2.1 Notation

Let ${\mathcal{U}}$ and ${\mathcal{H}}$ respectively denote the incident speed and depth of the liquid. We take Cartesian coordinates $\boldsymbol{x}=(x,z)$ such that the base is positioned along $z=0$ , with the transition point at $x=0$ . The free surface is at $z=h(x)$ and the liquid occupies the region $0<z<h$ . The gravitational acceleration is $\boldsymbol{g}=(0,-g)$ , and let $\boldsymbol{u}=(u,w)$ denote the components of velocity. The incident flow velocity and depth are given by $\boldsymbol{u}\rightarrow ({\mathcal{U}},0)$ and $h\rightarrow {\mathcal{H}}>0$ as $x\ll -{\mathcal{H}}$ .

2.2 Non-dimensionalisation

We shall henceforth work in rescaled units such that ${\mathcal{U}}=1$ and ${\mathcal{H}}=1$ , and write everything in terms of the Reynolds and Froude numbers, instead of $g$ and $\unicode[STIX]{x1D708}$ . In these units, the flow rate $q={\mathcal{U}}{\mathcal{H}}=1$ . We also take the density of the liquid as $\unicode[STIX]{x1D70C}=1$ , and assume that the air above has negligible density.

2.3 Governing equations and boundary conditions

In terms of the non-dimensionalised quantities, the steady Navier–Stokes equations read as

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}}+{\displaystyle \frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}\left[u{\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}}+w{\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}}\right]=-{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}+\frac{Fr^{2}}{Re}\left[{\displaystyle \frac{\unicode[STIX]{x2202}2u}{\unicode[STIX]{x2202}x2}}+{\displaystyle \frac{\unicode[STIX]{x2202}2u}{\unicode[STIX]{x2202}z2}}\right], & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}\left[u{\displaystyle \frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x}}+w{\displaystyle \frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}}\right]=-1-{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}+\frac{Fr^{2}}{Re}\left[{\displaystyle \frac{\unicode[STIX]{x2202}2w}{\unicode[STIX]{x2202}x2}}+{\displaystyle \frac{\unicode[STIX]{x2202}2w}{\unicode[STIX]{x2202}z2}}\right] & \displaystyle\end{eqnarray}$$

and the boundary conditions are

(2.4a,b ) $$\begin{eqnarray}\displaystyle u=1\quad \text{and}\quad w=0\quad \text{on }z=0\text{ for }x<0,\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle u=w=0\quad \text{on }z=0\text{ for }x>0, & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle w=u{\displaystyle \frac{ d\!h}{ d\!x}}\quad \text{on }z=h(x), & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle p=O(Fr^{2}/Re)\quad \text{on }z=h(x). & \displaystyle\end{eqnarray}$$

Equation (2.6) is the kinematic condition for the free surface, while the dynamic boundary condition (2.7) comes from the more general condition that , where  is the stress tensor and $\boldsymbol{n}$ is the normal to the free surface (Batchelor Reference Batchelor2000). For an inviscid liquid, this condition is equivalent to $p=0$ at the upper boundary, but, in general, the liquid viscosity offers a contribution of magnitude $O(Fr^{2}/Re)$ to . In § 3, we shall study the limit  $Re\rightarrow \infty$ , with $Fr$ fixed. Our first-order perturbation analysis will consider corrections of magnitude $O(Re^{-1/2})$ . The smaller $O(Fr^{2}/Re)$ contributions may therefore be ignored.

The problem as it has been formulated is in fact ill-posed at large $x$ , an issue that we shall explore in § 4.3, where we shall also discuss possible resolutions that make the problem physical without affecting the analysis below.

3.1 Leading-order outer solution

In the inviscid limit $Re=\infty$ , the change in basal conditions for $x>0$ has no effect on the flow, since the no-slip condition may not be applied in the absence of viscosity. The flow merely maintains its upstream values, and the leading-order outer solution is therefore given by

(3.1a-d ) $$\begin{eqnarray}\displaystyle h\sim h^{(0)}=1,\quad u\sim u^{(0)}=1,\quad w\sim w^{(0)}=0,\quad p\sim p^{(0)}=h^{(0)}-z. & & \displaystyle\end{eqnarray}$$

3.2 Leading-order boundary layer solution

At high Reynolds numbers, the effects of viscosity are confined to the boundary layer of thickness $O(Re^{-1/2})$ near the base, in which the solution adjusts between the outer solution ‘slip velocity’ and the zero velocity imposed at $z=0$ . On rescaling variables with $z=Re^{-1/2}\tilde{z}$ and $w=Re^{-1/2}\tilde{w}$ , the governing equations become

