Gravity Waves in a Flowing Fluid

Consider a gravity wave traveling through a fluid that is flowing horizontally at the uniform velocity ${\bf V}=V\,{\bf e}_x$. Let us write

$${\bf v}({\bf r},t) = {\bf V} + {\bf v}_1({\bf r},t),$$ (1190)

$$p({\bf r},t) = p_0 - \rho\,g\,z + p_1({\bf r},t),$$ (1191)

where ${\bf v}_1$ and $p_1$ are the small velocity and pressure perturbations, respectively, due to the wave. To first order in small quantities, the fluid equations of motion, (1107) and (1108), reduce to

$$\nabla \cdot{\bf v}_1 = 0,$$ (1192)

$$\left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right){\bf v}_1 = -\frac{\nabla p_1}{\rho},$$ (1193)

respectively. We can also define the displacement, $\mbox{\boldmath$\xi$}({\bf r},t)$, of a fluid particle due to the passage of the wave, as seen in a frame co-moving with the fluid, as

$$\left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right)\mbox{\boldmath$\xi$}= {\bf v}_1.$$ (1194)

The curl of Equation (1193) implies that $\nabla\times{\bf v}_1={\bf0}$. Hence, we can write ${\bf v}_1=\nabla\phi$, and (1193) yields

$$\left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right)\phi = -\frac{p_1}{\rho}.$$ (1195)

Finally, Equation (1192) gives

$$\nabla^2\phi = 0.$$ (1196)

The most general traveling wave solution to (1196), with wavevector ${\bf k}=k\,{\bf e}_x$, and angular frequency $\omega$, is

$$\phi(x,z,t)= \left[A\,\cosh(k\,z)+B\,\sinh(k\,z)\right]\cos(\omega\,t-k\,x).$$ (1197)

It follows from Equation (1195) that

$$p_1(x,z,t) = -\rho\,k\,(V-c)\left[A\,\cosh(k\,z) + B\,\sinh(k\,z)\right]\sin(\omega\,t-k\,x),$$ (1198)

and from Equation (1194) that

$$\xi_z(x,z,t,) = -(V-c)^{-1}\left[A\,\sinh(k\,z)+ B\,\cosh(k\,z)\right]\sin(\omega\,t-k\,x).$$ (1199)

Here, $c=\omega/k$ is the phase velocity of the wave.