant h_{\mathit{x}0}\leqslant h^{+}$ has been determined, a reasonable approximation of $h_{x}$ for any $\unicode[STIX]{x1D709}$ value is given by the following form

(5.1) $$\begin{eqnarray}\displaystyle h_{x}=\frac{1}{\unicode[STIX]{x1D709}/h^{-}+(1-\unicode[STIX]{x1D709})/h_{\mathit{x}0}}. & & \displaystyle\end{eqnarray}$$

These trends are illustrated in figure 8, where we report $h_{x}$ against $\unicode[STIX]{x1D709}$ for different spacing $\ell$ along with its estimate in (5.1) (using the numerical value of $h_{\mathit{x}0}$ ); in the reported cases, the agreement is better than 5 %. We also report $h_{\mathit{x}0}$ against $\ell$ for which we did not find a satisfactory estimate.

Figure 8.

Typical variations of  $h_{x}$ , (a)  $h_{x}$  against  $\unicode[STIX]{x1D709}$  for different  $\ell$ ( $h^{-}\!=0.05$ , $h^{+}=1$ ); blue plain lines show $h_{x}$ computed numerically and dotted red lines the estimate in (5.1). Dashed grey lines show $\langle h^{-1}\rangle ^{-1}$ and $\langle h\rangle$ ; (b) $h_{\mathit{x}0}={h_{x}}_{|\unicode[STIX]{x1D709}=0}$ against $\ell$ ( $h^{-}=0.05$ ).

5.2 The effective parameters $({\mathcal{B}},{\mathcal{C}})$

The effective parameters $({\mathcal{B}},{\mathcal{C}})$ entering in the transmission conditions depend on the geometry of the structuration of the ridge at its extremities. We begin with the parameter ${\mathcal{C}}$ which has an explicit dependence on this geometry, and use the parametrization $x_{\mathit{e}}\in (0,\ell )$ shown in figure 8. For $x_{\mathit{e}}=0$ the ridge starts with a vanishing thickness plate at depth $h^{-}$ and increasing $x_{\mathit{e}}$ produces a shift of the structuration toward the left. From (4.29), it is easy to see that ${\mathcal{C}}$ varies following

(5.2) $$\begin{eqnarray}\displaystyle {\mathcal{C}}\ell ^{2}=(h^{+}-h^{-})\left\{\begin{array}{@{}ll@{}}\displaystyle (1-\unicode[STIX]{x1D709})\left(x_{\mathit{e}}-\frac{\unicode[STIX]{x1D709}\ell }{2}\right), & x_{\mathit{e}}\in (0,\unicode[STIX]{x1D709}\ell ),\\[10.0pt] \displaystyle \unicode[STIX]{x1D709}\left(-x_{\mathit{e}}+\frac{(1+\unicode[STIX]{x1D709})\ell }{2}\right), & x_{\mathit{e}}\in (\unicode[STIX]{x1D709}\ell ,\ell ).\end{array}\right. & & \displaystyle\end{eqnarray}$$

Unlike ${\mathcal{C}}$ , ${\mathcal{B}}$ requires the resolution of an elementary problem which is quite involved. Indeed, the problem on $\unicode[STIX]{x1D6F9}$ in (4.17) is a problem of potential flow ending on one side with a semi-infinite periodic medium, and this is much more demanding numerically, see chapter 9 in Vinoles (Reference Vinoles2016). Fortunately, the evanescent field that we want to capture is localized in the close vicinity of this ending layer; this means that, in practice, only the ending layer (plus one or two additional layers) can be considered in the numerical resolution and a constant depth can be considered afterwards. It turns out that for the simple layered structure considered here, an accurate approximate expression for ${\mathcal{B}}$ is provided which avoids any numerical resolution. To build such an estimate, we make use of two ingredients, (i) the exact result ${\mathcal{B}}_{0}$ for ridges of constant immersion depth $h_{r}$ provided by Tuck (Reference Tuck1976) (in this case, it is easy to see that $\unicode[STIX]{x1D6F7}=0$ is a solution of (3.18) with (3.20); we also get from (3.26) that $\tilde{\unicode[STIX]{x1D6F7}}(x_{\mathit{r}},z_{\mathit{r}})=\hat{\unicode[STIX]{x1D6F7}}(x_{\mathit{r}},z_{\mathit{r}})=-z_{\mathit{r}}(z_{\mathit{r}}+2h_{c})/2$ ; this is expected since a constant bathymetry does not produce rapid variations of the field. Also expected is the fact that, from (3.17), $\unicode[STIX]{x1D711}^{1}(\boldsymbol{r})$ does not depend on $(x_{\mathit{r}},z_{\mathit{r}})$ as we have obtained in § 3.1, and from (3.22), $h_{x}=h_{y}=h_{r}$ . The transmission conditions at first order are unchanged and at second order simply follow from the problem set in (4.17), with $\unicode[STIX]{x1D6F7}=0$ , and ${\mathcal{C}}=0$ in agreement with Tuck (Reference Tuck1976))