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}}+{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{w}}{\unicode[STIX]{x2202}\tilde{z}}}=0, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}\left[u{\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}}+\tilde{w}{\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}\tilde{z}}}\right]=-{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}+Fr^{2}\left[\frac{1}{Re}{\displaystyle \frac{\unicode[STIX]{x2202}2u}{\unicode[STIX]{x2202}x2}}+{\displaystyle \frac{\unicode[STIX]{x2202}2u}{\unicode[STIX]{x2202}\tilde{z}2}}\right], & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{Fr^{2}}{Re^{1/2}}\left[u{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{w}}{\unicode[STIX]{x2202}x}}+\tilde{w}{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{w}}{\unicode[STIX]{x2202}z}}\right]=-1-Re^{1/2}{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\tilde{z}}}+\frac{Fr^{2}}{Re^{1/2}}\left[\frac{1}{Re}{\displaystyle \frac{\unicode[STIX]{x2202}2\tilde{w}}{\unicode[STIX]{x2202}x2}}+{\displaystyle \frac{\unicode[STIX]{x2202}2\tilde{w}}{\unicode[STIX]{x2202}\tilde{z}2}}\right]. & \displaystyle\end{eqnarray}$$

The boundary conditions at $\tilde{z}=0$ are the no-penetration and no-slip conditions. There is also the matching condition (Hinch Reference Hinch1991) that the boundary layer and outer solutions match as $z\rightarrow 0$ and $\tilde{z}\rightarrow \infty$ . For the leading-order solutions in both regions, this can be expressed by means of van Dyke’s matching rule,

(3.5a,b ) $$\begin{eqnarray}\displaystyle \lim _{z\rightarrow 0}u^{(0)}=1=\lim _{\tilde{z}\rightarrow \infty }u_{(0)},\quad \lim _{z\rightarrow 0}p^{(0)}=1=\lim _{\tilde{z}\rightarrow \infty }p_{(0)}. & & \displaystyle\end{eqnarray}$$

At leading order in $O(Re^{1/2})$ , equation (3.4) implies that the pressure must be independent of $\tilde{z}$ in the boundary layer, and is therefore determined by the outer solution pressure evaluated at $z=0$ , according to the matching condition. Since the leading-order outer solution predicts no streamwise pressure gradient, the pressure term in (3.3) vanishes. The leading-order boundary layer solution is the Blasius solution (Acheson Reference Acheson1989; Schlichting & Gersten Reference Schlichting and Gersten2003), given by

(3.6a,b ) $$\begin{eqnarray}\displaystyle u\sim u_{(0)}=B^{\prime }(s),\quad \tilde{w}\sim \tilde{w}_{(0)}=\left({\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\displaystyle \right)_{\tilde{z}}(2x)^{1/2}B(s), & & \displaystyle\end{eqnarray}$$

where $s=\tilde{z}/(2x)^{1/2}$ is a similarity variable and $B$ is the unique solution to the system

(3.7a-c ) $$\begin{eqnarray}\displaystyle B^{\prime \prime \prime }+{\textstyle \frac{1}{2}}BB^{\prime \prime }=0,\quad B(0)=B^{\prime }(0)=0,\quad \lim _{s\rightarrow \infty }B^{\prime }(s)=1. & & \displaystyle\end{eqnarray}$$

3.3 First-order correction to the outer flow

The boundary conditions on $B$ ensure that the leading-order boundary layer solution both satisfies the no-slip condition on $z=0$ and matches the leading-order outer solution. However, while the outer solution has $w^{(0)}=0$ , the boundary layer solution predicts that

(3.8) $$\begin{eqnarray}\displaystyle \tilde{w}_{(0)}\rightarrow c\,(2x)^{-1/2}\quad \text{as }\tilde{z}\rightarrow \infty , & & \displaystyle\end{eqnarray}$$

where

(3.9) $$\begin{eqnarray}\displaystyle c=\lim _{s\rightarrow \infty }(sB^{\prime }-B)\approx 1.72. & & \displaystyle\end{eqnarray}$$

In other words, the outer flow experiences a displacement velocity perpendicular to the base, due to the base slowing the flow down in the streamwise direction. This displacement velocity is $O(Re^{-1/2})$ , which does not affect the leading-order outer solution, but suggests that the outer flow must have an $O(Re^{-1/2})$ correction. We therefore proceed by writing

