3.1 Outer region for $\text{[IBVP]}^{\prime }$ when $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$

We begin in an outer region in which $(\overline{x},y)(\in \overline{{\mathcal{D}}}(\unicode[STIX]{x1D70F}))=O(1)$ and $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , after which we will require an inner region, in which $(\overline{x},y)=o(1)$ and $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , in order to capture the initial interaction between the plate and the perturbed fluid free surface. We first observe that the initial conditions (3.11) and (3.12) require that $\unicode[STIX]{x1D702}$ , $\unicode[STIX]{x1D719}=o(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the outer region. Specifically, conditions (3.4) and (3.6) require that $\unicode[STIX]{x1D719}=O(\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ and $\unicode[STIX]{x1D702}=O(\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . Thus we introduce the asymptotic expansions

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(\overline{x},y,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}(\unicode[STIX]{x1D707}-1)\cos \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D719}^{\prime }(\overline{x},y,\unicode[STIX]{x1D70F})+O(\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}), & \displaystyle\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}(\overline{x},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}\unicode[STIX]{x1D702}^{\prime }(\overline{x},\unicode[STIX]{x1D70F})+O(\unicode[STIX]{x1D6FF}^{3\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}), & \displaystyle\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the outer region, with the factor $(\unicode[STIX]{x1D707}-1)\cos \unicode[STIX]{x1D6FC}$ in (3.13) included for algebraic convenience at a later stage. Here we note that the notation $^{\prime }$ is not to be confused with the indication of dimensionality in GNB, nor does it refer to differentiation. Substituting the expansions (3.13) and (3.14) into $\text{[IBVP]}^{\prime }$ , we obtain the leading-order evolution problem in the outer region for $\unicode[STIX]{x1D719}^{\prime }$ , namely

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}^{\prime }=0,\quad (\overline{x},y)\in {\mathcal{D}}(0),\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\unicode[STIX]{x1D719}^{\prime }\boldsymbol{\cdot }\hat{\boldsymbol{n}}=\unicode[STIX]{x1D70F},\quad y=-\overline{x}\tan \unicode[STIX]{x1D6FC},0<\overline{x}<\cot \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{y}^{\prime }=0,\quad y=-1,\overline{x}>\cot \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\prime }=0,\quad y=0,\overline{x}>0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle |\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}^{\prime }|\rightarrow 0\quad \text{as }\overline{x}\rightarrow \infty ,\text{ uniformly for }-1\leqslant y\leqslant 0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}^{\prime }(\overline{x},y,0)=0,\quad (\overline{x},y)\in \overline{{\mathcal{D}}}(0); & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D702}^{\prime }$ then given by

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70F}}^{\prime }(\overline{x},\unicode[STIX]{x1D70F})={\textstyle \frac{1}{2}}(\unicode[STIX]{x1D707}-1)\cos \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D719}_{y}^{\prime }(\overline{x},0,\unicode[STIX]{x1D70F}),\quad \overline{x}\geqslant 0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}^{\prime }(\overline{x},0)=0,\quad \overline{x}\geqslant 0. & \displaystyle\end{eqnarray}$$

Note that the free surface perturbation is absent in (3.22) as the perturbation is localised in the inner region. We next introduce $\overline{\unicode[STIX]{x1D719}}^{\prime }(\overline{x},y)$ and $\overline{\unicode[STIX]{x1D702}}^{\prime }(\overline{x})$ by writing

(3.23a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D719}^{\prime }(\overline{x},y,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70F}\overline{\unicode[STIX]{x1D719}}^{\prime }(\overline{x},y),\quad \unicode[STIX]{x1D702}^{\prime }(\overline{x},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70F}^{2}\overline{\unicode[STIX]{x1D702}}^{\prime }(\overline{x}).\end{eqnarray}$$

The resulting problem for $\overline{\unicode[STIX]{x1D719}}^{\prime }$ and $\overline{\unicode[STIX]{x1D702}}^{\prime }$ , obtained after substituting (3.23) into (3.15)–(3.22), is the same as the outer region problem which was solved and discussed in GNB, and as such details are omitted here. We thus have that $\overline{\unicode[STIX]{x1D719}}^{\prime }$ is given, for some real constants $A_{n}$ , $B_{n}$ and $C_{n}(n=0,1,2,\ldots )$ , by

(3.24) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D719}}^{\prime }(r,\unicode[STIX]{x1D703})=\frac{r\sin \unicode[STIX]{x1D703}}{\cos \unicode[STIX]{x1D6FC}}+\mathop{\sum }_{n=0}^{\infty }A_{n}r^{(n+1/2)\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}}\sin ((n+1/2)\unicode[STIX]{x03C0}\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FC}),\end{eqnarray}$$

for $0\leqslant r<\text{cosec}\,\unicode[STIX]{x1D6FC}$ , $-\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D703}\leqslant 0$ , where $\overline{x}=r\cos \unicode[STIX]{x1D703}$ and $y=r\sin \unicode[STIX]{x1D703}$ , whilst

(3.25) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D719}}^{\prime }(\unicode[STIX]{x1D70C},\overline{\unicode[STIX]{x1D703}})=\frac{\unicode[STIX]{x1D70C}\cos \overline{\unicode[STIX]{x1D703}}}{\sin \unicode[STIX]{x1D6FC}}+\mathop{\sum }_{n=0}^{\infty }B_{n}\unicode[STIX]{x1D70C}^{n\unicode[STIX]{x03C0}/(\unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC})}\cos \left(\frac{n\unicode[STIX]{x03C0}\overline{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC}}\right),\end{eqnarray}$$

for $0\leqslant \unicode[STIX]{x1D70C}<1$ , $0\leqslant \overline{\unicode[STIX]{x1D703}}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC}$ , where $\overline{x}-\cot \unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70C}\cos \overline{\unicode[STIX]{x1D703}}$ , and $y+1=\unicode[STIX]{x1D70C}\sin \overline{\unicode[STIX]{x1D703}}$ , and finally

(3.26) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D719}}^{\prime }(\overline{x},y)=\mathop{\sum }_{n=0}^{\infty }C_{n}\text{e}^{-(n+1/2)\unicode[STIX]{x03C0}\overline{x}}\sin ((n+1/2)\unicode[STIX]{x03C0}y),\end{eqnarray}$$

for $\overline{x}>\cot \unicode[STIX]{x1D6FC}$ , $-1\leqslant y\leqslant 0$ . It then follows from (3.21), (3.24) and (3.26), that

(3.27) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D702}}^{\prime }(\overline{x})=\frac{1}{2}\unicode[STIX]{x1D70E}\tan \unicode[STIX]{x1D6FC}+\frac{1}{2}\unicode[STIX]{x1D70E}\sin \unicode[STIX]{x1D6FC}\mathop{\sum }_{n=0}^{\infty }(n+1/2)\frac{\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FC}}A_{n}\overline{x}^{(n+1/2)\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-1},\end{eqnarray}$$

for $0\leqslant \overline{x}<\text{cosec}\,\unicode[STIX]{x1D6FC}$ , and

(3.28) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D702}}^{\prime }(\overline{x})=\frac{1}{2}\unicode[STIX]{x1D70E}\sin \unicode[STIX]{x1D6FC}\mathop{\sum }_{n=0}^{\infty }(n+1/2)\unicode[STIX]{x03C0}C_{n}\text{e}^{-(n+1/2)\unicode[STIX]{x03C0}\overline{x}},\end{eqnarray}$$

for $\overline{x}\geqslant \text{cosec}\,\unicode[STIX]{x1D6FC}$ . We observe from (3.24) and (3.27) that, as $(\overline{x},y)\rightarrow (0,0)$ ,

(3.29) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D719}}^{\prime }(r,\unicode[STIX]{x1D703}) & = & \displaystyle \frac{r\sin \unicode[STIX]{x1D703}}{\cos \unicode[STIX]{x1D6FC}}+A_{0}(\unicode[STIX]{x1D6FC})r^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}}\sin \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D703}}{2\unicode[STIX]{x1D6FC}}+O(r^{3\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}}),\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }r\rightarrow 0^{+},\text{ with }-\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D703}\leqslant 0,\end{eqnarray}$$

(3.30) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D702}}^{\prime }(\overline{x}) & = & \displaystyle \frac{1}{2}(\unicode[STIX]{x1D707}-1)+A_{0}(\unicode[STIX]{x1D6FC})\frac{\unicode[STIX]{x03C0}}{4\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D707}-1)\cos \unicode[STIX]{x1D6FC}\overline{x}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}+O(\overline{x}^{3\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}),\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }\overline{x}\rightarrow 0^{+},\end{eqnarray}$$

where $r$ and $\unicode[STIX]{x1D703}$ are the polar coordinates defined above, and $A_{0}(\unicode[STIX]{x1D6FC})({<}\!0)$ as given in GNB (see GNB (3.10)). Equation (3.30) reveals a weak singularity in $\overline{\unicode[STIX]{x1D702}}_{\overline{x}}^{\prime }(\overline{x})$ as $\overline{x}\rightarrow 0^{+}$ . This singular behaviour as $\overline{x}\rightarrow 0^{+}$ is compounded in higher-order terms in the outer expansions (3.13) and (3.14), and so the regularity conditions (see GNB (2.16)) fail to be satisfied by the outer region asymptotic expansions in a neighbourhood of the initial intersection point of the plate and the free surface, where $(\overline{x},y)=o(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . Therefore, in order to capture the full regularity in the neighbourhood of the intersection point of the plate and the free surface, the introduction of an inner region is required.

