Steady Flow over a Corrugated Bottom

Consider a stream of water of mean depth $d$ , and uniform horizontal velocity ${\bf V}=V\,{\bf e}_x$ , that flows over a corrugated bottom whose elevation is $z=-d+ a\,\sin(k\,x)$ , where $a$ is much smaller than $d$ . Let the elevation of the free surface of the water be $z=b\,\sin(k\,x)$ . We wish to determine the relationship between $a$ and $b$ .

We expect the velocity potential, perturbed pressure, and vertical displacement of the water to be of the form (11.91), (11.92), and (11.93), respectively, with $\omega=c=0$ , because we are looking for a stationary (i.e., non-propagating) perturbation driven by the static corrugations in the bottom. The boundary condition at the bottom is

$$\xi_z(x,-d) = a\,\sin(k\,x),$$ (11.104)

which yields

$$V^{\,-1}\left[-A\,\sinh(k\,d)+ B\,\cosh(k\,d)\right] = a.$$ (11.105)

At the free surface, we have

$$\xi_z(x,0) = b\,\sin(k\,x),$$ (11.106)

which gives

$$b= V^{\,-1}\,B.$$ (11.107)

In addition, pressure balance across the free surface yields

$$\rho\,g\,b\,\sin(k\,x) = p_1(x,0) = \rho\,k\,V\,A\,\sin(k\,x),$$ (11.108)

which leads to

$$g\,b = k\,V\,A.$$ (11.109)

Hence, from Equations (11.105), (11.107), and (11.109),

$$b = \frac{a}{\cosh(k\,d) - (g/k\,V^{\,2})\,\sinh(k\,d)},$$ (11.110)

or

$$b = \frac{a}{\cosh(k\,d)\,(1-c^{\,2}/V^{\,2})},$$ (11.111)

where $c=[(g/k)\,\tanh(k\,d)]^{1/2}$ is the phase velocity of a gravity wave of wave number $k$ . [See Equation (11.21).] It follows that the peaks and troughs of the free surface coincide with those of the bottom when $\vert V\vert> \vert c\vert$ , and the troughs coincide with the peaks, and vice versa, when $\vert V\vert<\vert c\vert$ . If $\vert V\vert=\vert c\vert$ then the ratio $b/a$ becomes infinite, implying that the oscillations driven by the corrugations are not of small amplitude, and, therefore, cannot be described by linear theory.