2.1. Formulation

We consider a two-dimensional incompressible, irrotational, inviscid flow through a channel of finite depth over a submerged step. The upstream channel depth is given by $L$, and the upstream flow velocity is given by $U$. An elastic sheet with flexural rigidity $D$ rests on the surface of the flow. The position of the elastic sheet is denoted as $\xi (x)$. A schematic of this flow behaviour is shown in figure 1. We now non-dimensionalize the lengths of the system by the upstream depth $L$, and the velocities by the upstream flow velocity $U$.

Figure 1.

Mapping of a fluid domain underneath an elastic sheet to a fixed known region of the complex potential plane. The elastic sheet is shown as a dashed line, and the rigid base is shown as an unbroken line. Steady flow follows streamlines, which are curves in the potential plane with constant $\psi$. The free surface maps to the top streamline, typically labelled $\psi = 0$, while the lower boundary typically maps to $\psi = -{\rm \pi}$; consequently, the flow region is known completely. The fluid velocity is singular at the points labelled $B$ and $C$, shown as black circles. (a) Physical fluid domain, $z=x+\mathrm {i} y$. (b) Complex potential domain, $w=\phi +\mathrm {i}\psi$.

The fluid potential satisfies Laplace's equation

(2.1)\begin{equation} \nabla^2 \phi = 0. \end{equation}

As the flow is steady, we apply a kinematic boundary condition on all boundaries,

(2.2)\begin{equation} \frac{\partial \phi}{\partial n} = 0, \end{equation}

where $n$ is the unit normal direction. On the free surface, we have the dynamic boundary condition, obtained from the Bernoulli equation,

(2.3)\begin{equation} \frac{F^2}{2}\,(|\boldsymbol{\nabla} \phi^2| - 1) + y + \frac{D}{\rho g L^4} \left(\kappa_{ss} + \frac{1}{2}\,\kappa^3\right) = 0\quad \mathrm{on} \ y = \xi(x), \end{equation}

where $\kappa$ is the curvature, defined to be positive if the centre of curvature lies within the fluid, $s$ is the arc length along the surface after non-dimensionalization, $\rho$ is the density of the fluid, $g$ is the acceleration due to gravity, $F$ is the Froude number defined in (1.5a,b), and $D$ is the flexural rigidity of the plate. The upstream flow is uniform with velocity $U$, such that

(2.4)\begin{equation} (\phi_x, \phi_y) \to (1,0) \quad \mathrm{as} \ x \to -\infty. \end{equation}

Using the quantities defined in (1.5a,b), we may rewrite (2.3) as

(2.5)\begin{equation} \frac{\beta \epsilon}{2}\,(|\boldsymbol{\nabla} \phi|^2 - 1) + y + \beta \tau \epsilon^4 \left(\kappa_{ss} + \frac{1}{2}\,\kappa^3\right) = 0\quad \mathrm{on} \ y = \xi(x), \end{equation}

where $\epsilon$ is a small parameter, and $\beta$ and $\tau$ determine the ratio between the Froude number $F$ and the elastic length ratio $l_D/L$. We note that the $\tau = 0$ problem corresponds to pure gravity waves, studied in Chapman & Vanden-Broeck (Reference Chapman and Vanden-Broeck2006), while $\beta \to 0$ and $\tau = 1/\beta$ corresponds to pure elastic waves, studied in Lustri et al. (Reference Lustri, Koens and Pethiyagoda2020). In § 2.3, we will find our results to be consistent with these prior studies.

Differentiating (2.5) with respect to $s$ gives

(2.6)\begin{equation} \beta\epsilon q\,\frac{\mathrm{d} q}{\mathrm{d} s} + \frac{\mathrm{d} y}{\mathrm{d} s} + \beta \tau \epsilon^4 \left(\kappa_{sss} + \frac{3}{2}\,\kappa_s \kappa^2\right) = 0\quad \mathrm{on} \ y = \xi(x). \end{equation}

We define a complex potential $w = \phi + \mathrm {i} \psi$, where $\phi$ is the fluid potential, and $\psi$ is the streamfunction. This maps the fluid region to an infinite strip bounded by $\psi = -{\rm \pi}$ and $\psi = 0$. Noting that

(2.7a–c)\begin{equation} \kappa = \frac{\mathrm{d} \theta}{\mathrm{d} s},\quad \frac{\mathrm{d} y}{\mathrm{d} s} = \sin\theta, \quad \frac{\mathrm{d} }{\mathrm{d} s} = q\,\frac{\mathrm{d} }{\mathrm{d} \phi}, \end{equation}

we write the Bernoulli condition (2.6) in terms of $\phi$:

(2.8)\begin{align} &\beta\epsilon q^2\,\frac{\mathrm{d} q}{\mathrm{d} \phi} + \sin\theta + \beta \tau \epsilon^4 \left[ q \left(\frac{\mathrm{d} q}{\mathrm{d} \phi}\right)^3 \frac{\mathrm{d} \theta}{\mathrm{d} \phi} + 4q^2\,\frac{\mathrm{d} q}{\mathrm{d} \phi}\, \frac{\mathrm{d} \theta}{\mathrm{d} \phi}\,\frac{\mathrm{d} ^2 q}{\mathrm{d} \phi^2}+ 7q \left(\frac{\mathrm{d} q}{\mathrm{d} \phi}\right)^2\frac{\mathrm{d} ^2\theta}{\mathrm{d} \phi^2}\right.\nonumber\\ &\quad \left.{}+ 4 q^3\,\frac{\mathrm{d} ^2 q}{\mathrm{d} \phi^2}\,\frac{\mathrm{d} ^2\theta}{\mathrm{d} \phi^2} + q^3\,\frac{\mathrm{d} \theta}{\mathrm{d} \phi}\,\frac{\mathrm{d} ^3\theta}{\mathrm{d} \phi^3} + q^4\,\frac{\mathrm{d} ^4\theta}{\mathrm{d} \phi^4}\!+\! \frac{3}{2}\,q^3\,\frac{\mathrm{d} q}{\mathrm{d} \phi}\left(\frac{\mathrm{d} \theta}{\mathrm{d} \phi}\right)^3 \!+\! \frac{3}{2}\,q^4\left(\frac{\mathrm{d} \theta}{\mathrm{d} \phi}\right)^2\frac{\mathrm{d} ^2\theta}{\mathrm{d} \phi}\right] \!=\! 0. \end{align}

We also define the complex velocity $\mathrm {d} w/\mathrm {d} z = u - \mathrm {i} v$, written as $q\, \mathrm {e}^{-\mathrm {i}\theta }$. In this formulation, $q$ is the flow velocity at a point, and $\theta$ is the angle that the streamlines make with the horizontal axis. Analytically continuing (2.8) allows us to replace $\phi$ with the complex potential $w$ in (2.8) to obtain the analytically continued free-surface condition