3.2 Inner region for $\text{[IBVP]}^{\prime }$ when $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$

We have $(\overline{x},y)=O(\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FF}))$ , with $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FF})=o(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner region. It then follows from (3.14), (3.23) and (3.30) that $\unicode[STIX]{x1D702}=O(\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner region, so that, to capture the free surface in the inner region, we must take $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FF})=O(\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ ; therefore, without loss of generality, we set $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FF})=\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}$ . An examination of (3.13), (3.23), and (3.29) then requires that $\unicode[STIX]{x1D719}=O(\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner region. Thus we introduce scaled inner region coordinates $(X,Y)$ by

(3.31a,b ) $$\begin{eqnarray}\overline{x}=\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}X,\quad y=\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}Y,\end{eqnarray}$$

with $(X,Y)=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ ( $\dagger$ ). The plate is located in the inner region at $Y=-X\tan \unicode[STIX]{x1D6FC}$ , with the contact point denoted by $(X,Y)=(X_{p}(\unicode[STIX]{x1D70F}),Y_{p}(\unicode[STIX]{x1D70F}))$ , where $\overline{x}_{p}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}X_{p}(\unicode[STIX]{x1D70F})$ and $y_{p}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}Y_{p}(\unicode[STIX]{x1D70F})$ ( $\dagger$ ). We now write the free surface and velocity potential in the inner region as

(3.32) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}(X,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}\unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F}),\quad X\geqslant X_{p}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0; & \displaystyle\end{eqnarray}$$

(3.33) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(X,Y,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{3\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}\unicode[STIX]{x1D719}_{I}(X,Y,\unicode[STIX]{x1D70F}),\quad X\geqslant X_{p}(\unicode[STIX]{x1D70F}),-X\tan \unicode[STIX]{x1D6FC}\leqslant Y\leqslant \unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0;\quad & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F})$ , $\unicode[STIX]{x1D719}_{I}(X,Y,\unicode[STIX]{x1D70F})=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ ( $\dagger$ ). The inner region expansions are introduced as

(3.34) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F})=\widetilde{\unicode[STIX]{x1D702}}_{0}^{\prime }(X,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime }(X,\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}), & \displaystyle\end{eqnarray}$$

(3.35) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{I}(X,Y,\unicode[STIX]{x1D70F})=\widetilde{\unicode[STIX]{x1D719}}_{0}^{\prime }(X,Y,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime }(X,Y,\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}), & \displaystyle\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ with $(X,Y)=O(1)$ , where the form of the correction terms have been deduced from (3.31), together with (3.14), (3.23) and (3.30). It follows from (3.31) and (3.34) that the free surface in the inner region is located at

(3.36) $$\begin{eqnarray}Y(X,\unicode[STIX]{x1D70F})=\widetilde{\unicode[STIX]{x1D702}}_{0}^{\prime }(X,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime }(X,\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}),\quad X>X_{p}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0,\end{eqnarray}$$

Hence we must expand

(3.37) $$\begin{eqnarray}X_{p}(\unicode[STIX]{x1D70F})=X_{0}(\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}X_{1}(\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}}),\quad \unicode[STIX]{x1D70F}\geqslant 0,\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , after which $Y_{p}(\unicode[STIX]{x1D70F})=-X_{p}(\unicode[STIX]{x1D70F})\tan \unicode[STIX]{x1D6FC}$ ( $\dagger$ ).

We now write $\text{[IBVP]}^{\prime }$ in terms of the inner variables and substitute from (3.34) and (3.35). The problem obtained at leading order, when supplemented with matching conditions to the outer region (obtained through Van Dyke’s matching principle (Van Dyke Reference Van Dyke1964)), has the solution

(3.38) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D719}}_{0}^{\prime }(X,Y,\unicode[STIX]{x1D70F}) & = & \displaystyle \unicode[STIX]{x1D70F}(\unicode[STIX]{x1D707}-1)Y+{\textstyle \frac{1}{6}}\unicode[STIX]{x1D70F}^{3}(1-\unicode[STIX]{x1D707})(2\unicode[STIX]{x1D707}-1),\nonumber\\ \displaystyle & & \displaystyle \quad X\geqslant X_{0}(\unicode[STIX]{x1D70F}),-X\tan \unicode[STIX]{x1D6FC}\leqslant Y\leqslant {\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}(\unicode[STIX]{x1D707}-1),\unicode[STIX]{x1D70F}\geqslant 0;\end{eqnarray}$$

(3.39) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D702}}_{0}^{\prime }(X,\unicode[STIX]{x1D70F})={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}(\unicode[STIX]{x1D707}-1),\quad X\geqslant X_{0}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0; & \displaystyle\end{eqnarray}$$

(3.40) $$\begin{eqnarray}\displaystyle & \displaystyle X_{0}(\unicode[STIX]{x1D70F})=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}(\unicode[STIX]{x1D707}-1)\cot \unicode[STIX]{x1D6FC},\quad \unicode[STIX]{x1D70F}\geqslant 0. & \displaystyle\end{eqnarray}$$

It follows from (3.38) and (3.39), with (3.31)–(3.35), that the regularity conditions (see GNB (2.16)) are satisfied at leading order in the inner region. We now formulate the problem at $O(\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ , where it is convenient to introduce the coordinates $(\overline{X},\overline{Y})$  as

(3.41a,b ) $$\begin{eqnarray}X=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}(\unicode[STIX]{x1D707}-1)\cot \unicode[STIX]{x1D6FC}+\overline{X},\quad Y={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}(\unicode[STIX]{x1D707}-1)+\overline{Y},\end{eqnarray}$$

which is simply a shift of origin from the original inner coordinates $(X,Y)$ . We next write

(3.42a,b ) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime }(\overline{X},\unicode[STIX]{x1D70F})=\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime }(\overline{X},\unicode[STIX]{x1D70F})|_{0}+H^{\prime }(\overline{X},\unicode[STIX]{x1D70F}),\quad \widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime }(\overline{X},\overline{Y},\unicode[STIX]{x1D70F})=\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime }(\overline{X},\overline{Y},\unicode[STIX]{x1D70F})|_{0}+\unicode[STIX]{x1D6F7}^{\prime }(\overline{X},\overline{Y},\unicode[STIX]{x1D70F}),\end{eqnarray}$$

where $\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime }|_{0}$ and $\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime }|_{0}$ are given by

(3.43) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime }(\overline{X},\unicode[STIX]{x1D70F})|_{0}=A_{0}(\unicode[STIX]{x1D6FC})(\unicode[STIX]{x1D707}-1)\cos \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70F}^{\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2}\widehat{\unicode[STIX]{x1D702}}\left(\frac{\overline{X}}{\unicode[STIX]{x1D70F}^{2}}\right),\quad \overline{X}\geqslant 0,\unicode[STIX]{x1D70F}>0, & & \displaystyle\end{eqnarray}$$

(3.44) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime }(\overline{X},\overline{Y},\unicode[STIX]{x1D70F})|_{0} & = & \displaystyle A_{0}(\unicode[STIX]{x1D6FC})(\unicode[STIX]{x1D707}-1)\cos \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70F}^{\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2}\widehat{\unicode[STIX]{x1D719}}\left(\frac{\overline{X}}{\unicode[STIX]{x1D70F}^{2}},\frac{\overline{Y}}{\unicode[STIX]{x1D70F}^{2}}\right),\nonumber\\ \displaystyle & & \displaystyle \quad \overline{X}\geqslant 0,-\overline{X}\tan \unicode[STIX]{x1D6FC}\leqslant \overline{Y}\leqslant 0,\unicode[STIX]{x1D70F}>0,\end{eqnarray}$$

and $\widehat{\unicode[STIX]{x1D702}}$ and $\widehat{\unicode[STIX]{x1D719}}$ are solutions to the boundary value problem in GNB (see (3.10)–(4.16)) when written in the similarity coordinates $(\overline{X},\overline{Y})\rightarrow \unicode[STIX]{x1D70F}^{-2}(\overline{X},\overline{Y})$ . Rewriting the problem at $O(\unicode[STIX]{x1D6FF}^{1-2\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ in terms of $H^{\prime }$ and $\unicode[STIX]{x1D6F7}^{\prime }$ we obtain