(5.3) $$\begin{eqnarray}\displaystyle {\mathcal{B}}_{0}(\unicode[STIX]{x1D707})=\frac{1}{\unicode[STIX]{x03C0}}\left(\frac{\unicode[STIX]{x1D707}^{2}+1}{\unicode[STIX]{x1D707}}\log \frac{\unicode[STIX]{x1D707}+1}{|\unicode[STIX]{x1D707}-1|}-2\log \frac{4\unicode[STIX]{x1D707}}{|\unicode[STIX]{x1D707}^{2}-1|}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D707}=h/h_{r}$ , (ii) explicit lower bounds for ${\mathcal{B}}$ , see § B.2. Playing with these two ingredients, an estimate of ${\mathcal{B}}$ is found of the form

(5.4) $$\begin{eqnarray}\displaystyle {\mathcal{B}}\simeq \left\{\begin{array}{@{}l@{}}\displaystyle \left(x_{\mathit{e}}-\frac{\unicode[STIX]{x1D709}\ell }{2}\right)\left(\frac{1}{h^{-}}-\frac{1}{h_{x}}\right)+{\mathcal{B}}_{0}\left(\frac{h}{h^{-}}\right),\quad x_{\mathit{e}}\in (0,\unicode[STIX]{x1D709}\ell ),\\[10.0pt] \displaystyle (x_{\mathit{e}}-\unicode[STIX]{x1D709}\ell )\left(\frac{1}{h^{+}}-\frac{1}{h_{x}}\right)+\frac{\unicode[STIX]{x1D709}\ell }{2}\left(\frac{1}{h^{-}}-\frac{1}{h_{x}}\right)\\[10.0pt] \quad \displaystyle +\,\hat{{\mathcal{B}}}_{0}+\left({\mathcal{B}}_{0}\left(\frac{h}{h^{-}}\right)-\hat{{\mathcal{B}}}_{0}\right)\text{e}^{-((x_{\mathit{e}}-\unicode[STIX]{x1D709}\ell )/(h^{+}-h^{-}))},\quad x_{\mathit{e}}\in (\unicode[STIX]{x1D709}\ell ,\ell ),\end{array}\right. & & \displaystyle\end{eqnarray}$$

with $\hat{{\mathcal{B}}}_{0}={\mathcal{B}}_{0}(h/h^{+})+{\mathcal{B}}_{0}(h^{+}/h^{-})$ . We have accounted for the fact that $h^{+}$ and $h^{-}$ do not play the same role. Indeed, a first layer at immersion depth $h^{-}$ blocks efficiently the flow which does not feel the following layers. On the contrary, when the first layer is at immersion depth $h^{+}$ , the effect of the two first layers with decreasing depth immersions have to be accounted for.

Figure 9.

Variations of ${\mathcal{B}}$ and ${\mathcal{C}}$ against $x_{\mathit{e}}$ ; we used $h=2$ , $h^{-}=0.05$ , $\unicode[STIX]{x1D709}=0.1$ $\ell =2$ , and $h^{+}=0.06$ , 0.1 and 1. For ${\mathcal{B}}$ the plain lines show ${\mathcal{B}}$ computed numerically and the dashed lines show the approximate expression in (5.4) (here ${\mathcal{B}}_{0}(h/h^{-})=2.10$ ).

Typical variations of ${\mathcal{B}}$ and ${\mathcal{C}}$ against $x_{\mathit{e}}$ are shown in figure 9 (in the reported cases, the accuracy of the estimate of ${\mathcal{B}}$ in (5.4) is better than 0.2 %). It is worth noting that ${\mathcal{B}}$ is discontinuous at $x_{\mathit{e}}=0,\ell$ since the two situations are not identical; for $x_{\mathit{e}}=0$ the ridge starts on a vanishing thickness plate at immersion depth $h^{-}$ while for $x_{\mathit{e}}=\ell$ it starts with an entire layer at immersion depth $h^{+}$ . Also unusual is the fact that ${\mathcal{B}}$ can be negative, while ${\mathcal{B}}_{0}\geqslant 0$ for a ridge of constant height.

