# Homework Help: The dispersion relation of waves between two layers of varying density

1. Oct 31, 2009

### Desperate1

1. The problem statement, all variables and given/known data
Show that c^2 = g/k*(rho1 - rho2)/ (rho1 + rho2)
where rho1 and rho 2 are the different densities, k is the constant from solving the PDE (separation of variables).

2. Relevant equations

Use the fact that phi(1) --> 0 as y --> neg. infinity and phi(2) --> 0 as y --> infinity
Also, use the:

The pressure condition:
rho1*partial phi(1)/dt + rho1*g*eta = rho2*partial phi(2)/dt + rho2*g*eta

where phi is the velocity potential or phi =f(y)*sin(kx-wt), and eta = A*cos (kx-wt)

other useful equations:

at y=0:

partial phi/dy =partial eta/dt (after linearizing and ommiting higher quadratic terms)

partial phi/ft + g*eta =0 after a similar treatment of the pressure condition

3. The attempt at a solution

I first attempted to find the coefficients for the "function of 'y' portion" of the separation of variables: f(1)(y) = Ee^ky + De^-ky and f(2)(y)= Ge^ky + He^-ky

the condition of y --> infinity and neg. infinity tells me that two of these 4 constants must be equal to 0. From here I think that I need to replace these new values into the definition of phi so I can take partials of eta and phi to somehow use it in the pressure condition and then find the dispersion relation based on c^2 = w^2/ k^2. How this is to happen? I am am at a loss
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution