Help - Missing Step - Gas in a Tube

[Sol_Solved]

Help ASAP --- Missing Step --- Gas in a Tube


I am dealing with finding normal modes of oscillations in a continuum

I have no problems with the string example, but now I have a gas in a tube, one side of the gas has pressure= p_0 and the other side p_0+delta p, they're separated by a wall that is lifted at t=0. This is a tube closed in the extremes.

So, I've started with the classic wave equation for the tube
rho is the density, P is the pressure, psi the movement
(second partial derivatire of psi on t = second partial derivative of psi on x, times - rho_0 )and proposing the typical solution psi(x, t)= A cos(kx)cos(omega*t+phi). I also know the relationship between the speed of sound and pressure and density, and the value of delta rho.

I start with the boundary conditions to solve for k, and now it is time to write out the initial conditions, and from there solve for A and phi. According to the string example, one of these conditions will end up as a function f(x) to be solved by Fourier.

The problem is, I can't see for the life of me which are the initial conditions in a gas! And I can't see what kind of function it would be either. In a string I just see where the string is at time=0 and what velocity it has (usually 0), and according to its shape, I find the f(x) to use Fourier with.

Please, anyone answer this, I've been trying to find it for the last 5 hours with no luck, the library is closed and my exam is tomorrow morning. And I apologize for the lack of LATeX.

Thanks,

Sol.

 

[quantumdude]

It would help if you would state the problem as it appears in the book.

 

[Sol_Solved]

It's not a problem from a book, it's just the physical description of of oscillations of gas in a tube solved using Fourier analysis. It is not explained in the books I have, and I couldn't find the step by step analysis online (the usual example is a string) so that's why I asked. Anyway, my exam was this morning. Thanks.

Sol.