6.1 The scattering problem for an incident plane wave

In the homogenized problem, the potential $\unicode[STIX]{x1D711}(x,y)$ reads

(6.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(x,y)=\text{e}^{\text{i}k\sin \unicode[STIX]{x1D703}y}\left\{\begin{array}{@{}ll@{}}\displaystyle \text{e}^{\text{i}k_{x}x}+R\text{e}^{-\text{i}k_{x}x}, & x<0,\\ \displaystyle a\text{e}^{\text{i}K_{x}x}+b\text{e}^{-\text{i}K_{x}x}, & 0<x<L,\\ \displaystyle T\text{e}^{\text{i}k_{x}(x-L)}, & L<x,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where $k_{x}$ and $K_{x}$ satisfy the dispersion relations, from (2.2), $\unicode[STIX]{x1D714}^{2}=gh(k_{x}^{2}+k_{y}^{2})$ for $x<0$ and $x>L$ , and $\unicode[STIX]{x1D714}^{2}=g(h_{x}K_{x}^{2}+h_{y}k_{y}^{2})$ for $0<x<L$ and in the present case $k_{y}=K_{y}=k\sin \unicode[STIX]{x1D703}$ . It is worth noting that such dispersion relations miss the dispersion inherent to water waves. When needed, and in order to inspect the influence of the transmission conditions only, we use the following relations

(6.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D714}^{2}=gk\tan (kh),\\ \unicode[STIX]{x1D714}^{2}=g(K_{x}^{2}h_{x}+K_{y}^{2}h_{y})\tanh (X)/X,\quad X\equiv \sqrt{K_{x}^{2}h_{x}^{2}+K_{y}^{2}h_{y}^{2}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where the former is that valid for the flat bathymetry and the latter has been shown to be valid over a layered bathymetry up to relatively high frequencies (Maurel et al. Reference Maurel, Marigo, Cobelli, Petitjeans and Pagneux2017). The expressions of $(R,T,a,b)$ are given in appendix C, where the scattering properties in the homogenized problem are discussed.

Figure 10.

Accuracy of the different models; (a) to (c) report $\unicode[STIX]{x1D711}(x,y)$ to be compared with $\unicode[STIX]{x1D711}_{\mathit{num}}(x,y,0)$ in figure 1(a) and the profile $\unicode[STIX]{x1D711}(x,0)$ (dashed line) along with $\unicode[STIX]{x1D711}_{\mathit{num}}(x,0,0)$ (plain blue line). (a) Two-step shallow water model, with $h_{x}=\langle h^{-1}\rangle ^{-1}$ and $\unicode[STIX]{x27E6}\unicode[STIX]{x1D711}\unicode[STIX]{x27E7}=\unicode[STIX]{x27E6}D\unicode[STIX]{x27E7}=0$ . (b) First-order model, with $h_{x}$ corrected, equation (2.3), and $\unicode[STIX]{x27E6}\unicode[STIX]{x1D711}\unicode[STIX]{x27E7}=\unicode[STIX]{x27E6}D\unicode[STIX]{x27E7}=0$ ). (c) Second-order model, with $h_{x}$ corrected and $\unicode[STIX]{x27E6}\unicode[STIX]{x1D711}\unicode[STIX]{x27E7}$ , $\unicode[STIX]{x27E6}D\unicode[STIX]{x27E7}$ from (2.5).

6.2 Comparison of the different models

To begin with, we come back to the illustration of figure 1. Figure 10 shows the fields $\unicode[STIX]{x1D711}(x,y)$ and the profiles $\unicode[STIX]{x1D711}(x,0)$ given by the different models. Panel (a) shows the two-step homogenized model using $h_{x}=\langle h^{-1}\rangle ^{-1}=0.345$ ( $h_{y}=0.905$ ) and applying the continuities of the surface elevation and of the flux; the error in the field $|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathit{num}}|/|\unicode[STIX]{x1D711}_{\mathit{num}}|$ and on the reflection coefficient $|R-R_{\mathit{num}}|/|R_{\mathit{num}}|$ is larger than 100 % and by inspection of the profiles, the error is clearly attributable to an effective wavelength largely overestimated (we have seen that $h_{x}<\langle h^{-1}\rangle ^{-1}$ ). In panel (b), the homogenized model at first order corrects this drawback by considering the right $h_{x}=0.165$ from (2.3) and applying the continuities of the surface elevation and of the flux; although it is visible that the wavenumber over the ridge has been corrected, the scattering properties of the ridge are still not correctly accounted for. The errors remain high with an approximately 50 % discrepancy on the field and 80 % on the reflection coefficient. Eventually in panel (c), the model at second order appears to reproduce with a good accuracy the field and the scattering properties of the ridge, with errors on the field and on $R$ of approximately 10 %, which is satisfactory at this relatively high frequency ( $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D714}\sqrt{h/g}=0.4$ , and decreasing the frequency would produce lower errors). We used the effective parameters ${\mathcal{B}}=2.356$ computed numerically (the estimate in (5.4) provides ${\mathcal{B}}=2.358$ ) and ${\mathcal{C}}\ell ^{2}=0.043$ .

