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:
If we use equation 4.6 we get:
this implies that w^2=gk and this is equation 4.9.
If we use equation 4.3 we get:
The potential is:
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.
Phase speed definition: http://www.thefreedictionary.com/phase+speed