The dispersion relation of waves between two layers of varying density

[Desperate1]

Homework Statement

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).

Homework 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

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

 

[Jhero]

at this point.

Hello there! It seems like you are on the right track. Let's break down the steps to solving this problem:

Step 1: Find the coefficients for the functions of 'y'

As you mentioned, using the boundary conditions of y --> infinity and y --> neg. infinity, we can determine that two of the four coefficients must be equal to 0. This leaves us with two unknown coefficients for each function of 'y'. To find these coefficients, we can use the given equations for phi(1) and phi(2) and their partial derivatives with respect to y. By equating the coefficients of the exponential terms, we can solve for the remaining coefficients.

Step 2: Use the dispersion relation to find c^2

As you mentioned, we can use the dispersion relation c^2 = w^2/ k^2 to find the value of c^2. To do this, we need to determine the values of w and k. We can use the given equations for phi and eta to find the partial derivatives with respect to t. Then, by equating the coefficients of the sine and cosine terms, we can solve for w and k.

Step 3: Substitute the values of c^2, w, and k into the given equation

Now that we have determined the values of c^2, w, and k, we can substitute them into the given equation c^2 = g/k*(rho1 - rho2)/ (rho1 + rho2). This will give us a final equation that relates the densities and the constant k.

I hope this helps guide you in solving the problem. Remember to keep track of your units and double check your calculations. Good luck!