(3.45) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D6FB}}^{2}\unicode[STIX]{x1D6F7}^{\prime }=0,\quad \overline{X}>0,-\overline{X}\tan \unicode[STIX]{x1D6FC}<\overline{Y}<0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.46) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D6F7}^{\prime }\boldsymbol{\cdot }\hat{\boldsymbol{n}}=0,\quad \overline{X}>0,\overline{Y}=-\overline{X}\tan \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.47) $$\begin{eqnarray}\displaystyle & \displaystyle H_{\unicode[STIX]{x1D70F}}^{\prime }-\unicode[STIX]{x1D6F7}_{\overline{Y}}^{\prime }=0,\quad \overline{X}>0,\overline{Y}=0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.48) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70F}}^{\prime }+\unicode[STIX]{x1D707}H^{\prime }=0,\quad \overline{X}>0,\overline{Y}=0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.49) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{\prime }(\overline{R},\unicode[STIX]{x1D703},\unicode[STIX]{x1D70F})\rightarrow 0\quad \text{as }\overline{R}\rightarrow \infty ,-\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D703}<0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.50) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime }(\overline{X},\unicode[STIX]{x1D70F})\rightarrow 0\quad \text{as }\overline{X}\rightarrow \infty ,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(3.51) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{\prime }(\overline{R},\unicode[STIX]{x1D703},0)=0,\quad \overline{R}\geqslant 0,-\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D703}\leqslant 0; & \displaystyle\end{eqnarray}$$

(3.52) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime }(\overline{X},0)=\unicode[STIX]{x1D702}_{0}(\overline{X}),\quad \overline{X}\geqslant 0. & \displaystyle\end{eqnarray}$$

Here $\overline{\unicode[STIX]{x1D735}}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\overline{X},\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\overline{Y})$ and we have, for convenience, introduced polar coordinates $(\overline{R},\unicode[STIX]{x1D703})$ , given by $\overline{X}=\overline{R}\cos \unicode[STIX]{x1D703}$ , $\overline{Y}=\overline{R}\sin \unicode[STIX]{x1D703}$ ( $\dagger$ ). We begin our investigation of the problem (3.45)–(3.52) in the case $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times (\mathbb{R}\backslash \{0\})$ . We seek solutions of the form

(3.53a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{\prime }(\overline{X},\overline{Y},\unicode[STIX]{x1D70F})=\text{e}^{-\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}}\overline{\unicode[STIX]{x1D6F7}}^{\prime }(\overline{X},\overline{Y}),\quad H^{\prime }(\overline{X},\unicode[STIX]{x1D70F})=\text{e}^{-\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}}\overline{H}^{\prime }(\overline{X}),\end{eqnarray}$$

where $\unicode[STIX]{x1D706}\in \mathbb{C}$ , which leads to a decoupled linear harmonic boundary value problem for $\overline{\unicode[STIX]{x1D6F7}}^{\prime }$ , given by

(3.54) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D6FB}}^{2}\overline{\unicode[STIX]{x1D6F7}}^{\prime }=0,\quad \overline{X}>0,-\overline{X}\tan \unicode[STIX]{x1D6FC}<\overline{Y}<0; & \displaystyle\end{eqnarray}$$

(3.55) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D735}}\overline{\unicode[STIX]{x1D6F7}}^{\prime }\boldsymbol{\cdot }\hat{\boldsymbol{n}}=0,\quad \overline{X}>0,\overline{Y}=-\overline{X}\tan \unicode[STIX]{x1D6FC}; & \displaystyle\end{eqnarray}$$

(3.56) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D6F7}}_{\overline{Y}}^{\prime }-k\overline{\unicode[STIX]{x1D6F7}}^{\prime }=0,\quad \overline{X}>0,\overline{Y}=0; & \displaystyle\end{eqnarray}$$

(3.57) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D6F7}}^{\prime },\overline{\unicode[STIX]{x1D735}}\;\overline{\unicode[STIX]{x1D6F7}}^{\prime }\quad \text{bounded as }\overline{R}\rightarrow \infty ,\text{ uniformly for }-\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D703}<0; & \displaystyle\end{eqnarray}$$

where $k=-\unicode[STIX]{x1D706}^{2}/\unicode[STIX]{x1D707}$ (for $\unicode[STIX]{x1D707}\neq 0$ ). Although unnecessary, if required we obtain $\overline{H}^{\prime }$ from

(3.58) $$\begin{eqnarray}\overline{H}^{\prime }(\overline{X})=\frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D707}}\overline{\unicode[STIX]{x1D6F7}}^{\prime }(\overline{X},0),\quad \overline{X}\geqslant 0.\end{eqnarray}$$

Here, according to the regularity conditions (see GNB (2.16)), we require that solutions to (3.54)–(3.57), $\overline{\unicode[STIX]{x1D6F7}}^{\prime }:\overline{{\mathcal{G}}}_{\infty }\rightarrow \mathbb{C}$ , have regularity

(3.59) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6F7}}^{\prime }\in C^{1}(\overline{{\mathcal{G}}}_{\infty })\cap C^{2}({\mathcal{G}}_{\infty }),\end{eqnarray}$$

where

(3.60) $$\begin{eqnarray}{\mathcal{G}}_{\infty }=\{(\overline{X},\overline{Y}):-\overline{X}\tan \unicode[STIX]{x1D6FC}<\overline{Y}<0,\overline{X}>0\}.\end{eqnarray}$$

The linear harmonic problem (3.54)–(3.57) is a spectral problem with spectral parameter $k\in \mathbb{C}$ . We henceforth refer to this spectral problem as [SP $(k)$ ]. It is clear that [SP $(k)$ ] has the trivial solution for each $k\in \mathbb{C}$ . Further, let $\overline{\unicode[STIX]{x1D6F7}}_{k}^{\prime }:{\mathcal{G}}_{\infty }\rightarrow \mathbb{C}$ be a non-trivial solution to [SP $(k)$ ], then it is straightforward to establish the following.

1. (i) When $k\in \mathbb{R}^{+}$

   (3.61) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D6F7}}_{k}^{\prime }(\overline{R},\unicode[STIX]{x1D703}) & = & \displaystyle (c_{k}\text{e}^{\text{i}k\overline{R}\cos \unicode[STIX]{x1D703}}+d_{k}\text{e}^{-\text{i}k\overline{R}\cos \unicode[STIX]{x1D703}})\text{e}^{k\overline{R}\sin \unicode[STIX]{x1D703}}\nonumber\\ \displaystyle & & \displaystyle +\,a_{k}\frac{1}{\overline{R}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}}}\sin \frac{\unicode[STIX]{x03C0}}{2\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D703}+O\left(\frac{1}{\overline{R}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}+1}}\right),\end{eqnarray}$$
   as $\overline{R}\rightarrow \infty$ uniformly for $-\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D703}\leqslant 0$ . Here, $c_{k}$ , $d_{k}$ , $a_{k}\in \mathbb{C}$ are not all zero constants.

2. (ii) When $k=0$

   (3.62) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6F7}}_{0}^{\prime }(\overline{R},\unicode[STIX]{x1D703})=a_{0}+b_{0}\frac{1}{\overline{R}^{\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}}}\cos \frac{\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D703}+O\left(\frac{1}{\overline{R}^{2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}}}\right),\end{eqnarray}$$
   as $\overline{R}\rightarrow \infty$ uniformly for $-\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D703}\leqslant 0$ . Here $a_{0}$ , $b_{0}\in \mathbb{C}$ are not both zero constants.

3. (iii) When $k\in \mathbb{C}\backslash (\mathbb{R}^{+}\cup \{0\})$

   (3.63) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6F7}}_{k}^{\prime }(\overline{R},\unicode[STIX]{x1D703})=a_{k}\frac{1}{\overline{R}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}}}\sin \frac{\unicode[STIX]{x03C0}}{2\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D703}+O\left(\frac{1}{\overline{R}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}+1}}\right),\end{eqnarray}$$
   as $\overline{R}\rightarrow \infty$ uniformly for $-\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D703}\leqslant 0$ . Here, $a_{k}\in \mathbb{C}$ is a non-zero constant.

