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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

$\displaystyle {\bf v}({\bf r},t)$ $\displaystyle = {\bf V} + {\bf v}_1({\bf r},t),$ (11.84)
$\displaystyle p({\bf r},t)$ $\displaystyle =p_0 - \rho\,g\,z + p_1({\bf r},t),$ (11.85)

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, (11.1) and (11.2), reduce to

$\displaystyle \nabla \cdot{\bf v}_1$ $\displaystyle = 0,$ (11.86)
$\displaystyle \left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right){\bf v}_1$ $\displaystyle = -\frac{\nabla p_1}{\rho},$ (11.87)

respectively. We can also define the displacement, $ \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

$\displaystyle \left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right)$$\displaystyle \mbox{\boldmath$\xi$}$$\displaystyle = {\bf v}_1.$ (11.88)

The curl of Equation (11.87) implies that $ \nabla\times{\bf v}_1={\bf0}$ . Hence, we can write $ {\bf v}_1=-\nabla\phi$ , and Equation (11.87) yields

$\displaystyle \left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right)\phi = \frac{p_1}{\rho}.$ (11.89)

Finally, Equation (11.86) gives

$\displaystyle \nabla^{\,2}\phi = 0.$ (11.90)

The most general traveling wave solution to Equation (11.90), with wave vector $ {\bf k}=k\,{\bf e}_x$ , and angular frequency $ \omega$ , is

$\displaystyle \phi(x,z,t)= \left[A\,\cosh(k\,z)+B\,\sinh(k\,z)\right]\cos(\omega\,t-k\,x).$ (11.91)

It follows from Equation (11.89) that

$\displaystyle p_1(x,z,t) = \rho\,k\,(V-c)\left[A\,\cosh(k\,z) + B\,\sinh(k\,z)\right]\sin(\omega\,t-k\,x),$ (11.92)

and from Equation (11.88) that

$\displaystyle \xi_z(x,z,t,) = (V-c)^{-1}\left[A\,\sinh(k\,z)+ B\,\cosh(k\,z)\right]\sin(\omega\,t-k\,x).$ (11.93)

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


next up previous
Next: Gravity Waves at an Up: Waves in Incompressible Fluids Previous: Ship Wakes
Richard Fitzpatrick 2016-01-22