Equation (4.2) in Grue (Reference Grue2017) employs the approximation
$\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{F}/\unicode[STIX]{x2202}t\simeq -p/\unicode[STIX]{x1D70C}$
(
$\unicode[STIX]{x1D719}_{F}$
the potential at the surface,
$p$
the given surface pressure,
$\unicode[STIX]{x1D70C}$
the density). This is a valid approximation in the supercritical case with
$Fr^{2}=U^{2}/gh\gg 1$
(
$U$
the speed of the moving pressure,
$g$
the acceleration due to gravity,
$h$
the water depth,
$Fr$
the depth Froude number). In the subcritical case with
$Fr^{2}=U^{2}/gh<1$
the hydrostatic term in the dynamic boundary condition at the free surface gives a significant contribution, however. This results in a multiplicative factor of
$-Fr^{2}/(1-Fr^{2})$
of the asymptotic upstream wave elevations derived in § 4. The revised result is obtained from the linear free surface boundary condition:
$\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}t^{2}+g\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}y=-\unicode[STIX]{x2202}(p/\unicode[STIX]{x1D70C})/\unicode[STIX]{x2202}t$
, at
$y=0$
, where
$\unicode[STIX]{x1D719}$
is the velocity potential. In a frame of reference moving with the speed
$U$
of the surface pressure, along the
$x_{1}$
-direction, where
$\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t=-U\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{1}$
, the free surface boundary condition becomes
$U^{2}\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}_{F}/\unicode[STIX]{x2202}x_{1}^{2}+g\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}y=(U/\unicode[STIX]{x1D70C})\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x_{1}$
. Fourier transformation gives
$$\begin{eqnarray}-U^{2}k_{1}^{2}\hat{\unicode[STIX]{x1D719}}_{F}+g\widehat{\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}y}}=\text{i}k_{1}U\,\frac{\hat{p}}{\unicode[STIX]{x1D70C}},\end{eqnarray}$$
where a hat denotes a Fourier transform and
$\boldsymbol{k}=(k_{1},k_{2})$
is the wavenumber in Fourier space. The normal velocity
$\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}y$
is connected to the velocity potential at the free surface by the solution of the Laplace equation, expressed in terms of the Fourier transform, for a fluid layer of constant depth
$h$
by
$\widehat{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}y}=k\tanh kh\,\hat{\unicode[STIX]{x1D719}}_{F}=(\unicode[STIX]{x1D714}^{2}/g)\hat{\unicode[STIX]{x1D719}}_{F}$
, giving
$$\begin{eqnarray}(\unicode[STIX]{x1D714}^{2}-U^{2}k_{1}^{2})\hat{\unicode[STIX]{x1D719}}_{F}=\text{i}k_{1}U\,\frac{\hat{p}}{\unicode[STIX]{x1D70C}},\end{eqnarray}$$
and yielding
$$\begin{eqnarray}-\text{i}k_{1}U\hat{\unicode[STIX]{x1D719}}=Fr^{2}{\mathcal{C}}\,\frac{\hat{p}}{\unicode[STIX]{x1D70C}},\quad {\mathcal{C}}=\frac{k_{1}^{2}/k^{2}}{1-(k_{1}^{2}/k^{2})Fr^{2}},\end{eqnarray}$$
where
$k=|\boldsymbol{k}|$
and the spectral wave speed
$c=\unicode[STIX]{x1D714}/k\rightarrow \sqrt{gh}$
for
$k\rightarrow 0$
. The constant
${\mathcal{C}}$
is positive in the subcritical case (
$Fr^{2}<1$
) and negative for
$(k_{1}^{2}/k^{2})Fr^{2}>1$
. The product
$Fr^{2}{\mathcal{C}}\rightarrow -1$
for
$Fr^{2}\gg 1$
. The two-dimensional case gives the coefficient as
${\mathcal{C}}_{0}=1/(1-Fr^{2})$
, yielding
$\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{F}/\unicode[STIX]{x2202}t=Fr^{2}{\mathcal{C}}_{0}\,p/\unicode[STIX]{x1D70C}$
.
Figure 1. Asymptotic upstream waves. Delta function moving over a bottom step,
$\unicode[STIX]{x0394}h>0$
,
$Fr=0.5$
,
$t^{\ast }=t\sqrt{g/h}=30$
, 90. Stationary phase, ——; wave front expressions, – – –. (a) Two dimensions. (b) Three dimensions.
