Finding Density as a Function of Space and Time for 1D Wave Equation Problem

[jlee07]

Homework Statement

Hello-

I'm having trouble understanding a problem:

Consider a sealed 1D pipe of length L. At t=0, v=0 everywhere and the pressure is given by: P=P_0 +δP

and δP = (p-bar)x/L

P_0 and (p-bar) are both constants.

and I'm supposed to find density (ϱ) as a function of x and t.

I don't understand why I'm given the pressure, and how to find δϱ.

Homework Equations

The Attempt at a Solution

Using the boundary conditions, I have an expression for the velocity with sin terms. But I don't know how to use that to find the density as a function of (x,t).

 

[mfb]

The gas laws give you a relation between pressure and density.