Continuing on from last week, we looked at water waves, and I will start with the example that we did last week. We have to make assumptions with this:

• v=0; and the partial derivative with respect to y=0.
• It is inviscid.
• It is incompressible.
• It is irrotational. If it’s incompressible and irrotatinal, this implies that nabla^2*Phi=0.
• g is the only body ‘force’.
• no surface tension.
• the displacement and the velocities are small (ignore products).
• water at the free surface stays there.
• Starts from rest.

We use these equations to find the solution for both finite depth and infinite depth: This is equation 4.1 This is equation 4.6 This is equation 4.5 This is equation 4.3

We can work out the solution by following these steps: We can now imply the boundary conditions: Implying that B=0, we get the solution to be: Dispersion Relation

If we use equation 4.6 we get: this implies that w^2=gk and this is equation 4.9.

Surface Displacement

If we use equation 4.3 we get: Potential

The potential is: Phase Speed

The phase speed is the speed at which the phase of a wave is propagated, the product of the frequency times the wavelength.

In our solution, this is given by: c=w/k.

The following is the example of finite depth:  This example is to do with infinite depth:    This week, going through the examples I’m not sure I fully understand it, so I aim to go back over the examples till I am 100% sure I know where we get the solution from and the stages at each step.

Sources:

Phase speed definition: http://www.thefreedictionary.com/phase+speed