In three dimensions, in the subcritical case, we obtain
$$\begin{eqnarray}-\text{i}k_{1}U\hat{\unicode[STIX]{x1D719}}_{F}=\frac{\hat{p}}{\unicode[STIX]{x1D70C}}\left(Fr^{2}\frac{k_{1}^{2}}{k^{2}}+Fr^{4}\frac{k_{1}^{4}}{k^{4}}+Fr^{6}\frac{k_{1}^{6}}{k^{6}}+\cdots +\right).\end{eqnarray}$$
Assuming that the pressure distribution is a delta function in the two horizontal directions with
$p(x_{1},x_{2})=\unicode[STIX]{x1D70C}gV_{0}\unicode[STIX]{x1D6FF}(x_{1})\unicode[STIX]{x1D6FF}(x_{2})$
where
$V_{0}$
is the volume of the pressure distribution, its Fourier transform becomes
$\hat{p}=\unicode[STIX]{x1D70C}gV_{0}$
. Evaluating the inverse Fourier transform of the first term on the right-hand side of (0.4) gives
$$\begin{eqnarray}\unicode[STIX]{x1D719}_{F}\simeq \frac{V_{0}U}{(2\unicode[STIX]{x03C0})^{2}h}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\frac{\text{i}k_{1}\text{e}^{\text{i}(k_{1}x_{1}+k_{2}x_{2})}}{k_{1}^{2}+k_{2}^{2}}\,\text{d}k_{1}\,\text{d}k_{2}=-\frac{V_{0}U}{2\unicode[STIX]{x03C0}h}\frac{x_{1}}{x_{1}^{2}+x_{2}^{2}}.\end{eqnarray}$$
The contribution to the right-hand side of (4.1) becomes
$$\begin{eqnarray}{\hat{h}}_{1}\simeq -\text{i}k_{1}\unicode[STIX]{x0394}h\int _{-\infty }^{\infty }\unicode[STIX]{x1D719}_{F}|_{x_{1}=-Ut}\,\text{d}x_{2}=\frac{\text{i}k_{1}\unicode[STIX]{x0394}hV_{0}U}{2\unicode[STIX]{x03C0}h}\int _{-\infty }^{\infty }\frac{-Ut}{(Ut)^{2}+x_{2}^{2}}\,\text{d}x_{2}=-\frac{\text{i}k_{1}\unicode[STIX]{x0394}hV_{0}U}{2h}\frac{t}{|t|}.\end{eqnarray}$$
The contribution to the Fourier-transformed wave elevation becomes
$$\begin{eqnarray}\hat{\unicode[STIX]{x1D702}}_{0}=-\frac{\text{i}k_{1}\unicode[STIX]{x0394}hV_{0}U}{2h}\int _{t_{0}}^{t}\cos \unicode[STIX]{x1D714}(s-t)\frac{s}{|s|}\,\text{d}s=-\frac{\text{i}k_{1}\unicode[STIX]{x0394}hV_{0}\,Fr^{2}}{U\unicode[STIX]{x1D714}/g}\sin \unicode[STIX]{x1D714}t.\end{eqnarray}$$
The successive contributions from the expansion give
$$\begin{eqnarray}\hat{\unicode[STIX]{x1D702}}_{0}=-\frac{\text{i}k_{1}\unicode[STIX]{x0394}hV_{0}Fr^{2}(1+Fr^{2}+Fr^{4}\ldots )}{U\unicode[STIX]{x1D714}/g}\sin \unicode[STIX]{x1D714}t=-\frac{\text{i}k_{1}\unicode[STIX]{x0394}hV_{0}Fr^{2}{\mathcal{C}}_{0}}{U\unicode[STIX]{x1D714}/g}\sin \unicode[STIX]{x1D714}t,\end{eqnarray}$$
which replaces (4.4) in § 4, where
${\mathcal{C}}_{0}=1+Fr^{2}+Fr^{4}+\cdots =1/(1-Fr^{2})$
(
$Fr^{2}<1$
). The subsequent results for the elevations in § 4 are multiplied by the factor
$-Fr^{2}{\mathcal{C}}_{0}$
. The asymptotics show a leading wave of elevation, for a positive step with
$\unicode[STIX]{x0394}h>0$
; see the corrected figure 1. This correction does not affect the results of the fully dispersive calculations.
