Desperate1
Oct31-09, 11:26 AM
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
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