Navier Stokes Equation - Flow of waves

[unscientific]

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?

Homework Equations

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?

 

[unscientific]

bumpp

 

[unscientific]

Any help for part (b)??

 

[unscientific]

bumpp

 

[paranormal]

For part (b), you can use the boundary condition at $z=0$ to solve for the constant in the potential equation. This will give you a relation between the potential and the boundary conditions. Then, you can use this relation to verify that the given form of the potential satisfies the boundary conditions.

For part (c), the frequency of the waves will decrease after the boat is gone because the boat's motion will no longer be a source of energy for the waves. This means that the energy of the waves will decrease, causing a decrease in frequency. Additionally, the boat's motion may have caused some dispersion of the waves, which can also contribute to a decrease in frequency.