Wave Propagation solution for a variable area 1D duct

domainxh

In the problem I am suppose to use the wave equation to solve it.

I assume 1D plane wave duct,

u(x,t) = 1/(rho*C)*real((Aexp(ikx)-Bexp(-ikx))exp(iwt))
where C is the speed of sound, u is the velocity, p is the pressure, w is the angular frequency, t is time, rho is the density, and both A and B are unknown constants which I have to find.

The equation for Pressure is similar except without density and speed of sound term,
p(x,t) = real((Aexp(ikx)-Bexp(-ikx))exp(iwt))

At the area changing point, I am suppose to assume the same pressure and Aa*U1 = Ab*U2 (essentially matching the conditions)

After applying the left end B.C, I get A = B
But I can't move any further.... Any help would be greatly appreciated.

Attached Files

CD_HW-2.pdf (61.4 KB, 3 views)