Our objective now is to classify the spectrum of [SP $(k)$ ]. We define the spectrum of [SP $(k)$ ] to be $\boldsymbol{S}$ , where

(3.64) $$\begin{eqnarray}\boldsymbol{S}=\{k\in \mathbb{C}:\text{[SP}(k)\text{] has a non-trivial solution}\}.\end{eqnarray}$$

The set of eigenvalues of [SP $(k)$ ] is $\boldsymbol{S}^{d}$ , where

(3.65) $$\begin{eqnarray}\boldsymbol{S}^{d}=\{k\in \mathbb{C}:\exists \text{a non-trivial solution to }\text{[SP}(k)\text{] with }\overline{\unicode[STIX]{x1D6F7}}^{\prime }\rightarrow 0\text{ as }\overline{R}\rightarrow \infty \},\end{eqnarray}$$

and the continuous spectrum of [SP $(k)$ ] is $\boldsymbol{S}^{c}$ , where

(3.66) $$\begin{eqnarray}\boldsymbol{S}^{c}=\{k\in \mathbb{C}:\exists \text{a non-trivial solution to [SP}(k)\text{] with }\overline{\unicode[STIX]{x1D6F7}}^{\prime }{\nrightarrow}0\text{ as }\overline{R}\rightarrow \infty \},\end{eqnarray}$$

with the limits as $\overline{R}\rightarrow \infty$ considered as uniform for $-\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D703}\leqslant 0$ . We observe that

(3.67) $$\begin{eqnarray}\boldsymbol{S}=\boldsymbol{S}^{c}\cup \boldsymbol{S}^{d}.\end{eqnarray}$$

Following Needham (Reference Needham2012), together with the use of (3.61)–(3.63) above we can establish that (see appendix A)

(3.68a,b ) $$\begin{eqnarray}\boldsymbol{S}^{c}=\mathbb{R}^{+}\cup \{0\},\quad \boldsymbol{S}^{d}=\varnothing ,\end{eqnarray}$$

and so

(3.69) $$\begin{eqnarray}\boldsymbol{S}=\mathbb{R}^{+}\cup \{0\}.\end{eqnarray}$$

We now introduce $\boldsymbol{S}_{\unicode[STIX]{x1D706}}\subseteq \mathbb{C}$ , where

(3.70) $$\begin{eqnarray}\boldsymbol{S}_{\unicode[STIX]{x1D706}}=\{\unicode[STIX]{x1D706}\in \mathbb{C}:\unicode[STIX]{x1D706}^{2}=-\unicode[STIX]{x1D707}k,k\in \boldsymbol{S}\}.\end{eqnarray}$$

Following (3.69), for $\unicode[STIX]{x1D707}>0$ , we have

(3.71) $$\begin{eqnarray}\boldsymbol{S}_{\unicode[STIX]{x1D706}}=\boldsymbol{S}_{\unicode[STIX]{x1D706}}^{+}=\{\unicode[STIX]{x1D706}\in \mathbb{C}:\unicode[STIX]{x1D706}=\pm \text{i}\sqrt{\unicode[STIX]{x1D707}}k^{1/2},k\in \mathbb{R}^{+}\cup \{0\}\},\end{eqnarray}$$

and, for $\unicode[STIX]{x1D707}<0$ , we have

(3.72) $$\begin{eqnarray}\boldsymbol{S}_{\unicode[STIX]{x1D706}}=\boldsymbol{S}_{\unicode[STIX]{x1D706}}^{-}=\{\unicode[STIX]{x1D706}\in \mathbb{C}:\unicode[STIX]{x1D706}=\pm \sqrt{(-\unicode[STIX]{x1D707})}k^{1/2},k\in \mathbb{R}^{+}\cup \{0\}\}.\end{eqnarray}$$

Now, the linear evolution problem (3.45)–(3.52) has a solution represented as a Fourier type integral with respect to $\unicode[STIX]{x1D706}$ over the elementary solutions (3.53), with $\unicode[STIX]{x1D706}\in \boldsymbol{S}_{\unicode[STIX]{x1D706}}^{+}$ ( $\unicode[STIX]{x1D707}>0$ ) or $\unicode[STIX]{x1D706}\in \boldsymbol{S}_{\unicode[STIX]{x1D706}}^{-}$ ( $\unicode[STIX]{x1D707}<0$ ) and density functions chosen to satisfy the initial conditions (3.51) and (3.52). We conclude that the linear evolution problem (3.45)–(3.52) is:

1. (i) well-posed and stable when $\text{Re}(\unicode[STIX]{x1D706})\geqslant 0$ for all $\unicode[STIX]{x1D706}\in \boldsymbol{S}_{\unicode[STIX]{x1D706}}$ ;

2. (ii) well-posed and unstable when there exists $M\in \mathbb{R}$ such that $\text{Re}(\unicode[STIX]{x1D706})\geqslant M$ for all $\unicode[STIX]{x1D706}\in \boldsymbol{S}_{\unicode[STIX]{x1D706}}$ , and there exists $\unicode[STIX]{x1D706}^{\ast }\in \boldsymbol{S}_{\unicode[STIX]{x1D706}}$ such that $\text{Re}(\unicode[STIX]{x1D706}^{\ast })<0$ ;

3. (iii) ill-posed when there exists a sequence $\{\unicode[STIX]{x1D706}_{n}\}_{n\in \mathbb{N}}$ , with $\unicode[STIX]{x1D706}_{n}\in \boldsymbol{S}_{\unicode[STIX]{x1D706}}$ for all $n\in \mathbb{N}$ , and such that $\text{Re}(\unicode[STIX]{x1D706}_{n})\rightarrow -\infty$ as $n\rightarrow \infty$ .

It now follows directly from (3.71) and (3.72) that the linear evolution problem (3.45)–(3.52) is:

1. (i) well-posed and stable when $\unicode[STIX]{x1D707}>0$ ;

2. (ii) ill-posed when $\unicode[STIX]{x1D707}<0$ .

We conclude that [IBVP] is well-posed and stable when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$ , whilst [IBVP] is ill-posed when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{-}$ . We should note here that the time scale for ill-posed growth or stable decay for the perturbation close to the initial contact point corresponds to $\unicode[STIX]{x1D70F}\gg 1$ , which in the original time variable has $t\gg O(\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FC}/\unicode[STIX]{x03C0}})$ .

We next consider the problem (3.45)–(3.52) in the case $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/4]\times \{0\}$ , where it is straightforward to show that the only solution is given by

(3.73) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{\prime }(\overline{X},\overline{Y},\unicode[STIX]{x1D70F})=0,\quad \overline{X}\geqslant 0,-\overline{X}\tan \unicode[STIX]{x1D6FC}\leqslant \overline{Y}\leqslant 0,\unicode[STIX]{x1D70F}\geqslant 0, & \displaystyle\end{eqnarray}$$

(3.74) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime }(\overline{X},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D702}_{0}(\overline{X}),\quad \overline{X}\geqslant 0,\unicode[STIX]{x1D70F}\geqslant 0, & \displaystyle\end{eqnarray}$$

and thus the linear evolution problem (3.45)–(3.52) is well-posed and stable for all pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/4]\times \{0\}$ . We conclude that [IBVP] is well-posed and stable when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/4]\times \{0\}$ .

It remains to consider the well-posedness and stability of [IBVP] for those pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (\unicode[STIX]{x03C0}/4,\unicode[STIX]{x03C0}/2)\times {\mathcal{G}}(\unicode[STIX]{x1D6FF})$ , with ${\mathcal{G}}(\unicode[STIX]{x1D6FF})$ being an $o(1)$ neighbourhood of $\unicode[STIX]{x1D707}=0$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ .

