# Navier Stokes Equation - Flow of waves

## Homework Statement

[/B]
(a) Show that for an incompressible flow the velocity potential satisfies $\nabla^2 \phi = 0$. Show further the relation for the potential to be $\frac{\partial \phi}{\partial t} + \frac{\nabla \phi \dot \nabla \phi}{2} + \frac{p}{\rho} + gz = const.$
(b)Write out the boundary conditions on velocity and show $\phi$ is of the form: $\phi = -\frac{\omega n_0}{k} cos(kx-\omega t) \frac{cosh kz}{sinh kH}$ subject to $\omega^2 = gk \space tanh \space (kH)$
(c) How does the frequency of the waves vary after the boat is gone? ## The Attempt at a Solution

Part (a)[/B]

For an irrotational flow, $\nabla \cdot \vec u = 0$ so this means that velocity can be written as gradient of some potential $u = \nabla \phi$.

Therefore $\nabla \cdot \vec u = \nabla^2 \phi = 0$.

Navier stokes equation is given by:
$$\frac{\partial \vec u}{\partial t} + (\vec u \cdot \nabla) \vec u + \frac{1}{\rho} \nabla p + g \vec k = \nu \nabla^2 \vec u$$

Using the identity and $\nabla^2 \vec u = 0$, $\nabla \times \vec u \times \vec u = 0$ I get the answer.

Part (b)

The boundary condition is velocity at $z=0$ is zero (non-slip).

I don't think this question wants me to simply substitute the given form of $\phi$ to verify it is right. I am expected to derive it.

Substituting, I get
$$Z \frac{\partial X}{\partial t} + \frac{ (Z\nabla X + X \nabla Z) \cdot (Z\nabla X + X \nabla Z) }{2} + \frac{p}{\rho} + gz = const.$$

This leads to nowhere. Any idea how to proceed?