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle h\sim h^{(0)}+Re^{-1/2}h^{(1)}=1+Re^{-1/2}h^{(1)}, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle u\sim u^{(0)}+Re^{-1/2}u^{(1)}=1+Re^{-1/2}u^{(1)}, & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle w\sim w^{(0)}+Re^{-1/2}w^{(1)}=Re^{-1/2}w^{(1)}, & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle p\sim p^{(0)}+Re^{-1/2}p^{(1)}=h-z+Re^{-1/2}p^{(1)}. & \displaystyle\end{eqnarray}$$

We now expand the equations of motion to $O(Re^{-1/2})$ , treating $Fr$ as a constant. The viscous terms are $O(Fr^{2}/Re)$ and still do not affect the outer solution at this order. The equations of motion (2.1)–(2.3) become

(3.14a-c ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}u^{(1)}}{\unicode[STIX]{x2202}x}}=-{\displaystyle \frac{\unicode[STIX]{x2202}w^{(1)}}{\unicode[STIX]{x2202}z}},\quad Fr^{2}{\displaystyle \frac{\unicode[STIX]{x2202}u^{(1)}}{\unicode[STIX]{x2202}x}}=-{\displaystyle \frac{\unicode[STIX]{x2202}p^{(1)}}{\unicode[STIX]{x2202}x}},\quad Fr^{2}{\displaystyle \frac{\unicode[STIX]{x2202}w^{(1)}}{\unicode[STIX]{x2202}x}}=-{\displaystyle \frac{\unicode[STIX]{x2202}p^{(1)}}{\unicode[STIX]{x2202}z}}. & & \displaystyle\end{eqnarray}$$

Together, these imply that $w^{(1)}$ satisfies Laplace’s equation,

(3.15) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}2w^{(1)}}{\unicode[STIX]{x2202}x2}}+{\displaystyle \frac{\unicode[STIX]{x2202}2w^{(1)}}{\unicode[STIX]{x2202}z2}}=0. & & \displaystyle\end{eqnarray}$$

The boundary conditions at the free surface may be found by linearising the exact boundary conditions, applying them at $z=h^{(0)}=1$ instead of $z=h(x)$ . These give

(3.16a,b ) $$\begin{eqnarray}\displaystyle w^{(1)}={\displaystyle \frac{ d\!h^{(1)}}{ d\!x}}\quad \text{and}\quad p^{(1)}=h^{(1)}\quad \text{on}~z=1, & & \displaystyle\end{eqnarray}$$

which may be combined with the approximate equations of motion (3.14) to give a Robin boundary condition on $w^{(1)}$ ,

(3.17) $$\begin{eqnarray}\displaystyle w^{(1)}=Fr^{2}{\displaystyle \frac{\unicode[STIX]{x2202}w^{(1)}}{\unicode[STIX]{x2202}z}}\quad \text{on }z=1. & & \displaystyle\end{eqnarray}$$

Meanwhile, the boundary condition on $w^{(1)}$ at $z=0$ for $x>0$ comes from matching with the displacement velocity from the boundary layer solution, while there is no displacement velocity for $x<0$ . Hence,

(3.18) $$\begin{eqnarray}\displaystyle w^{(1)}=c\,(2xH(x))^{-1/2}\quad \text{on }z=0, & & \displaystyle\end{eqnarray}$$

where $H(x)$ is the Heaviside step function. This boundary condition implies that $w^{(1)}$ is large near $x=0$ , violating the asymptoticity of the expansion when $w^{(1)}=O(Re^{1/2})$ ; we discuss this problem in § 4.3.

Equations (3.15)–(3.18) together give a linear system for the perturbation velocity $w^{(1)}$ . The solution may be found by a Fourier transform with respect to $x$ . Let us first write $f(x)=c\,(2x)^{-1/2}H(x)$ . Since the Fourier transform of $f(x)$ is given by

(3.19) $$\begin{eqnarray}\displaystyle \tilde{f}(k)=\int _{-\infty }^{\infty }f(x)\text{e}^{-\text{i}kx}\, d\!x=\frac{1}{2}c\unicode[STIX]{x03C0}^{1/2}(1-\text{i}\,\text{sgn}(k))|k|^{-1/2}, & & \displaystyle\end{eqnarray}$$

the solution for $w^{(1)}$ can then be written in integral form as the real part of