4.1 Outer region for $\text{[IBVP]}^{\prime \prime }$ when $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$

We begin in an outer region in which $(\overline{x},y)\in \overline{{\mathcal{D}}}(\unicode[STIX]{x1D70F})=O(1)$ , when $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , after which we will require two additional regions, in which $(\overline{x},y)=o(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , in order to capture the initial interaction between the plate and the free surface. We observe that conditions (4.5) and (4.7) require $\unicode[STIX]{x1D719}=O(\unicode[STIX]{x1D6FF}^{1/\unicode[STIX]{x1D6FE}})$ and $\unicode[STIX]{x1D702}=O(\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the outer region. Thus we introduce the asymptotic expansions

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(\overline{x},y,\unicode[STIX]{x1D70F})=-\unicode[STIX]{x1D6FF}^{1/\unicode[STIX]{x1D6FE}}\cos \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D719}^{\prime \prime }(\overline{x},y,\unicode[STIX]{x1D70F})+O(\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}), & \displaystyle\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}(\overline{x},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D702}^{\prime \prime }(\overline{x},\unicode[STIX]{x1D70F})+O(\unicode[STIX]{x1D6FF}^{3/\unicode[STIX]{x1D6FE}}), & \displaystyle\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the outer region, with the factor $-\cos \,\unicode[STIX]{x1D6FC}$ in (4.14) included for algebraic convenience at a later stage. Substituting the expansions (4.14) and (4.15) into $\text{[IBVP]}^{\prime \prime }$ we obtain, at leading order, a problem equivalent to (3.15)–(3.22) in the case $\unicode[STIX]{x1D707}=0$ . It follows that, after writing

(4.16a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D719}^{\prime \prime }(\overline{x},y,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70F}\overline{\unicode[STIX]{x1D719}}^{\prime \prime }(\overline{x},y),\quad \unicode[STIX]{x1D702}^{\prime \prime }(\overline{x},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70F}^{2}\overline{\unicode[STIX]{x1D702}}^{\prime \prime }(\overline{x}),\end{eqnarray}$$

then $\overline{\unicode[STIX]{x1D719}}^{\prime \prime }$ and $\overline{\unicode[STIX]{x1D702}}^{\prime \prime }$ are given (for some real constants $A_{n}$ , $B_{n}$ and $C_{n}$ ) by (3.24)–(3.28). The form of $\overline{\unicode[STIX]{x1D719}}^{\prime \prime }$ and $\overline{\unicode[STIX]{x1D702}}^{\prime \prime }$ as $(\overline{x},y)\rightarrow (0,0)$ is given by (3.29) and (3.30), where we again have singular behaviour as $(\overline{x},y)\rightarrow (0,0)$ , which motivates the introduction of an inner region in which $(\overline{x},y)=o(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ .

4.2 Inner region for $\text{[IBVP]}^{\prime \prime }$ when $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$

We set $(\overline{x},y)=O(\overline{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D6FF}))$ with $\overline{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D6FF})=o(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner region. It follows from (4.15), and (4.16), along with (3.23) and (3.30), that $\unicode[STIX]{x1D702}=O(\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner region, and so, in order to capture the free surface in the inner region, we must take $\overline{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D6FF})=O(\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ ; thus without loss of generality, we set $\overline{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D6FF})=\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}$ . An examination of (4.14), together with (3.23) and (3.29), then requires that $\unicode[STIX]{x1D719}=O(\unicode[STIX]{x1D6FF}^{3/\unicode[STIX]{x1D6FE}})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner region. Thus we introduce scaled inner region coordinates $(X,Y)$ by

(4.17a,b ) $$\begin{eqnarray}\overline{x}=\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}X,\quad y=\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}Y,\end{eqnarray}$$

with $(X,Y)=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner region ( $\dagger$ ). The location of the plate in the inner region is given by $Y=-X\tan \unicode[STIX]{x1D6FC}$ , whilst the contact point is denoted by $(X,Y)=(X_{p}(\unicode[STIX]{x1D70F}),Y_{p}(\unicode[STIX]{x1D70F}))$ , where $\overline{x}_{p}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}X_{p}(\unicode[STIX]{x1D70F})$ and $y_{p}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}Y_{p}(\unicode[STIX]{x1D70F})$ ( $\dagger$ ). We now write the free surface and velocity potential in the inner region as

(4.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}(X,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F}),\quad X\geqslant X_{p}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0; & \displaystyle\end{eqnarray}$$

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(X,Y,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{3/\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D719}_{I}(X,Y,\unicode[STIX]{x1D70F}),\quad X\geqslant X_{p}(\unicode[STIX]{x1D70F}),-X\tan \unicode[STIX]{x1D6FC}\leqslant Y\leqslant \unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0;\quad & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F})$ , $\unicode[STIX]{x1D719}_{I}(X,Y,\unicode[STIX]{x1D70F})=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ ( $\dagger$ ). The inner region expansions are then introduced as

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{I}(X,\unicode[STIX]{x1D70F})=\widetilde{\unicode[STIX]{x1D702}}_{0}^{\prime \prime }(X,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime \prime }(X,\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}), & \displaystyle\end{eqnarray}$$

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{I}(X,Y,\unicode[STIX]{x1D70F})=\widetilde{\unicode[STIX]{x1D719}}_{0}^{\prime \prime }(X,Y,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }(X,Y,\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}), & \displaystyle\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ where $(X,Y)=O(1)$ , with the form of the correction terms having been deduced from (4.17), together with (3.14), (3.23) and (3.30). It also follows from (4.17) and (4.20) that the free surface in the inner region is located at

(4.22) $$\begin{eqnarray}Y=\widetilde{\unicode[STIX]{x1D702}}_{0}^{\prime \prime }(X,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime \prime }(X,\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}),\quad X>X_{p}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0,\end{eqnarray}$$

and hence we must expand

(4.23) $$\begin{eqnarray}X_{p}(\unicode[STIX]{x1D70F})=X_{0}(\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}X_{1}(\unicode[STIX]{x1D70F})+o(\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}),\quad \unicode[STIX]{x1D70F}\geqslant 0,\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , after which $Y_{p}(\unicode[STIX]{x1D70F})=-X_{p}(\unicode[STIX]{x1D70F})\tan \unicode[STIX]{x1D6FC}$ ( $\dagger$ ).

We can now write $\text{[IBVP]}^{\prime \prime }$ in terms of the inner variables and substitute from (4.20) and (4.21). The problem obtained at leading order, when supplemented with matching conditions to the outer region (via Van Dyke’s matching principle, (Van Dyke Reference Van Dyke1964)), has the solution

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D719}}_{0}^{\prime \prime }(X,Y,\unicode[STIX]{x1D70F})=-\unicode[STIX]{x1D70F}Y-{\textstyle \frac{1}{6}}\unicode[STIX]{x1D70F}^{2},\quad X\geqslant X_{0}(\unicode[STIX]{x1D70F}),-X\tan \unicode[STIX]{x1D6FC}\leqslant Y\leqslant -{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2},\unicode[STIX]{x1D70F}\geqslant 0; & \displaystyle\end{eqnarray}$$

(4.25) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D702}}_{0}^{\prime \prime }(X,\unicode[STIX]{x1D70F})=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2},\quad X\geqslant X_{0}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0; & \displaystyle\end{eqnarray}$$

(4.26) $$\begin{eqnarray}\displaystyle & \displaystyle X_{0}(\unicode[STIX]{x1D70F})={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}\cot \unicode[STIX]{x1D6FC},\quad \unicode[STIX]{x1D70F}\geqslant 0. & \displaystyle\end{eqnarray}$$

It follows from (4.24) and (4.25), using (4.17)–(4.21), that the regularity conditions (see GNB (2.16)) are satisfied at leading order in the inner region. We now formulate the problem at $O(\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}})$ , where it is convenient to introduce the coordinates $(\overline{X},\overline{Y})$ according to

(4.27a,b ) $$\begin{eqnarray}X={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}\cot \unicode[STIX]{x1D6FC}+\overline{X},\quad Y=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}+\overline{Y},\end{eqnarray}$$

which is simply a shift of origin from the original inner region coordinates $(X,Y)$ ( $\dagger$ ). We now obtain the problem for $\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }$ , $\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime \prime }$ and $X_{1}$ , as

(4.28) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D6FB}}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }=0,\quad \overline{X}>0,-\overline{X}\tan \unicode[STIX]{x1D6FC}<\overline{Y}<0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.29) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D735}}\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }\boldsymbol{\cdot }\hat{\boldsymbol{n}}=0,\quad \overline{X}>0,\overline{Y}=-\overline{X}\tan \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.30) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D702}}_{1,\unicode[STIX]{x1D70F}}^{\prime \prime }-\widetilde{\unicode[STIX]{x1D719}}_{1,\overline{Y}}^{\prime \prime }=0,\quad \overline{X}>0,\overline{Y}=0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.31) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D719}}_{1,\unicode[STIX]{x1D70F}}^{\prime \prime }=0,\quad \overline{X}>0,\overline{Y}=0,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.32) $$\begin{eqnarray}\displaystyle & & \displaystyle \widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }(\overline{R},\unicode[STIX]{x1D703})=\unicode[STIX]{x1D70F}A_{0}(\unicode[STIX]{x1D6FC})\cos \unicode[STIX]{x1D6FC}\overline{R}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}}\cos \frac{\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FC})+o(\overline{R}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}})\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }\overline{R}\rightarrow \infty ,-\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D703}<0,\unicode[STIX]{x1D70F}>0;\end{eqnarray}$$