6.3 Influence of the ending layers

To illustrate further the influence of the transmission conditions, we shall now focus on the geometry of the ending layers while keeping constant the other parameters. To do so, we set the ridge structuration: $h^{+}=1$ , $h^{-}=0.05$ , $h=2$ , $\unicode[STIX]{x1D709}=0.1$ , $\ell =4$ and a constant extent $L=36.4$ . It follows that in the models at the orders 1 and 2, $h_{x}=0.268$ and $h_{y}=0.905$ are fixed. Next, the ridge occupies the region $x\in (0,L)$ and the form of the first layer is parametrized by $x_{\mathit{e}}\in (0,\ell )$ defined in figure 9. Using $e=\unicode[STIX]{x1D709}\ell$ as reference length, figure 11 shows the scattering properties of ridges for $x_{\mathit{e}}\simeq 0,$ $e$ and $2.5e$ . Specifically, we report (i) the variations of $R_{\mathit{num}}$ against $\unicode[STIX]{x1D703}$ (plain blue lines) and the corresponding variations of $R$ given by the models at the first (dashed grey lines) and at the second orders (black dashed lines), (ii) the wavefields $\unicode[STIX]{x1D711}_{\mathit{num}}(x,y,0)$ and $\unicode[STIX]{x1D711}(x,y)$ from the model at the second order. The results are given for a frequency $\unicode[STIX]{x1D714}\sqrt{h/g}=0.07$ . It is visible that the model at order 2 reproduces accurately the significant variations in the scattering properties of the ridges. These variations are not predicted by the model at first order which see erroneously the same effective structure.

Figure 11.

Influence of the layers terminating a ridge of extent $L=36.4$ ( $h^{+}=1$ , $h^{-}=0.05$ , $h=2$ , $\unicode[STIX]{x1D709}=0.1$ and $\ell =4$ ). The three panels correspond to different layer arrangements at the entrance (see inset, with $x_{\mathit{e}}$ defined in figure 9). (a) Reflection coefficient against $\unicode[STIX]{x1D703}$ , $R_{\mathit{num}}$ from direct numerics (blue plain lines) and $R$ from the model at order 1 (dashed grey lines) and from the model at order 2 (dashed black lines). (b) Corresponding fields at incidence $\unicode[STIX]{x1D703}=40^{\circ }$ ; for comparison, $\unicode[STIX]{x1D711}_{\mathit{num}}$ is shown for $y>0$ and $\unicode[STIX]{x1D711}$ from the model at second order for $y<0$ . The frequency is $\unicode[STIX]{x1D714}\sqrt{h/g}=0.07$ .

Eventually, figure 12 offers a continuous version of these variations by reporting $|R_{\mathit{num}}|$ in colour scale against $\unicode[STIX]{x1D703}$ and $x_{\mathit{e}}/e$ . The model at second order is accurate with an overall error of approximately 1 %. The corresponding variations of $({\mathcal{B}},{\mathcal{C}})$ denoted $({\mathcal{B}}_{e},{\mathcal{C}}_{e})$ at the entrance and $({\mathcal{B}}_{o},{\mathcal{C}}_{o})$ at the exit are reported and allow us to interpret the observed variations. For $x_{\mathit{e}}\in (0,e)$ the parameters vary rapidly, resulting in rapid variations of the scattering properties. Due to the symmetry of the ridge, $x_{\mathit{e}}\in (e,2e)$ exchanges the role of the entrance and exit, hence $R_{\mathit{num}}$ follows the reverse course. For $x_{\mathit{e}}=2e=0.8$ , ${\mathcal{B}}_{o}$ experiences a jump corresponding to the disappearance of a layer at immersion depth $h^{-}$ . Eventually, for $x_{\mathit{e}}>2e$ , the end boundaries of the ridge are formed by a layer at immersion depth $h^{+}$ which produces ${\mathcal{B}}_{o}\simeq -{\mathcal{B}}_{e}$ and ${\mathcal{C}}_{o}\simeq -{\mathcal{C}}_{e}$ . Inspecting $R$ in (C 6) shows that this produces the cancellation of the dominant corrections in ${\mathcal{B}}(kh)$ , ${\mathcal{C}}(kh)$ ; thus, although each extremity of the ridge is the place of significant transition effects, their collective effect tends to cancel. As a consequence, the scattering properties for $x_{\mathit{e}}>2e$ are correctly described by the model at first order.

Figure 12.

(a) Variations of $R_{\mathit{num}}$ and $R$ in the model at first and second orders against $x_{\mathit{e}}$ and $\unicode[STIX]{x1D703}$ in colour scale. (b) Corresponding variations of the parameters ( ${\mathcal{B}},{\mathcal{C}}$ ) against $x_{\mathit{e}}$ (index $e$ refers to the ridge entrance at $o$ to the ridge exit).