(2.9)\begin{align} &\beta\epsilon q^2\,\frac{\mathrm{d} q}{\mathrm{d} w} + \sin\theta + \beta \tau \epsilon^4 \left[ q \left(\frac{\mathrm{d} q}{\mathrm{d} w}\right)^3 \frac{\mathrm{d} \theta}{\mathrm{d} w} + 4q^2\,\frac{\mathrm{d} q}{\mathrm{d} w}\,\frac{\mathrm{d} \theta}{\mathrm{d} w}\,\frac{\mathrm{d} ^2 q}{\mathrm{d} w^2}+ 7q \left(\frac{\mathrm{d} q}{\mathrm{d} w}\right)^2\frac{\mathrm{d} ^2\theta}{\mathrm{d} w^2}\right. \nonumber\\ &\quad \left. {}+ 4 q^3\,\frac{\mathrm{d} ^2 q}{\mathrm{d} w^2}\,\frac{\mathrm{d} ^2\theta}{\mathrm{d} w^2} + q^3\,\frac{\mathrm{d} \theta}{\mathrm{d} w}\,\frac{\mathrm{d} ^3\theta}{\mathrm{d} w^3} \!+\! q^4\,\frac{\mathrm{d} ^4\theta}{\mathrm{d} w^4} \!+\! \frac{3}{2}\,q^3\frac{\mathrm{d} q}{\mathrm{d} w}\left(\frac{\mathrm{d} \theta}{\mathrm{d} w}\right)^3 \!+\! \frac{3}{2}\,q^4\left(\frac{\mathrm{d} \theta}{\mathrm{d} w}\right)^2\frac{\mathrm{d} ^2\theta}{\mathrm{d} w}\right] \!=\! 0. \end{align}

We apply a conformal map $\zeta = \mathrm {e}^{-w}$ in order to map the fluid region from a strip in the complex potential plane to the upper half $\zeta$-plane. This map is illustrated in figure 2. We also define $\zeta = \xi + \mathrm {i} \eta$, where $\xi$ and $\eta$ are real quantities. Notably, the free surface maps to $\xi > 0$ and the base of the flow region maps to $\xi < 0$. In the mapped plane, we can apply Cauchy's theorem to obtain

(2.10)\begin{equation} \log q ={-} \frac{1}{\rm \pi} {{\int\hskip -1,05em -\,}_{-\infty}^{\infty}} \frac{\theta(\xi')}{\xi' - \xi}\,\mathrm{d}\xi'. \end{equation}

Analytically continuing this expression into the upper half-plane gives

(2.11)\begin{equation} q - \mathrm{i} \theta ={-}\frac{1}{\rm \pi}\int_{-\infty}^{\infty} \frac{\theta(\xi')}{\xi' - \zeta}\,\mathrm{d}\xi'. \end{equation}

We will define the base of the flow as a step, where $\theta = {\rm \pi}/2$ for $-(1+\delta ) < \zeta < -1$, and $\theta = 0$ for $\zeta < -(1+\delta )$ and $-1 < \zeta < 0$. Hence

(2.12)\begin{equation} q - \mathrm{i} \theta = \frac{1}{2}\log\left(\frac{\zeta+b}{\zeta+1}\right)-\frac{1}{\rm \pi}\int_{0}^{\infty} \frac{\theta(\xi')}{\xi' - \zeta}\,\mathrm{d}\xi'.\end{equation}

We note that the strip in the complex potential plane may also be analytically continued into the lower half $\zeta$-plane, which will produce complex conjugate behaviour. The full behaviour of the elastic sheet can be obtained by taking the sum of both the upper and lower half-plane contributions. We will not perform the lower half-plane calculations explicitly, but will instead add the appropriate complex conjugate contribution to the results of the upper half-plane analysis.

Figure 2.

This schematic illustrates the effect of the mapping $w \mapsto \zeta$ between (a) the fluid potential domain $w=\phi +\mathrm {i}\psi$, and (b) the mapped domain $\zeta =\xi +\mathrm {i}\eta$. The mapping takes the fluid region to the entire upper half mapped plane. The elastic sheet $\psi = 0$ maps to the line $\xi > 0$, and the submerged boundary $\psi = -{\rm \pi}$ maps to the line $\xi < 0$. The elastic sheet is shown as a dashed line, and the rigid base is shown as an unbroken line. The singularities map to points that will be labelled $\xi = -b$ and $\xi = -1$.

This completes the governing equations. We treat (2.9) and (2.12) as the equations governing the analytically continued free surface. We will subsequently express the Bernoulli equation in terms of the mapped variable $\zeta$, but we will hold off until after the linearization step.

2.2. Linearization

We can linearize around the free stream for small step height. We set $b = 1 + \delta$ (assuming $0 < \delta \ll \epsilon$) and set $q = 1 + \delta \hat {q}$ and $\theta = \delta \hat {\theta }$. The linearized fluid equation is now given by

(2.13)\begin{equation} \hat{q} - \mathrm{i} \hat{\theta} = \frac{1}{2(\zeta+1)}-\frac{1}{\rm \pi}\int_{0}^{\infty} \frac{\hat{\theta}(\xi')}{\xi' - \zeta}\,\mathrm{d}\xi'. \end{equation}

The linearized Bernoulli equation is given by

(2.14)\begin{equation} \beta\epsilon\,\frac{\mathrm{d} \hat{q}}{\mathrm{d} w} + \hat{\theta} + \beta\tau\epsilon^4\,\frac{\mathrm{d} ^4\hat{\theta}}{\mathrm{d} w^4} = 0. \end{equation}

We could apply the mapping $\zeta = \mathrm {e}^{-w}$ to the Bernoulli equation, but the analysis is more straightforward in the complex potential plane. We have now fixed the problem so that the boundary follows a known curve ($\zeta > 0$ in the mapped plane, $\psi = 0$ in the complex potential plane). The linearization step is not necessary; we could apply this exponential asymptotic analysis to the fully nonlinear problem. In this case, the existence of both upstream and downstream waves on the free boundary mean that it is difficult to verify the results computationally. Progress on studying these systems has been made in the context of gravity-capillary waves in Jamshidi & Trinh (Reference Jamshidi and Trinh2020), but this remains a challenging numerical problem. We will discuss briefly the asymptotics of nonlinear geometries in § 2.5.

2.3. Exponential asymptotics

We write the series expression in the limit that $\epsilon \to 0$,

(2.15a,b)\begin{equation} \hat{q} \sim \sum_{n=0}^{\infty}\epsilon^{n}q^{(n)},\quad \hat{\theta} \sim \sum_{n=0}^{\infty}\epsilon^{n}\theta^{(n)}. \end{equation}

Note that including both gravity and elastic waves means that the form of the late-order terms requires powers of $\epsilon$ rather than $\epsilon ^3$, unlike in Lustri et al. (Reference Lustri, Koens and Pethiyagoda2020). Note that retaining the full power series in $\epsilon$ in a system that has been linearized in $\delta$ implies that we are considering the regime $0 < \delta \ll \epsilon \ll 1$.