(3.20) $$\begin{eqnarray}\displaystyle w^{(1)} & = & \displaystyle \frac{1}{2\unicode[STIX]{x03C0}}\int _{-\infty }^{\infty }\text{e}^{\text{i}kx}\tilde{f}(k)\frac{Fr^{2}k\cosh (k(1-z))-\sinh (k(1-z))}{Fr^{2}k\cosh k-\sinh k}\, d\!k\nonumber\\ \displaystyle & = & \displaystyle \frac{c\text{e}^{-\text{i}\unicode[STIX]{x03C0}/4}}{(2\unicode[STIX]{x03C0})^{1/2}}\int _{0}^{\infty }\text{e}^{\text{i}kx}k^{-1/2}\frac{Fr^{2}k\cosh (k(1-z))-\sinh (k(1-z))}{Fr^{2}k\cosh k-\sinh k}\, d\!k.\end{eqnarray}$$

It should be noted that for $Fr>1$ , the singularity of the integrand at $k=0$ is integrable, and (3.20) is convergent for all $x$ and all $0\leqslant z\leqslant 1$ , except at the point $x=z=0$ . However, for $Fr<1$ , the integrand has a non-integrable pole at the positive solution to the equation $Fr^{2}k=\tanh k$ . This results from the ill-posedness of the original formulation of the problem if $Fr<1$ (see § 4.1).

3.4 Pressure gradient induced by correction

The boundary layer solution in § 3.2 assumes that the pressure in the boundary layer is constant, since the leading-order outer solution gives a constant pressure on $z=0$ . However, the perturbation flow (3.20) found in § 3.3 induces a pressure gradient. We now consider the size of this pressure gradient and, in particular, determine whether it is comparable to the $O(Fr^{2})$ inertia terms in (3.3). If so, then the solution found in § 3.2 becomes invalid. The value of this pressure gradient may be evaluated using (3.14),

(3.21) $$\begin{eqnarray}\displaystyle \left.{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}\right|_{z=0} & = & \displaystyle Re^{-1/2}\left.{\displaystyle \frac{\unicode[STIX]{x2202}p^{(1)}}{\unicode[STIX]{x2202}x}}\right|_{z=0}=Re^{-1/2}Fr^{2}\left.{\displaystyle \frac{\unicode[STIX]{x2202}w^{(1)}}{\unicode[STIX]{x2202}z}}\right|_{z=0}\nonumber\\ \displaystyle & = & \displaystyle -\frac{Fr^{2}c\text{e}^{-\text{i}\unicode[STIX]{x03C0}/4}}{Re^{1/2}(2\unicode[STIX]{x03C0})^{1/2}}\int _{0}^{\infty }\text{e}^{\text{i}kx}k^{1/2}\frac{Fr^{2}k\sinh k-\cosh k}{Fr^{2}k\cosh k-\sinh k}\, d\!k\nonumber\\ \displaystyle & = & \displaystyle -\frac{Fr^{2}c\text{e}^{-\text{i}\unicode[STIX]{x03C0}/4}}{Re^{1/2}(2\unicode[STIX]{x03C0})^{1/2}}I(x,Fr^{2}),\end{eqnarray}$$

where the real part is to be taken throughout. The integrand of $I(x,Fr^{2})$ is unbounded as $k\rightarrow \infty$ , but the integral is nonetheless convergent for $x\neq 0$ . We separate the integrand into two parts, writing

(3.22) $$\begin{eqnarray}\displaystyle I(x,Fr^{2}) & = & \displaystyle \int _{0}^{\infty }\text{e}^{\text{i}kx}k^{1/2}\, d\!k+\int _{0}^{\infty }\text{e}^{\text{i}kx}k^{1/2}\left\{\frac{Fr^{2}k\sinh k-\cosh k}{Fr^{2}k\cosh k-\sinh k}-1\right\}\,d\!k\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x03C0}^{1/2}}{2}\text{e}^{3\text{i}\,\text{sgn}(x)\unicode[STIX]{x03C0}/4}|x|^{-3/2}-\int _{0}^{\infty }\text{e}^{\text{i}kx}k^{1/2}\frac{(Fr^{2}k+1)(\cosh k-\sinh k)}{Fr^{2}k\cosh k-\sinh k}\, d\!k\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x03C0}^{1/2}}{2}\text{e}^{3\text{i}\,\text{sgn}(x)\unicode[STIX]{x03C0}/4}x^{-3/2}-J(x,Fr^{2}),\end{eqnarray}$$

where the first term has been evaluated using complex analytical methods (Ablowitz & Fokas Reference Ablowitz and Fokas2003). When the real part is taken, this term contributes nothing to $\unicode[STIX]{x2202}p^{(1)}/\unicode[STIX]{x2202}x$ for $x>0$ , but results in a singularity as $x\rightarrow 0^{-}$ , which will be discussed in § 4.3.

