Waves: Particle Paths

Continuing on from last week, we looked at water waves, and I will start with the example that we did last week.

Image

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:

waterequations This is equation 4.1

waterequations2 This is equation 4.6

waterequations3 This is equation 4.5

waterequations4 This is equation 4.3

We can work out the solution by following these steps:

Image

We can now imply the boundary conditions:

Image

Implying that B=0, we get the solution to be:

solutiontoinfinite

Dispersion Relation

If we use equation 4.6 we get:

dispersionrelation

this implies that w^2=gk and this is equation 4.9.

Surface Displacement

If we use equation 4.3 we get:

surfacedisplacement

Potential

The potential is:

potential

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:

Image

 

 

Image

This example is to do with infinite depth:

Image

Image

Image

Image

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

 

 

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