The leading-order behaviour of the flow on the complex free surface is found by direct substitution, giving

(2.16a,b)\begin{equation} q^{(0)} = \frac{1}{2(\zeta+1)} = \frac{1}{2(\mathrm{e}^{{-}w}+1)},\quad \theta^{(0)} = 0. \end{equation}

The leading-order behaviour is singular at $w = \pm (2M+1) {\rm \pi}\mathrm {i}$, for $M \in \mathbb {Z}$. The singularities that matter are located at $M = \pm 1$. We will concentrate on the contributions due to the singularity at $w ={\rm \pi} \mathrm {i}$, adding the corresponding contributions afterwards. At higher orders, we obtain recurrence expressions for the complexified free surface

(2.17)\begin{gather} q^{(n)} - \mathrm{i} \theta^{(n)} ={-}\frac{1}{\rm \pi}\int_{0}^{\infty} \frac{\theta^{(n)}(\xi')}{\xi' - \zeta}\,\mathrm{d}\xi', \end{gather}

(2.18)\begin{gather}\beta\,\frac{\mathrm{d} q^{(n-1)}}{\mathrm{d} w} + \theta^{(n)} + \beta \tau\,\frac{\mathrm{d} ^4\theta^{(n-4)}}{\mathrm{d} w^4}= 0. \end{gather}

We are most interested in the form of the late-order terms, so we apply the late-order ansatz as $n \to \infty$ from (1.10a,b). In the first equation from (2.17), we neglect the integral expression, as it must be exponentially subdominant to the remaining terms in the expression. This simplification was used in Chapman & Vanden-Broeck (Reference Chapman and Vanden-Broeck2002, Reference Chapman and Vanden-Broeck2006), and discussed in detail for the case of gravity waves past a ship in Trinh, Chapman & Vanden-Broeck (Reference Trinh, Chapman and Vanden-Broeck2011). A similar justification can be made here. At leading order as $n \to \infty$, we find $Q = \mathrm {i} \varTheta$, and

(2.19)\begin{equation} 1 - \mathrm{i}\beta\,\frac{\mathrm{d} \chi}{\mathrm{d} w} + \beta \tau \left(\frac{\mathrm{d} \chi}{\mathrm{d} w}\right)^4 = 0. \end{equation}

We can determine the form of the exponentially small contributions by solving the singulant equation (2.19). We recall that $\chi = 0$ at $w = {\rm \pi}\mathrm {i}$. This expression has four solutions, of the form

(2.20)\begin{equation} \chi = k_j (w + \mathrm{i} {\rm \pi}),\end{equation}

where $k_j$ for $j = 1,\ldots, 4$ depends on $\beta$ and $\tau$, but not $\zeta$. We denote the specific singulants as $\chi _j$ for $j = 1,\ldots,4$.

Continuing to the next order, corresponding to ${O}(q^{(n-1)})$ as $n \to \infty$, gives $Q$ and $\varTheta$ constant. To denote this clearly, we write $\varTheta = \varLambda$ and $Q = \mathrm {i}\varLambda$, where $\varLambda$ is constant in $w$, although it does depend on $\beta$ and $\tau$. This term must be determined by comparing the late-order terms with the inner problem in the neighbourhood of the singularity at $\zeta = -\mathrm {i}{\rm \pi}$. We perform this inner analysis in § A.1, and find that the prefactors are given by

(2.21)\begin{equation} \varLambda_j = \frac{k_j^8\left(k_j A_5 + k_j^2 A_6 + k_j^3 A_7 -\dfrac{1}{\beta\tau}\,A_8 \right)}{ \dfrac{2}{\beta\tau} - 6 k_j^4}, \end{equation}

where $A_j = \mathrm {i}^j \beta ^{j-3}(\beta ^3 + (4-j)\tau )$ for $j = 5, \ldots, 8$. The index choice is related to the analysis in § A.1.

Knowing that $\varTheta$ and $Q$ are constants, we can determine the value of $\gamma$ that is required for the late-order terms to be consistent with the leading-order behaviour near the singularity. The singularity in the leading order has strength 1, and this will increase by 1 at each iteration. Hence the strength of the singularity in the series term $q^{(n)}$ will be $n+1$, indicating that $\gamma = 1$. Consequently, we have fully determined the late-order asymptotic series terms (1.10a,b).

Using the methods of Olde Daalhuis et al. (Reference Olde Daalhuis, Chapman, King, Ockendon and Tew1995) and Chapman et al. (Reference Chapman, King and Adams1998), shown in detail in Trinh & Chapman (Reference Trinh and Chapman2013a,Reference Trinh and Chapmanb), we may determine the behaviour of the surface waves that correspond with each of the four solutions of (2.20). This analysis is presented in § A.2, and gives the exponentially small wave contribution from $\chi _j$, which we denote as $\theta _{{exp},j}$, as

(2.22)\begin{equation} \theta_{{exp},j} \sim 2\, \mathrm{Re}\left[\left(\frac{1}{1 - 4 \tau \mathrm{i} k_j^3}\right)\frac{2{\rm \pi} k_j \varLambda_j}{\beta\epsilon}\exp({-k_j(w+\mathrm{i}{\rm \pi})/\epsilon})\right],\end{equation}

with a similar expression for $q_{{exp},j}$. The real part is obtained by taking the sum of the upper and lower half $\zeta$-plane contributions, which are complex conjugate values. We have therefore calculated the form of the waves, and can determine the regions in which they are present by studying the Stokes phenomenon in the system.

From the value of $\chi$ in (2.20) and the form of the exponential oscillations (1.14a,b), it is apparent that non-decaying wave behaviour can exist only if $k_j$ takes a purely imaginary value. In figures 3(a,c), we illustrate the solutions for $\tau = 1$ over a range of $\beta$, while in figures 3(b,d), we illustrate the solutions for $\beta = 1$ over a range of $\tau$. For fixed $\tau$, there exists some critical $\beta$, denoted $\beta _{{c}}(\tau )$, such that there are two values of $k_j$ with no real component for $\beta > \beta _{{c}}(\tau )$, and there are no values of $k_j$ that are purely imaginary if $\beta$ is less than this critical value. Conversely, for fixed $\beta$, there exists a critical value of $\tau$, denoted $\tau _{{c}}(\beta )$, such that there are two values of $k_j$ with no real component for $\tau < \tau _{{c}}(\beta )$, and no purely imaginary values of $k_j$ if $\tau$ exceeds this critical value. Hence non-decaying wave behaviour exists in the solution only if $\beta > \beta _{{c}}(\tau )$, or equivalently, $\tau < \tau _{{c}}(\beta )$. The wave behaviour in the $\beta$–$\tau$ parameter space is illustrated in figure 4.

Figure 3.

Real and imaginary components of $k_1$ (red) and $k_2$ (black), $k_3$ (blue) and $k_4$ (magenta) for (a,c) $\tau = 1$, and (b,d) $\beta = 1$. Dashed lines indicate multiple solutions $k_j$ taking identical values. In (a,c), there is a critical value of $\beta$ above which $\mathrm {Re}(k_{1,2}) = 0$. In (b,d), there is a critical value of $\tau$ below which $\mathrm {Re}(k_{1,2}) = 0$.

Figure 4.

