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.

 

[Navin A S]

When dealing with wave equations, one of the most important steps is to apply the initial and boundary conditions to determine the constants A and B. In this case, you can use the assumption that the pressure and velocity at the area changing point are the same. This means that Aa*U1 = Ab*U2. You can then rearrange this equation to get A/B = U2/U1. Since you also know that A = B, this means that U2/U1 must be equal to 1. This implies that U1 = U2. Once you have determined U1 and U2, you can use the initial and boundary conditions (such as u(x=0,t)=0) to solve for A and B.