(4.33) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime \prime }(\overline{X})=-\unicode[STIX]{x1D70F}^{2}A_{0}(\unicode[STIX]{x1D6FC})\frac{\unicode[STIX]{x03C0}}{4\unicode[STIX]{x1D6FC}}\overline{X}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}+o(\overline{X}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1})\quad \text{as }\overline{X}\rightarrow \infty ,\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.34) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }(\overline{X},\overline{Y},0)=0,\quad \overline{X}\geqslant 0,-\overline{X}\tan \unicode[STIX]{x1D6FC}\leqslant \overline{Y}\leqslant 0; & \displaystyle\end{eqnarray}$$

(4.35) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime \prime }(\overline{X},0)=0,\quad \overline{X}\geqslant 0. & \displaystyle\end{eqnarray}$$

Here $\overline{\unicode[STIX]{x1D735}}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\overline{X},\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\overline{Y})$ , (4.32) and (4.33) are the matching conditions with the outer region, $A_{0}(\unicode[STIX]{x1D6FC})({<}\!0)$ is given in GNB (see (3.10)), and we have introduced polar coordinates $(\overline{R},\unicode[STIX]{x1D703})$ , given by $\overline{X}=\overline{R}\cos \unicode[STIX]{x1D703}$ , $\overline{Y}=\overline{R}\sin \unicode[STIX]{x1D703}$ ( $\dagger$ ). Additionally we have

(4.36) $$\begin{eqnarray}X_{1}(\unicode[STIX]{x1D70F})=-\widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime \prime }(0,\unicode[STIX]{x1D70F})\cot \unicode[STIX]{x1D6FC},\quad \unicode[STIX]{x1D70F}>0.\end{eqnarray}$$

It is straightforward to show that, in this degenerate case, the solution to the problem (4.28)–(4.35), which has least singular behaviour at $(\overline{X},\overline{Y})=(0,0)$ is simply given exactly by the far-field functions, that is

(4.37) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }(\overline{R},\unicode[STIX]{x1D703},\unicode[STIX]{x1D70F}) & = & \displaystyle \unicode[STIX]{x1D70F}A_{0}(\unicode[STIX]{x1D6FC})\cos \unicode[STIX]{x1D6FC}\overline{R}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}}\cos \frac{\unicode[STIX]{x03C0}}{2\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FC}),\nonumber\\ \displaystyle & & \displaystyle \quad \overline{R}>0,-\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D703}<0,\unicode[STIX]{x1D70F}>0;\end{eqnarray}$$

(4.38) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D702}}_{1}^{\prime \prime }(\overline{X},\unicode[STIX]{x1D70F})=-\unicode[STIX]{x1D70F}^{2}A_{0}(\unicode[STIX]{x1D6FC})\frac{\unicode[STIX]{x03C0}}{4\unicode[STIX]{x1D6FC}}\cos \unicode[STIX]{x1D6FC}\overline{X}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1},\quad \overline{X}>0,\unicode[STIX]{x1D70F}>0. & & \displaystyle\end{eqnarray}$$

We see from (4.37) and (4.38) that a weak singularity in derivatives (that is $\widetilde{\unicode[STIX]{x1D702}}_{1,\overline{X}}^{\prime \prime }$ , and $\overline{\unicode[STIX]{x1D735}}\widetilde{\unicode[STIX]{x1D719}}_{1}^{\prime \prime }$ ) persists close to the contact point and, in particular, when $(\overline{X},\overline{Y})=O(\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}})$ . This requires the introduction of an inner–inner region, in which $(\overline{X},\overline{Y})=o(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ and within which the full regularity conditions at the contact point are satisfied.

4.3 Inner–inner region for $\text{[IBVP]}^{\prime \prime }$ when $\unicode[STIX]{x1D70F}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$

We set $(\overline{X},\overline{Y})=O(\unicode[STIX]{x0394}(\unicode[STIX]{x1D6FF}))$ , with $\unicode[STIX]{x0394}(\unicode[STIX]{x1D6FF})=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . It then follows from (4.20), (4.25) and (4.38) that the free surface is located at $\overline{Y}=\unicode[STIX]{x1D70F}^{2}/2+\unicode[STIX]{x1D702}_{I}=O(\unicode[STIX]{x1D6FF}^{(\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2)/\unicode[STIX]{x1D6FE}}\unicode[STIX]{x0394}(\unicode[STIX]{x1D6FF})^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner–inner region. Therefore, to capture the free surface in the inner–inner region, we must take $\unicode[STIX]{x0394}(\unicode[STIX]{x1D6FF})=O(\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1})$ , and so, without loss of generality, we set $\unicode[STIX]{x0394}(\unicode[STIX]{x1D6FF})=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}$ . We introduce scaled inner–inner region coordinates $(\widetilde{x},\widetilde{y})$ by

(4.39a,b ) $$\begin{eqnarray}\overline{X}=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}\widetilde{x},\quad \overline{Y}=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}\widetilde{y},\end{eqnarray}$$

where $(\widetilde{x},\widetilde{y})=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the inner–inner region ( $\dagger$ ). The location of the plate in the inner–inner region is given by $\widetilde{y}=-\widetilde{x}\tan \unicode[STIX]{x1D6FC}$ , with the plate and contact point denoted by $(\widetilde{x},\widetilde{y})=(\widetilde{x}_{p}(\unicode[STIX]{x1D70F}),\widetilde{y}_{p}(\unicode[STIX]{x1D70F}))$ , with $\widetilde{x}_{p}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}\overline{X}_{p}(\unicode[STIX]{x1D70F})$ and $\widetilde{y}_{p}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}\overline{Y}_{p}(\unicode[STIX]{x1D70F})$ ( $\dagger$ ), where

(4.40a,b ) $$\begin{eqnarray}\overline{X}_{p}(\unicode[STIX]{x1D70F})=X_{p}(\unicode[STIX]{x1D70F})-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}\cot \unicode[STIX]{x1D6FC},\quad \overline{Y}_{p}(\unicode[STIX]{x1D70F})=Y_{p}(\unicode[STIX]{x1D70F})+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}.\end{eqnarray}$$

An examination of (4.20), (4.21), (4.24), (4.25), (4.37) and (4.38) requires that $\unicode[STIX]{x1D702}_{I}=-\unicode[STIX]{x1D70F}^{2}/2+O(\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1})$ and $\unicode[STIX]{x1D719}_{I}=\unicode[STIX]{x1D70F}^{3}/3-\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}\unicode[STIX]{x1D70F}\widetilde{y}+O(\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . We thus write the free surface and velocity potential in the inner–inner region as

(4.41) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{I}(\widetilde{x},\unicode[STIX]{x1D70F})=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70F}^{2}+\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}\unicode[STIX]{x1D702}_{II}(\widetilde{x},\unicode[STIX]{x1D70F}),\quad \widetilde{x}>\widetilde{x}_{p}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0; & & \displaystyle\end{eqnarray}$$

(4.42) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{I}(\widetilde{x},\widetilde{y},\unicode[STIX]{x1D70F}) & = & \displaystyle {\textstyle \frac{1}{3}}\unicode[STIX]{x1D70F}^{3}-\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}\unicode[STIX]{x1D70F}\widetilde{y}+\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}-2}\unicode[STIX]{x1D719}_{II}(\widetilde{x},\widetilde{y},\unicode[STIX]{x1D70F}),\nonumber\\ \displaystyle & & \displaystyle \quad \widetilde{x}\geqslant \widetilde{x}_{p}(\unicode[STIX]{x1D70F}),-\widetilde{x}\tan \unicode[STIX]{x1D6FC}\leqslant \widetilde{y}\leqslant \unicode[STIX]{x1D702}_{II}(\widetilde{x},\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0;\end{eqnarray}$$

where $\unicode[STIX]{x1D702}_{II}$ , $\unicode[STIX]{x1D719}_{II}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ ( $\dagger$ ). The inner–inner region asymptotic expansions are then introduced as

(4.43a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{II}(\widetilde{x},\unicode[STIX]{x1D70F})=\widehat{\unicode[STIX]{x1D702}}_{0}^{\prime \prime }(\widetilde{x},\unicode[STIX]{x1D70F})+o(1),\quad \unicode[STIX]{x1D719}_{II}(\widetilde{x},\widetilde{y},\unicode[STIX]{x1D70F})=\widehat{\unicode[STIX]{x1D719}}_{0}^{\prime \prime }(\widetilde{x},\widetilde{y},\unicode[STIX]{x1D70F})+o(1),\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , where $(\widetilde{x},\widetilde{y})=O(1)$ . It follows from (4.39) and (4.43) that the free surface in the inner–inner region is located at

