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Homework Statement
Show that the solution of the form ρ1 = ρ1(x±a0t) satisfy the equation:
∂2ρ1/∂t2 - a02∂2ρ1/∂x2 = 0
and that they correspond to waves propagating in the directions x increasing or decreasing.
Homework Equations
P = P0 + P1
ρ = ρ0 + ρ1
u = u1
The Attempt at a Solution
P1 = nKρ0n-1ρ1 ≡ nP0/ρ0 * ρ1
We shall define a quantity a02 = n * P0/ρ0.
a0 has dimensions of velocity and is a constant.
The linearized form of the continuity equation is:
1/ρ0 * ∂ρ1/∂t + ∂u1/∂x = 0
The linearized form of the momentum equation is:
∂u1/∂t = -1/ρ0 * ∂P1/∂x
This is as far as I have gotten with the problem. This problem can be found in the book of the physics of the interstellar medium by Dyson and Williams. Any help and I would be grateful!