Illustration of the wave behaviour as $\beta$ and $\tau$ are varied. In the unshaded region, corresponding to $\beta < \beta _{{c}}$, all of the surface waves decay spatially away from the obstacle. In the shaded region, corresponding to $\beta > \beta _{{c}}$, the surface behaviour contains non-decaying waves in the downstream and upstream directions, driven by gravitational and elastic restoring forces, respectively.

As (2.19) is a quartic equation, the four solutions may be computed exactly. We denote the solutions according to their asymptotic behaviour in the limit that $\tau \rightarrow 0$ for fixed $\beta$, which corresponds to the gravity wave limit. The four solutions have the behaviour

(2.23a–c)\begin{equation} \left.\begin{gathered} k_1 ={-}\frac{\mathrm{i}}{\beta} + \mathcal{O}(\tau^{1/6}), \quad k_2 ={-}\frac{\mathrm{i}}{\tau^{1/3}} + \frac{\mathrm{i}}{3\beta} + \mathcal{O}(\tau^{1/6}),\\ k_{3,4} ={-}\frac{\mathrm{i} \pm\sqrt{3}}{2\tau^{1/3}} + \frac{\mathrm{i}}{3\beta}+ \mathcal{O}(\tau^{1/6}). \end{gathered}\right\} \end{equation}

For each of these values of $k_j$, we denote the corresponding singulant as $\chi _j$. The exponentially small waves given by the solution $\chi _1$ tend to known gravitational wave behaviour from Chapman & Vanden-Broeck (Reference Chapman and Vanden-Broeck2006) in this limit, while the amplitude of the remaining waves tends to zero.

The three wave contributions associated with $\chi _2$, $\chi _3$ and $\chi _4$ correspond to three elastic wave contributions, and are equivalent in the limit that $\beta \to 0$ and $\tau \to 1/\beta$ to those found in Lustri et al. (Reference Lustri, Pethiyagoda and Chapman2019). One of these wave contributions, corresponding to the singulant $\chi _2$, produces waves that do not decay in space away from the obstacle in this limit, while the remaining wave contributions decay exponentially in space. This is consistent with the behaviour identified in Lustri et al. (Reference Lustri, Pethiyagoda and Chapman2019).

The exact form of these solutions may be determined using a computational algebra package, allowing us to determine the critical values of $\beta$ and $\tau$ exactly. This bifurcation corresponds to a branch point in the exact solutions at $27\beta _{{c}}^3 - 256 t_{{c}} = 0$, or

(2.24)\begin{equation} 3\beta_{{c}} = 2^{8/3}\tau^{1/3}_{{c}}. \end{equation}

This curve in the $\beta$–$\tau$ parameter space divides solutions with non-decaying oscillations corresponding to gravity waves (corresponding to the solution $k_1$) and elastic waves (corresponding to the solution $k_2$), and solutions in which all four of the wave contributions decay. This result corresponds to setting the flow velocity $U$ to be equal to $c_{{min}}$ in (1.4a,b). It is not surprising that we recover the critical speed from the infinite-depth problem, as the linearization step requires the obstacle depth to be small compared to the channel depth. These results are therefore consistent with the phase velocity behaviour (1.3). It is impossible in this regime to choose parameters such that only one of the upstream elastic waves or downstream gravitational waves decays, while the other has constant amplitude. The solution either contains both non-decaying surface wave contributions, or neither. We will see later that this is not necessarily true for nonlinear geometries.

When non-decaying waves are present, and therefore $k_j$ is imaginary, the surface behaviour can be computed by integrating (2.22) with respect to $x$. Noting that $\phi = x$ to leading order in $\delta$, $\varLambda _j$ is real when $k_j$ is imaginary, and recalling that the wave amplitude is scaled by $\delta$, we can obtain a wave amplitude satisfying

(2.25)\begin{equation} \mathrm{amplitude} \sim \frac{2{\rm \pi} \varLambda_j \delta}{\beta(1-4\tau \mathrm{i} k_j)}\exp({-|k_j|{\rm \pi}/\epsilon}) \quad \mathrm{as} \ \epsilon \to 0. \end{equation}

From the leading-order behaviour $q^{(0)}$ in (2.16a,b), it can be determined that the non-dimensional step height is $\delta /2$ to leading order in $\epsilon$. Hence we are able to determine a relationship between the wave amplitude and the step height in the linearized regime.

From this expression, the amplitude can be expressed in terms of the upstream flow speed and the step height, rather than $\delta$ and $\epsilon$. The step height is $\delta /2$ in the limit that $\epsilon \to 0$. The upstream flow speed can be incorporated using (1.4a,b) and (1.5a,b) to give $U/c_{{min}} = 3^{1/4}/(4\tau \epsilon ^3)$, which can be solved for $\epsilon$. This allows the amplitude expression in (2.25) to be written in terms of physical properties of the flow geometry, and used to determine quantities such as the wave energy, which scales with the square of the amplitude.

2.4. Stokes structure

The Stokes structure of the analytically continued free surface is presented in figures 5(a) and 5(b) for $\beta > \beta _{{c}}$ and $\beta < \beta _{{c}}$, respectively. Schematics representing the physical flow behaviour are shown in figures 5(c) and 5(d) for $\beta > \beta _{{c}}$ and $\beta < \beta _{{c}}$, respectively.

Figure 5.

Schematics of the analytically continued free surface are presented in (a,b). Stokes curves are shown as blue lines that originate at the leading-order singularity, shown as a black circle at $w = -\mathrm {i}{\rm \pi}$. The points where the Stokes curve intersect the free surface are depicted as blue circles. If the waves are constant amplitude, then the blue circles are filled. If the waves decay in space, then the circles are unfilled. The regions in which the wave contributions are present are indicated by arrows above the schematics. The cases for $\beta > \beta _{{c}}$ and $\beta < \beta {{c}}$ are shown in (a,b), respectively. In (a), the free surface contains constant-amplitude upstream elastic waves and downstream gravity waves, which are both switched on across the Stokes curve that follows $\mathrm {Re}(w) = 0$. The surface intersects two other Stokes curves, which switch on decaying free-surface waves. In (b), the free surface contains only decaying waves in the far field. This physical behaviour is shown in (c,d) for $\beta > \beta _{{c}}$ and $\beta < \beta c$, respectively. In (c), the constant-amplitude upstream and downstream waves are depicted. The decaying waves are exponentially small compared to the constant-amplitude waves as $\epsilon \to 0$, and therefore not shown. In (d), all four wave contributions are shown, and it can be seen that they decay in space.

Figure 5(a) shows the Stokes structure on the analytically continued free surface for $\beta > \beta _{{c}}$. Here, elastic and gravity wave contributions are switched on across a Stokes curve that extends vertically from the singularity at $w = -\mathrm {i} {\rm \pi}$, corresponding to $\chi _1$ and $\chi _2$, respectively. The gravity waves extend downstream from the Stokes curve, while the elastic waves extend upstream. As $k_1$ and $k_2$ are imaginary, the waves do not decay in space, but rather persist with constant amplitude.