(4.44) $$\begin{eqnarray}\widetilde{y}(\unicode[STIX]{x1D70F})=\widehat{\unicode[STIX]{x1D702}}_{0}^{\prime \prime }(\widetilde{x},\unicode[STIX]{x1D70F})+o(1),\quad \widetilde{x}>\widetilde{x}_{p}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}\geqslant 0,\end{eqnarray}$$

and hence we must expand

(4.45) $$\begin{eqnarray}\widetilde{x}_{p}(\unicode[STIX]{x1D70F})=\widetilde{x}_{0}(\unicode[STIX]{x1D70F})+o(1),\quad \unicode[STIX]{x1D70F}\geqslant 0,\end{eqnarray}$$

as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , after which $\widetilde{y}_{p}(\unicode[STIX]{x1D70F})=-\widetilde{x}_{p}(\unicode[STIX]{x1D70F})\tan \unicode[STIX]{x1D6FC}$ ( $\dagger$ ). An examination of the plate boundary condition (4.5), (4.39) and (4.42), together with (4.1), then requires that the gauge function $\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FF})=O(\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1})$ (from which it follows that ${\mathcal{G}}(\unicode[STIX]{x1D700})=O(\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1})$ ), and so, without loss of generality, we set $\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FF})=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}$ . We are now able to write $\text{[IBVP]}^{\prime \prime }$ in terms of the inner–inner region variables and substitute from (4.43). It is convenient to first introduce the dependent variables $\unicode[STIX]{x1D6F7}^{\prime \prime }$ and $H^{\prime \prime }$ , according to

(4.46a,b ) $$\begin{eqnarray}\widehat{\unicode[STIX]{x1D719}}_{0}^{\prime \prime }(\widetilde{x},\widetilde{y},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6F7}^{\prime \prime }(\widetilde{x},\widetilde{y},\unicode[STIX]{x1D70F})+\overline{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D70F}\widetilde{y}-{\textstyle \frac{1}{3}}\overline{\unicode[STIX]{x1D707}}^{2}\unicode[STIX]{x1D70F}^{3},\quad \widehat{\unicode[STIX]{x1D702}}_{0}^{\prime \prime }(\widetilde{x},\unicode[STIX]{x1D70F})=H^{\prime \prime }(\widetilde{x},\unicode[STIX]{x1D70F})+{\textstyle \frac{1}{2}}\overline{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D70F}^{2},\end{eqnarray}$$

along with the translated coordinates $(\widehat{x},\widehat{y})$ , according to

(4.47a,b ) $$\begin{eqnarray}\widehat{x}=\widetilde{x}+{\textstyle \frac{1}{2}}\overline{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D70F}^{2}\cot \unicode[STIX]{x1D6FC},\quad \widehat{y}=\widetilde{y}-{\textstyle \frac{1}{2}}\overline{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D70F}^{2},\end{eqnarray}$$

which is simply a shift of origin from the original inner–inner coordinates $(\widetilde{x},\widetilde{y})$ ( $\dagger$ ). The variable changes (4.46) and (4.47) have been chosen for algebraic convenience at a later stage. We obtain at leading order the following nonlinear harmonic evolution free boundary problem for $\unicode[STIX]{x1D6F7}^{\prime \prime }$ , $H^{\prime \prime }$ and $\widehat{x}_{0}$ , namely

(4.48) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D6FB}}^{2}\unicode[STIX]{x1D6F7}^{\prime \prime }=0,\quad \widehat{x}>\widehat{x}_{0}(\unicode[STIX]{x1D70F}),-\widehat{x}\tan \unicode[STIX]{x1D6FC}<\widehat{y}<H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.49) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D6F7}^{\prime \prime }\boldsymbol{\cdot }\hat{\boldsymbol{n}}=0,\quad \widehat{x}>\widehat{x}_{0}(\unicode[STIX]{x1D70F}),\widehat{y}=-\widehat{x}\tan \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.50) $$\begin{eqnarray}\displaystyle & \displaystyle H_{\unicode[STIX]{x1D70F}}^{\prime \prime }+\unicode[STIX]{x1D6F7}_{\widehat{x}}^{\prime \prime }H_{\widehat{x}}^{\prime \prime }-\unicode[STIX]{x1D6F7}_{\widehat{y}}^{\prime \prime }=0,\quad \widehat{x}>\widehat{x}_{0}(\unicode[STIX]{x1D70F}),\widehat{y}=H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.51) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70F}}^{\prime \prime }+{\textstyle \frac{1}{2}}|\widehat{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D6F7}^{\prime \prime }|^{2}+\overline{\unicode[STIX]{x1D707}}H^{\prime \prime }=0,\quad \widehat{x}>\widehat{x}_{0}(\unicode[STIX]{x1D70F}),\widehat{y}=H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.52) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime \prime }(\widehat{x}_{0}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D70F})=-\widehat{x}_{0}(\unicode[STIX]{x1D70F})\tan \unicode[STIX]{x1D6FC},\quad \unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.53) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{\prime \prime }(\widehat{x},\widehat{y},0)=0,\quad \widehat{x}>0,-\widehat{x}\tan \unicode[STIX]{x1D6FC}<\widehat{y}<\unicode[STIX]{x1D702}_{0}(\widehat{x}); & \displaystyle\end{eqnarray}$$

(4.54) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime \prime }(\widehat{x},0)=\unicode[STIX]{x1D702}_{0}(\widehat{x}),\quad \widehat{x}>0; & \displaystyle\end{eqnarray}$$

(4.55) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6F7}^{\prime \prime }(\widehat{r},\unicode[STIX]{x1D703},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70F}A_{0}(\unicode[STIX]{x1D6FC})\cos \unicode[STIX]{x1D6FC}\widehat{r}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}}\cos \frac{\unicode[STIX]{x03C0}}{2\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FC})+o(\widehat{r}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}})\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }\widehat{r}\rightarrow \infty ,-\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D703}<0,\unicode[STIX]{x1D70F}>0;\end{eqnarray}$$

(4.56) $$\begin{eqnarray}\displaystyle H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})=-\unicode[STIX]{x1D70F}^{2}A_{0}(\unicode[STIX]{x1D6FC})\frac{\unicode[STIX]{x03C0}}{4\unicode[STIX]{x1D6FC}}\cos \unicode[STIX]{x1D6FC}\widehat{x}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1}+o(\widehat{x}^{\unicode[STIX]{x03C0}/2\unicode[STIX]{x1D6FC}-1})\quad \text{as }\widehat{x}\rightarrow \infty ,\unicode[STIX]{x1D70F}>0; & & \displaystyle\end{eqnarray}$$

where

(4.57) $$\begin{eqnarray}\widehat{x}_{0}(\unicode[STIX]{x1D70F})=\widetilde{x}_{0}(\unicode[STIX]{x1D70F})+{\textstyle \frac{1}{2}}\overline{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D70F}^{2}\cot \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

Here $\widehat{\unicode[STIX]{x1D735}}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\widehat{x},\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\widehat{y})$ , $A_{0}(\unicode[STIX]{x1D6FC})({<}0)$ is as given in GNB (see (3.10)), and we have introduced polar coordinates $(\widehat{r},\unicode[STIX]{x1D703})$ as $\widehat{x}=\widehat{r}\cos \unicode[STIX]{x1D703}$ , $\widehat{y}=\widehat{r}\sin \unicode[STIX]{x1D703}$ ( $\dagger$ ). The initial boundary value problem (4.48)–(4.56), henceforth referred to as [EBVP], can now be solved numerically using a boundary integral method, which follows the approach discussed in GNB, with implicit time stepping to evolve the solution in time $\unicode[STIX]{x1D70F}$ .

Figure 2. Graph of the evolution of $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ (related to the free surface displacement through (4.46), (4.43), (4.41) and (4.18)) against $\widehat{x}$ for the numerical solution of [EBVP], for $\unicode[STIX]{x1D70F}=0$ , $\unicode[STIX]{x1D70F}=0.75$ , $\unicode[STIX]{x1D70F}=1.5$ , $\unicode[STIX]{x1D70F}=2.25$ , $\unicode[STIX]{x1D70F}=3$ , $\unicode[STIX]{x1D70F}=3.75$ , $\unicode[STIX]{x1D70F}=4.5$ , $\unicode[STIX]{x1D70F}=5.25$ and $\unicode[STIX]{x1D70F}=6$ , (a–i) where $\overline{\unicode[STIX]{x1D707}}=1$ and $\unicode[STIX]{x1D6FC}=1.4$ . In each plot a dash-dot line shows the location of the plate, a dotted line shows the solution for the case of zero initial data and a solid line shows the solution when the initial data are as given in (4.61).