We now give approximations to the second term $J(x,Fr^{2})$ in two limiting cases. For $Fr\gg 1$ , we have the approximation

(3.23) $$\begin{eqnarray}\displaystyle J(x,Fr^{2}) & = & \displaystyle J(x,\infty )+O(Fr^{-2})\nonumber\\ \displaystyle & = & \displaystyle \int _{0}^{\infty }\text{e}^{\text{i}kx}k^{1/2}\frac{\cosh k-\sinh k}{\cosh k}\, d\!k+O(Fr^{-2})\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x03C0}^{1/2}}{8}\left[\unicode[STIX]{x1D701}\left(\frac{3}{2},\frac{1}{2}-\frac{1}{4}\text{i}x\right)-\unicode[STIX]{x1D701}\left(\frac{3}{2},1-\frac{1}{4}\text{i}x\right)\right]+O(Fr^{-2}),\end{eqnarray}$$

where $\unicode[STIX]{x1D701}(s,a)$ is the generalised Riemann zeta function. It can be shown (Abramowitz & Stegun Reference Abramowitz and Stegun1970) that $J(x,\infty )$ is bounded in $x$ . It then follows that for $Fr\gg 1$ , the pressure gradient in $x>0$ is bounded and has magnitude

(3.24) $$\begin{eqnarray}\displaystyle \left.{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}\right|_{z=0}=Re^{-1/2}\left.{\displaystyle \frac{\unicode[STIX]{x2202}p^{(1)}}{\unicode[STIX]{x2202}x}}\right|_{z=0}=O(Fr^{2}Re^{-1/2}). & & \displaystyle\end{eqnarray}$$

Since $Re\gg 1$ , this pressure gradient is small compared with the $O(Fr^{2})$ inertia and viscosity terms in the leading-order boundary layer equation (3.3).

The other limit concerns a flow that is only just supercritical, with $\unicode[STIX]{x1D716}=Fr^{2}-1\ll 1$ . For $\unicode[STIX]{x1D716}=0$ , the integrand of $I$ has a non-integrable singularity at $k=0$ , suggesting that for $0<\unicode[STIX]{x1D716}\ll 1$ the integral is dominated by a contribution near $k=0$ . The denominator of the integrand has a distinguished scaling (Hinch Reference Hinch1991) for $k=O(\unicode[STIX]{x1D716}^{1/2})$ , and

(3.25) $$\begin{eqnarray}\displaystyle J(x,1+\unicode[STIX]{x1D716})\sim \unicode[STIX]{x1D716}^{-3/4}\int _{0}^{\infty }\frac{s^{1/2}}{s+s^{3}/3}\, d\!s\approx 2.92\,\unicode[STIX]{x1D716}^{-3/4}. & & \displaystyle\end{eqnarray}$$

The pressure gradient in this limit therefore has magnitude

(3.26) $$\begin{eqnarray}\displaystyle \left.{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}\right|_{z=0}=O(Re^{-1/2}(Fr^{2}-1)^{-3/4}), & & \displaystyle\end{eqnarray}$$

which becomes comparable to the inertia and viscosity terms appearing in (3.3) in the distinguished limit

(3.27) $$\begin{eqnarray}\displaystyle Fr^{2}=1+O(Re^{-2/3}). & & \displaystyle\end{eqnarray}$$

The above discussion is summarised by the plot of $\unicode[STIX]{x2202}p^{(1)}/\unicode[STIX]{x2202}x$ in figure 2. In the limit $Re\rightarrow \infty$ with $Fr>1$ fixed, the induced pressure gradient is bounded for all $x>0$ and does not affect the leading-order boundary layer solution. However, this behaviour is not uniform in $Fr$ , and in the distinguished limit (3.27), the pressure gradient is large and comparable to the inertia and viscosity terms.

Figure 2.

Plots of $Fr^{-2}\unicode[STIX]{x2202}p^{(1)}/\unicode[STIX]{x2202}x$ against $x$ for $Fr^{2}=1.2$ , $2$ , $5$ and $10$ , from the real part of (3.21). It should be noted that dividing by $Fr^{2}$ allows a comparison with the other terms in (3.3) (see the text for details).