Two other Stokes curves are present on the analytically continued free surface, corresponding to $\chi _3$ and $\chi _4$. These contributions cause rapidly decaying waves to appear downstream and upstream from the obstacle, respectively. In this case, the direction of propagation is determined by the sign of $\mathrm {Re}(k_3)$ and $\mathrm {Re}(k_4)$. As $\mathrm {Re}(k_3) > 0$, the waves must appear on the downstream side of the Stokes curve, as they would grow exponentially in the upstream direction. Conversely, as $\mathrm {Re}(k_3) < 0$, the waves must appear only in the upstream direction, where they decay exponentially in space.

A schematic of the physical behaviour of this system is shown in figure 5(c), which depicts constant-amplitude gravity and elastic waves in the downstream and upstream directions, respectively. The decaying waves are exponentially small compared to the constant-amplitude waves as $\epsilon \to 0$ even at the point in the surface where they first appear, so they are not depicted in the schematic. In this schematic, the gravitational waves are represented with a larger amplitude than the elastic waves. From figures 3(a,c), we see that $0 > \mathrm {Im}(k_1) > \mathrm {Im}(k_2)$ in the region $\beta > \beta _{{c}}$. From the form of the exponentially small terms in (2.22), this implies that the gravity waves associated with $\chi _1$ must have a greater amplitude than the elastic waves associated with $\chi _2$.

Figure 5(b) shows the Stokes structure on the analytically continued free surface for $\beta < \beta _{{c}}$. As $\mathrm {Re}(k_j) < 0$ for $j = 1,3$, the waves associated with $\chi _1$ and $\chi _3$ must decay downstream from the corresponding Stokes curve. Conversely, as $\mathrm {Re}(k_j) > 0$ for $j = 2,4$, the associated waves must decay upstream from the corresponding Stokes curve. A schematic of this physical configuration is shown in figure 5(d). All four wave contributions decay exponentially in space, meaning that the surface far upstream and downstream from the obstacle must be flat, with no waves present.

2.5. Nonlinear problem

We note that an exponential asymptotic analysis of gravity-capillary waves in nonlinear regimes in Trinh & Chapman (Reference Trinh and Chapman2013b) revealed a complicated wave structure, including second-generation Stokes switching (see Chapman & Mortimer Reference Chapman and Mortimer2005), which was caused by interactions between gravity and capillary effects. We will outline the steps required in order to analyse flexural-gravity waves in a nonlinear regime, and determine the singulant equation for general flow over topography. We will then consider the singulant behaviour for flow over a step in a nonlinear regime. This geometry again corresponds to figure 1, although the step height is no longer small.

The formulation of this problem without linearization is largely analogous to the previous analysis, except that the analytically continued nonlinear dynamic boundary condition is given by (2.9). The remaining steps follow essentially the same format. We obtain a system of recurrence equations for the series terms that is similar to (2.17)–(2.18), which can be solved to determine the algebraic series terms for the flow behaviour. The leading-order behaviour of the flow is given by

(2.26a,b)\begin{equation} \theta^{(0)} = 0,\quad q^{(0)} = \left(\frac{\mathrm{e}^{{-}w}+b}{\mathrm{e}^{{-}w} + 1}\right)^{1/2}. \end{equation}

We now pose a late-order ansatz identical to (1.10a,b) and apply this to the recurrence relation. Matching the resultant expression in the limit that $n \rightarrow \infty$ gives a singulant equation at leading order:

(2.27a,b)\begin{equation} 1 - \beta (q^{(0)})^3\,\frac{\mathrm{d} \chi}{\mathrm{d} w} + \beta \tau (q^{(0)})^4 \left(\frac{\mathrm{d} \chi}{\mathrm{d} w}\right)^4 = 0,\quad \chi(-\mathrm{i}{\rm \pi}) = 0. \end{equation}

This is a nonlinear differential equation that depends on the leading-order flow behaviour. Even without solving this differential equation, we are able to make some observations regarding the upstream and downstream flow behaviour. We denote $q^{(0)}$ in the limit that $w \rightarrow \infty$ as $q_{{down}}$, corresponding to the velocity far downstream from the obstacle. Similarly, we denote upstream flow velocity, corresponding $q^{(0)}$ in the limit $w \to -\infty$, as $q_{{up}}$.

The nonlinear system has different critical values of $\beta$ and $\tau$ for upstream and downstream waves. The downstream critical values are given by

(2.28)\begin{equation} 3 q_{{down}}^{8/3} \beta_{{c,down}} = 2^{8/3}\tau^{1/3}_{{c,down}}, \end{equation}

while the upstream critical values are given by

(2.29)\begin{equation} 3q_{{up}}^{8/3} \beta_{{c,up}} = 2^{8/3}\tau^{1/3}_{{c,up}}, \end{equation}

where the subscripts indicate whether the critical value describes the upstream or downstream region. For the step geometry in figure 1, we have $q_{{down}} = b^{1/2}$, while $q_{{up}} = 1$. This gives

(2.30a,b)\begin{equation} \beta_{{c,up}} = \left(\frac{256}{27}\,\tau_{{c,up}}\right)^{1/3},\quad \beta_{{c,down}} = \left(\frac{256}{27b^4}\,\tau_{{c,down}}\right)^{1/3}. \end{equation}

There are three possible elastic sheet behaviours. If $\beta < \beta _{{c,down}}$, then all waves on the free surface must decay in space away from the step. If $\beta > \beta _{{c,up}}$, then the surface can contain non-decaying gravitational waves downstream from the step, and non-decaying elastic waves upstream from the step. These configurations were both possible in the linearized problem. However, if $\beta _{{c,down}} < \beta < \beta _{{c,up}}$, then any downstream gravitational waves have non-decaying amplitude, while all elastic effects must decay in space away from the step. The parameter regimes are illustrated for a step with $b = 2$ in figure 6(a).

Figure 6.

Possible wave behaviour as $\beta$ and $\tau$ are varied, for (a) an upwards step with $b = 2$, and (b) a downwards step with $b=2$. In (a,b), there are three regions. In the unshaded region, all elastic and gravitational wave contributions must decay in space away from the step. In the dark-shaded region, both elastic and gravitational waves can propagate indefinitely with constant amplitude. In (a), the lightly shaded region corresponds to $\beta _{{c, down}} < \beta < \beta _{{c, up}}$, and the surface can support non-decaying waves only in the downstream region, driven by gravity. Any upstream elastic waves must decay in space. In (b), the lightly shaded region corresponds to $\beta _{{c, up}} < \beta < \beta _{{c, down}}$, and the surface can support non-decaying waves only in the upstream region, driven by elastic forces. Panels (c,d) depict the one-sided wave profiles for (a,b), respectively. In (c), the schematic shows an upwards step that produces downstream gravity waves with constant amplitude, while the upstream waves decay. In (d), the schematic shows a downwards step that produces upstream elastic waves with constant amplitude, while the downstream gravity waves decay. Neither of these two behaviours is possible in the linearized system.