Figure 3. Graph of the evolution of $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ (related to the free surface displacement through (4.46), (4.43), (4.41) and (4.18)) against $\widehat{x}$ for the numerical solution of [EBVP], for $\unicode[STIX]{x1D70F}=0$ , $\unicode[STIX]{x1D70F}=0.13$ , $\unicode[STIX]{x1D70F}=0.25$ , $\unicode[STIX]{x1D70F}=0.38$ , $\unicode[STIX]{x1D70F}=0.5$ , $\unicode[STIX]{x1D70F}=0.63$ , $\unicode[STIX]{x1D70F}=0.75$ , $\unicode[STIX]{x1D70F}=0.88$ and $\unicode[STIX]{x1D70F}=1$ , (a–i) where $\overline{\unicode[STIX]{x1D707}}=0$ and $\unicode[STIX]{x1D6FC}=1.4$ . In each plot a dash-dot line shows the location of the plate, a dotted line shows the solution for the case of zero initial data and a solid line shows the solution when the initial data are as given in (4.61).

Figure 4. Graph of the evolution of $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ (related to the free surface displacement through (4.46), (4.43), (4.41) and (4.18)) against $\widehat{x}$ , showing agreement with the far-field asymptotic form (4.56) for the numerical solution of [EBVP], for $\unicode[STIX]{x1D70F}=0$ , $\unicode[STIX]{x1D70F}=0.75$ , $\unicode[STIX]{x1D70F}=1.5$ , $\unicode[STIX]{x1D70F}=2.25$ , $\unicode[STIX]{x1D70F}=3$ , $\unicode[STIX]{x1D70F}=3.75$ , $\unicode[STIX]{x1D70F}=4.5$ , $\unicode[STIX]{x1D70F}=5.25$ and $\unicode[STIX]{x1D70F}=6$ , where $\overline{\unicode[STIX]{x1D707}}=1$ , and $\unicode[STIX]{x1D6FC}=1.4$ . Here $\unicode[STIX]{x1D70F}$ increases reading from the bottom of the figure to the top. The dash-dot line shows the location of the plate, solid lines show the numerical solution when the initial data are as given in (4.61), and dotted lines plot the far-field asymptotic form (4.56).

Figure 5. Graph of the evolution of $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ (related to the free surface displacement through (4.46), (4.43), (4.41) and (4.18)) against $\widehat{x}$ , showing agreement with the far-field asymptotic form (4.56) for the numerical solution of [EBVP], for $\unicode[STIX]{x1D70F}=0$ , $\unicode[STIX]{x1D70F}=0.13$ , $\unicode[STIX]{x1D70F}=0.25$ , $\unicode[STIX]{x1D70F}=0.38$ , $\unicode[STIX]{x1D70F}=0.5$ , $\unicode[STIX]{x1D70F}=0.63$ , $\unicode[STIX]{x1D70F}=0.75$ , $\unicode[STIX]{x1D70F}=0.88$ and $\unicode[STIX]{x1D70F}=1$ , where $\overline{\unicode[STIX]{x1D707}}=0$ and $\unicode[STIX]{x1D6FC}=1.4$ . Here $\unicode[STIX]{x1D70F}$ increases reading from the bottom of the figure to the top. The dash-dot line shows the location of the plate, solid lines show the numerical solution when the initial data are as given in (4.61), and dotted lines plot the far-field asymptotic form (4.56).

The first thing to observe concerning [EBVP], is that in the case when $\unicode[STIX]{x1D702}_{0}(\widehat{x})=0$ for $\widehat{x}\geqslant 0$ , then the solution to [EBVP] has the similarity structure

(4.58) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{\prime \prime }(\widehat{x},\widehat{y},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}}\widehat{\unicode[STIX]{x1D6F7}}^{\prime \prime }\left(\frac{\widehat{x}}{\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}}},\frac{\widehat{y}}{\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}}}\right),\quad \widehat{x}>\widehat{c}\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}},-\widehat{x}\tan \unicode[STIX]{x1D6FC}<\widehat{y}<\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}}\widehat{H}^{\prime \prime }\left(\frac{\widehat{x}}{\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}}}\right),\unicode[STIX]{x1D70F}>0; & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.59) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}}\widehat{H}^{\prime \prime }\left(\frac{\widehat{x}}{\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}}}\right),\quad \widehat{x}>\widehat{c}\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}},\unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

(4.60) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{x}_{0}(\unicode[STIX]{x1D70F})=\widehat{c}\unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6E4}},\quad \unicode[STIX]{x1D70F}>0; & \displaystyle\end{eqnarray}$$

where $\widehat{c}$ is the constant satisfying $\widehat{H}^{\prime \prime }(\widehat{c})+\widehat{c}=0$ and $\unicode[STIX]{x1D6E4}$ as in GNB (see (5.1)). This is anticipated from, and conforms with, the details in GNB (§ 5). We now proceed with the numerical solution of [EBVP].

In solving [EBVP] numerically we must specify the initial free surface profile. For the following results we have taken $H^{\prime \prime }(\widehat{x},0)$ as

(4.61) $$\begin{eqnarray}H^{\prime \prime }(\widehat{x},0)=\unicode[STIX]{x1D702}_{0}(\widehat{x})=\left\{\begin{array}{@{}ll@{}}0.02(1-\cos 2\unicode[STIX]{x03C0}\widehat{x}), & 0\leqslant \widehat{x}\leqslant 1,\\ 0, & \widehat{x}>1.\end{array}\right.\end{eqnarray}$$

We note that the original free surface displacement is then given by

(4.62) $$\begin{eqnarray}\unicode[STIX]{x1D702}=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}^{2/\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D70F}^{2}+\unicode[STIX]{x1D6FF}\left(H^{\prime \prime }+{\textstyle \frac{1}{2}}\overline{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D70F}^{2}\right)+o(\unicode[STIX]{x1D6FF})\quad \text{as }\unicode[STIX]{x1D6FF}\rightarrow 0.\end{eqnarray}$$

Numerical solutions of [EBVP] are plotted in figures 2–5. In figure 2 we present the comparison between the free surface $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ calculated with zero initial data (shown in each plot as a dotted line) and the free surface $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ calculated with the initial free surface profile given in (4.61) (shown in each plot as a solid line) for the case $\unicode[STIX]{x1D6FC}=1.4$ , with $\overline{\unicode[STIX]{x1D707}}=1$ . It is clear to see, as may be anticipated from § 3.2, that, when initially perturbed, the free surface $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ collapses to the free surface corresponding to zero perturbation (which has the similarity solution (4.59)) as $\unicode[STIX]{x1D70F}\rightarrow \infty$ , indicating that the problem [EBVP] is well-posed and stable in this case. This behaviour is typical of all pairs $(\unicode[STIX]{x1D6FC},\overline{\unicode[STIX]{x1D707}})$ tested in the range $(\unicode[STIX]{x1D6FC},\overline{\unicode[STIX]{x1D707}})\in (\unicode[STIX]{x03C0}/4,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$ , and of all initial free surface profiles $\unicode[STIX]{x1D702}_{0}(\widehat{x})$ tested. As we decrease $\overline{\unicode[STIX]{x1D707}}$ and take values with $\overline{\unicode[STIX]{x1D707}}<0$ we are unable to obtain numerically any converged solutions to [EBVP]. This indicates that the problem [EBVP] is ill-posed, which may also be anticipated from the theory presented in § 3.2. In figure 3 we present the comparison between the free surface $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ calculated with zero initial data (shown in each plot as a dotted line) and the free surface $H^{\prime \prime }(\widehat{x},\unicode[STIX]{x1D70F})$ calculated with the initial free surface profile given in (4.61) (shown in each plot as a solid line) for the case $\unicode[STIX]{x1D6FC}=1.4$ , with $\overline{\unicode[STIX]{x1D707}}=0$ . The case $\overline{\unicode[STIX]{x1D707}}=0$ separates the regions on the $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})$ plane where the problem [EBVP] is well-posed and stable ( $\overline{\unicode[STIX]{x1D707}}>0$ ) from ill-posed ( $\overline{\unicode[STIX]{x1D707}}<0$ ), with $\unicode[STIX]{x1D707}=0$ falling into the well-posed and stable case. Finally, figures 4 and 5 demonstrate the agreement of the numerical solution with the far-field asymptotic form in [EBVP] (4.56), at least to the graphical scales shown.