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The dispersion relation of waves between two layers of varying density

  1. Oct 31, 2009 #1
    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
     
  2. jcsd
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