It is also possible to consider a downwards step, such that $\theta _0 = 0$ for $-b < \zeta < -1$. In this case, the leading-order solution is given by

(2.31a,b)\begin{equation} \theta^{(0)} = 0,\quad q^{(0)} = \left(\frac{\mathrm{e}^{{-}w}+1}{\mathrm{e}^{{-}w} + b}\right)^{1/2}. \end{equation}

The new critical values are instead given by

(2.32a,b)\begin{equation} \beta_{{c,up}} = \left(\frac{256}{27}\,\tau_{{c,up}}\right)^{1/3},\quad \beta_{{c,down}} =\left(\frac{256b^4}{27}\,\tau_{{c,down}}\right)^{1/3}. \end{equation}

If $\beta < \beta _{{c,up}}$, then all waves on the free surface must decay in space away from the step. If $\beta > \beta _{{c,down}}$, then the flow exceeds both critical values of $\beta$, and can support non-decaying gravitational waves downstream from the step, and non-decaying elastic waves upstream from the step. If $\beta _{{c,up}} < \beta < \beta _{{c,down}}$, then any downstream gravitational waves must decay in space, while non-decaying elastic effects are possible. The parameter regimes are illustrated for a downwards step with $b = 2$ in figure 6(b).

This behaviour is consistent with the dispersion relation for finite-depth flow (1.2). We denote the upstream depth of the channel as $L_{{u}}$ and the downstream depth as $L_{{d}}$; for an upwards step, $L_{{d}} < L_{{u}}$. This means that the minimum value for the phase velocity, $c_{{min}}$, differs on either side of the step. As $L$ increases, it can be seen from the dispersion relation (1.2) that the minimum value of the phase speed for flexural-gravity waves also increases. In figure 6(a), we see that the geometry permits an intermediate region in which there are only downstream gravity waves. This corresponds to the case where the flow velocity exceeds $c_{{min}}$ in the downstream region, but is lower than the higher value of $c_{{min}}$ obtained in the upstream region. The converse is true for downstream steps, where $L_{{u}} < L_{{d}}$; in this case, the minimum phase velocity for waves is greater in the downstream region than the upstream region, leading to flow geometries described in figure 6(b) that contain only upstream waves.

This is not a full analysis of the nonlinear problem. We would need to study $\chi$ in order to identify the position of Stokes curves in the problem, and therefore determine the location at which the waves appear. This requires determining the solution to (2.27a,b), which would likely necessitate a computational study. Nonlinear problems can also contain more complicated switching behaviour, such as second-generation Stokes switching; see Body, King & Tew (Reference Body, King and Tew2005) and Chapman & Mortimer (Reference Chapman and Mortimer2005). This was found to exist in some nonlinear gravity-capillary wave regimes in Trinh & Chapman (Reference Trinh and Chapman2013b). Any conclusions reached for the nonlinear system would require validation against numerical simulations, but the presence of surface waves in both directions far from the step makes it challenging to obtain sensible boundary conditions for the flow behaviour. For a detailed description of the numerical challenges involved in studying these systems, and substantial progress in overcoming these obstacles, see the numerical analysis of two-dimensional gravity-capillary waves in Jamshidi & Trinh (Reference Jamshidi and Trinh2020). A full analysis of the nonlinear problem is therefore beyond the scope of the present study.

An important difference between the analysis of the linear problem and any full nonlinear analysis is that the wavelength of the flexural-gravity waves will depend on $b$, and therefore the step height. This may be seen by the inclusion of $q^{(0)}$ in the singulant equation (2.27a,b). The explicit dependence can be computed by solving (2.27a,b) using the asymptotic behaviour of $q^{(0)}$ in the limit that $w \to -\infty$ for upstream waves, and $w \to \infty$ for downstream waves. This phenomenon is predicted by the dispersion relation (1.2), which depends explicitly on the depth of the channel, and is consistent with other related exponential asymptotic studies such as Chapman & Vanden-Broeck (Reference Chapman and Vanden-Broeck2002, Reference Chapman and Vanden-Broeck2006), Trinh & Chapman (Reference Trinh and Chapman2013b) and Lustri et al. (Reference Lustri, Koens and Pethiyagoda2020). This wavelength selection did not occur in the linearized problem, as the flow was linearized around an unperturbed flow of constant depth; this constant depth determines the wavelength of the flexural-gravity waves to leading order in $\delta$.

3.1. Governing equation

The flow is governed by Laplace's equation in three dimensions,

(3.1)\begin{equation} \nabla^2\phi = 0,\quad -\infty < z < \xi(x,y), \end{equation}

with kinematic boundary condition

(3.2)\begin{equation} \xi_x\phi_x + \xi_y\phi_y = \phi_z,\quad z = \xi(x,y). \end{equation}

As discussed in § 1.2, we apply a biharmonic model to describe the behaviour of the elastic sheet in the linear regime. This gives

(3.3)\begin{equation} \tfrac{1}{2}\left(|\boldsymbol{\nabla}\phi|^2-1\right) +\epsilon^3(\xi_{xxxx} + 2 \xi_{xxyy} + \xi_{yyyy}) = 0,\quad z = \xi(x,y), \end{equation}

where $\epsilon ^3 = D/(\rho U^2 L^3)$. This quantity corresponds to the ratio between the Froude number and bending length ratio presented in (1.5a,b). We are concerned with the free-surface behaviour in the limit $0 < \epsilon \ll 1$, corresponding to a regime in which gravity is neglected, and inertial effects are large compared to the elastic restoring force. Since the flow is uniform in the far field, $\phi _x \rightarrow 1$. The source condition is set to

(3.4)\begin{equation} \phi \sim \frac{\delta}{4{\rm \pi}\sqrt{x^2 + y^2 + (z+h)^2}} \quad \mathrm{as} \ (x,y,z) \rightarrow (0,0,-h). \end{equation}

Finally, we can prescribe that the solution satisfies a radiation condition, with waves present directly upstream from the singularity.

We are concerned with the limit $0 < \delta \ll \epsilon$, describing a weak source. In this case, the flow disturbance due to the source effect is small, and the equations may be linearized in $\delta$ about a uniform stream while retaining the full asymptotic behaviour in the small-$\epsilon$ limit. We are therefore studying the combined asymptotic parameter regime $0 < \delta \ll \epsilon \ll 1$.

3.2. Linearization

We linearize about uniform flow by setting

(3.5a,b)\begin{equation} \phi = x + \delta\tilde{\phi},\quad \xi = \delta\tilde{\xi}, \end{equation}

to give, at leading order in $\delta$,

(3.6)\begin{gather} \nabla^2\tilde{\phi} = 0 ,\quad -\infty < z < 0, \end{gather}

(3.7)\begin{gather} \tilde{\phi}_z - \tilde{\xi}_x = 0,\quad z= 0, \end{gather}

(3.8)\begin{gather} \tilde{\phi}_x -\epsilon^3 \left(\tilde{\xi}_{xxxx} +2 \tilde{\xi}_{xxyy} + \tilde{\xi}_{yyyy}\right)=0 ,\quad z = 0, \end{gather}

where the boundary conditions are now applied on the fixed surface $z = 0$. The far-field conditions imply that $\tilde {\phi } \rightarrow 0$ as $x^2 + y^2 + z^2 \rightarrow \infty$, while near the source, the singular behaviour is given by

(3.9)\begin{equation} \tilde{\phi} \sim \frac{1}{4{\rm \pi}\sqrt{x^2 + y^2 + (z+h)^2}} \quad \mathrm{as} \ (x,y,z) \rightarrow (0,0,-h). \end{equation}

3.3. Series expression

We first expand the fluid potential and free-surface position as power series in $\epsilon$,

(3.10a,b)\begin{equation} \tilde{\phi} \sim \sum_{n=0}^{\infty}\epsilon^{3n}\phi^{(n)},\quad \tilde{\xi} \sim \sum_{n=0}^{\infty}\epsilon^{3n}\xi^{(n)}, \end{equation}

to give for $n \geq 0$

(3.11)\begin{gather} \nabla^2 \phi^{(n)} = 0,\quad -\infty < z < 0, \end{gather}

(3.12)\begin{gather}\phi^{(n)}_z - \xi^{(n)}_x = 0,\quad z= 0, \end{gather}

(3.13)\begin{gather}{\phi}_x^{(n)} -{\xi}_{xxxx}^{(n-1)} -2 {\xi}_{xxyy}^{(n-1)} - {\xi}_{yyyy}^{(n-1)} =0 , \quad z = 0, \end{gather}

with the convention that $\xi ^{(-1)} = 0$. The far-field behaviour tends to zero at all orders of $n$, and the singularity condition (3.9) is applied to the leading-order expression, giving

(3.14)\begin{equation} \phi^{(0)} \sim \frac{1}{4{\rm \pi}\sqrt{x^2 + y^2 + (z+h)^2}} \quad \mathrm{as} \ (x,y,z) \rightarrow (0,0,-h). \end{equation}

The leading-order solution is given by

(3.15)\begin{gather} \phi^{(0)} = \frac{1}{4{\rm \pi}\sqrt{x^2 + y^2 + (z+h)^2}} - \frac{1}{4{\rm \pi}\sqrt{x^2 + y^2 + (z-h)^2}}, \end{gather}

(3.16)\begin{gather}\xi^{(0)} ={-}\frac{x h}{2 {\rm \pi}(y^2 + h^2)\sqrt{x^2+y^2+h^2}} - \frac{1}{2{\rm \pi}(y^2+h^2)}, \end{gather}

where the leading-order free-surface behaviour is set to be undisturbed far behind the source.

3.4. Late-order terms

In order to optimally truncate the asymptotic series prescribed in (3.10a,b), we must determine the form of the late-order terms. To accomplish this, we make a factorial-over-power ansatz with the form

(3.17a,b)\begin{equation} \phi^{(n)} \sim \frac{\varPhi(x,y,z)\,\varGamma(3n+\gamma)}{\chi(x,y,z)^{3n+\gamma}}\quad \mathrm{and} \quad \xi^{(n)} \sim \frac{\varXi(x,y)\,\varGamma(3n+\gamma)}{\chi(x,y,0)^{3n+\gamma}} \quad \mathrm{as} \ n \rightarrow \infty, \end{equation}

where $\gamma$ is a constant. In order that (3.17a,b) is the power series developed in § 3.3, we require that the singulant, $\chi$ satisfies

(3.18)\begin{equation} \chi = 0 \quad \mathrm{on} \ x^2 + y^2 + (z\pm h)^2 = 0, \end{equation}

where the sign chosen depends upon which of the two singularities is being considered. For complex values of $x$, $y$ and $z$, this defines a four-dimensional hypersurface. Irrespective of which singularity is under consideration, this hypersurface intersects the four-dimensional complexified free surface on the two-dimensional hypersurface satisfying $x^2 + y^2 + h^2 = 0$.

3.4.1. Calculating the singulant

Applying the ansatz expressions in (3.17a,b) to the governing equation (3.11) and taking the first two orders as $n \rightarrow \infty$ gives, for $z \leq 0$,

(3.19)\begin{gather} \chi_x^2 + \chi_y^2 + \chi_z^2 = 0, \end{gather}

(3.20)\begin{gather}2\varPhi_x\chi_x + 2\varPhi_y\chi_y + 2\varPhi_z\chi_z ={-}(\chi_{xx}+\chi_{yy}+\chi_{zz}), \end{gather}

while the boundary conditions on $z = 0$ at leading order become

(3.21)\begin{gather} -\chi_z\varPhi + \chi_x \varXi = 0, \end{gather}

(3.22)\begin{gather}\chi_x\varPhi + (\chi_x^4 +2 \chi_x^2\chi_y^2 + \chi_y^4)\varXi = 0. \end{gather}

The system in (3.21)–(3.22) must have non-zero solutions, which requires

(3.23a,b)\begin{equation} \chi_z ={-}\frac{\chi_x^2}{(\chi_x^2+\chi_y^2)^2}, \quad \varXi ={-}\frac{\chi_x}{\chi_x^2+\chi_y^2}\,\varPhi. \end{equation}

Applying (3.23a,b) to (3.19) evaluated on $z=0$ gives a singulant equation for $\chi$ on the free surface:

(3.24)\begin{equation} \chi_x^4 + \left(\chi_x^2 + \chi_y^2\right)^5 = 0. \end{equation}

This expression is similar to the capillary wave singulant equation from Lustri et al. (Reference Lustri, Pethiyagoda and Chapman2019), with a different power in the second term. The subsequent analysis is therefore similar, and we include an outline of the details. Because the singularity lies below the fluid surface, we must solve (3.24) for complex $x$ and $y$ with the boundary condition

(3.25)\begin{equation} \chi = 0 \quad \mathrm{on} \ x^2 + y^2 + h^2 = 0. \end{equation}

Solving (3.24)–(3.25) using Charpit's method gives

(3.26)\begin{equation} \chi ={\pm}\frac{3 z_j h^{1/3}s^{5/3}(s-x)}{2h^2+5s^2}, \end{equation}

where $z_j$ is one of the three solutions of $z_j^3 = 1$, and $s$ is one of the four solutions of

(3.27)\begin{equation} 25(x^2+y^2)s^4 + 20h^2 x s^3 + (4h^2+25x^2+20y^2)h^2 s^2 + 20h^4 x s + 4 h^4 (h^2+y^2) = 0. \end{equation}

This produces twenty-four potential late-order contributions, corresponding to the choice of sign and $z_j$ in (3.26) and the four solutions of (3.27). Twelve of these solutions are spurious, introduced by squaring both sides of an equation in the algebraic manipulations. This leaves twelve solutions, which appear as six complex conjugate pairs. Three of these pairs do not demonstrate Stokes switching, as they are exponentially large on the curve $\mathrm {Im}(\chi ) = 0$; they must therefore be inactive on the surface, and do not contribute to the free-surface behaviour. This leaves three singulant pairs that produce waves on the free surface. We will denote these as $\chi _j$ and $\overline {\chi }_j$ for $j = 1, 2, 3$, where the bar represents complex conjugation.

The behaviour of $\chi _1$ is illustrated in figure 9. The surface waves are absent directly downstream from the obstacle. The surface contains a Stokes curve that passes through the origin. This Stokes curve causes elastic waves to be switched on upstream from the obstacle. From direct algebraic computation, we find that $\chi \sim (1-\mathrm {i} x)/h$ in the limit that $x \rightarrow -\infty$. These elastic waves do not decay exponentially in space, although they will have algebraic spatial decay due to the prefactor, calculated below.

Figure 9.

The (a) real and (b) imaginary parts of the singulant $\chi _1$, corresponding to algebraically decaying elastic waves. The Stokes curve, satisfying $\mathrm {Re}(\chi_1 ) > 0$ and $\mathrm {Im}(\chi _1) = 0$, is depicted in (b). The waves are not present in a region downstream from the obstacle, and appear as the Stokes curve is crossed into the region ahead of the obstacle. The equal phase lines in (b) illustrate the shape of the surface waves.

The behaviour of $\chi _2$ is illustrated in figure 10. The surface waves are also absent directly downstream from the obstacle. In fact, we see that this does not depend on the radiation condition; instead, there is an anti-Stokes curve on the surface. If the surface waves were present on this side of the Stokes curve, then they would become exponentially large on the downstream side of the anti-Stokes curve. We therefore see that the waves are present only on the downstream side of the Stokes curve. Importantly, $\mathrm {Re}(\chi )$ grows monotonically in the negative $x$-direction, becoming arbitrarily large as $x \rightarrow -\infty$. This means that the waves decay exponentially in space. We will therefore not consider these waves in the subsequent analysis.

Figure 10.

The (a) real and (b) imaginary parts of the singulant $\chi _2$, corresponding to one set of exponentially decaying elastic waves. The anti-Stokes curve satisfies $\mathrm {Re}(\chi _2) = 0$, and is depicted as a dashed line. On the inside of this curve (where $\mathrm {Re}(\chi _2) < 0$), the exponential term would be large. Consequently, the remainder must be inactive in a region containing the anti-Stokes curve. The Stokes curve, satisfying $\mathrm {Re}(\chi _2) > 0$ and $\mathrm {Im}(\chi _2) = 0$, is depicted in (b). We see that the exponential must be switched on as this curve is crossed in a direction heading away from the origin. As the real part of $\chi _2$ increases without bound as $x \rightarrow -\infty$, these waves must decay exponentially in space, which is too rapid to have an observable physical effect. The behaviour of $\chi _3$ can be obtained by the mapping $x \mapsto -x$.

Finally, we find that $\chi _3$ is identical to $\chi _2$ reflected around the $y$-axis. This is consistent with the two-dimensional result, seen in Lustri et al. (Reference Lustri, Pethiyagoda and Chapman2019), in which the surface contains symmetric waves emerging in both directions above the obstacle, which decay exponentially in space.

3.4.2. Calculating the prefactor

We follow an analysis similar to previous work on gravity and capillary waves in order to determine the prefactor expression $\varPhi$, and hence $\varXi$. These calculations are quite technical, and the details are included in § B.1. The prefactor $\varPhi$ is given by

(3.28)\begin{equation} \varPhi\!=\! \frac{s^{1/3}\sqrt{2}}{4{\rm \pi}^{3/2} h^{5/6}}\left[1\!-\!\frac{5 h^2 (2 h^2 \!-\! s^2) (s \!-\! x)}{3s (2 h^2 \!+\! 5 s^2) }\right]^{{({-}1)^{1/3} (4h^{10}-5h^8s^2-30h^2s^8-48s^{10})}/{30 h^{8/3}s^{16/3}(2h^2-s^2)}}, \end{equation}

where $s$ is the solution of (3.27) corresponding to the singulant illustrated in figure 9.

Finally, to find $\gamma$, we ensure that the strength of the singularity in the late-order behaviour $\phi ^{(n)}$ given in (3.17a,b) is consistent with the leading-order behaviour $\phi ^{(0)}$, which has strength $1/2$. It is clear from the recurrence relation (3.13) that the strength of the singularity will increase by three between $\phi ^{(n-1)}$ and $\phi ^{(n)}$. This implies that near the singularity at $x^2 + y^2 + h^2 = 0$,

(3.29)\begin{equation} \frac{\varPhi\,\varGamma(\gamma)}{\chi^{\gamma}} \rightarrow \frac{\alpha(x,y)}{(x^2+y^2+h^2)^{1/2}}, \end{equation}

where $\alpha$ is of order 1 in the limit. From (3.28), we see that the prefactor is also order 1 in this limit. A local analysis near the singularity (performed in (B13)) shows that $1/\chi$ contains a singularity with strength one at $x^2 + y^2 + h^2 = 0$. Matching the order of the expressions in (3.29) therefore gives $\gamma = 1/2$. We have therefore completely described the late-order terms in (3.17a,b), where (3.23a,b) is used to determine the value of $\varXi$, and hence the behaviour of the free-surface waves.

In § B.3, we use the late-order terms in (3.17a,b) to apply the matched asymptotic expansion methodology of Olde Daalhuis et al. (Reference Olde Daalhuis, Chapman, King, Ockendon and Tew1995). We optimally truncate the asymptotic series and identify the Stokes curves. Finally, we use a matched asymptotic expansion analysis on the truncation remainder to compute the exponentially small contribution to the free-surface behaviour that appears across the Stokes lines. This analysis in § B.3 follows steps similar to the equivalent analysis in Lustri & Chapman (Reference Lustri and Chapman2013) and Lustri et al. (Reference Lustri, Pethiyagoda and Chapman2019).

Using this method, we find that the exponentially small contributions to the fluid potential (denoted $\phi _{{exp}}$) and free-surface position (denoted $\xi _{{exp}}$) as $\epsilon \to 0$ are switched in the region to the left of the Stokes curve shown in figure 9. In the region where the exponentially small contributions are present, they are given by

(3.30a,b)\begin{equation} \phi_{{exp}} \sim \frac{2{\rm \pi}\mathrm{i} \varPhi}{3\sqrt{\epsilon}}\,\mathrm{e}^{-\chi_1/\epsilon} + \mathrm{c.c.},\quad \xi_{{exp}} \sim \frac{2{\rm \pi}\mathrm{i} \varXi}{3\sqrt{\epsilon}}\,\mathrm{e}^{-\chi_1/\epsilon} + \mathrm{c.c.},\end{equation}

where c.c. denotes the complex conjugate contribution. In particular, the expression for $\xi _{{exp}}$ contains exponentially small oscillations as $\epsilon \to 0$, representing the elastic ripples on the free surface.

We note that (3.30b) is a leading-order expression for the exponentially small waves, with algebraic corrections to the prefactors $\varPhi$ and $\varXi$ omitted. The solution also contains contributions from $\chi _2$ and $\chi _3$ with the same form, but these decay exponentially in space, and are therefore exponentially smaller in amplitude as $\epsilon \to 0$ than the elastic waves caused by $\chi _1$, as well as the neglected correction terms. We therefore omit these contributions from the asymptotic expression.