The propagation of disturbances in interstellar gases

[p834]

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 = nKρ₀^n-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!

 

[mark726]

To show that the solution of the form ρ1 = ρ1(x±a0t) satisfies the equation ∂2ρ1/∂t2 - a02∂2ρ1/∂x2 = 0, we can start by substituting ρ1 = ρ1(x±a0t) into the linearized continuity equation:

1/ρ0 * ∂(ρ1(x±a0t))/∂t + ∂u1/∂x = 0

Using the chain rule, we get:

1/ρ0 * (∂ρ1/∂x * a0 ± ∂ρ1/∂t) + ∂u1/∂x = 0

Next, we can substitute in the linearized momentum equation:

∂u1/∂t = -1/ρ0 * ∂P1/∂x

to get:

1/ρ0 * (∂ρ1/∂x * a0 ± ∂ρ1/∂t) - 1/ρ0 * ∂P1/∂x = 0

Using the definition of P1 from the given equations, we have:

1/ρ0 * (∂ρ1/∂x * a0 ± ∂ρ1/∂t) - 1/ρ0 * nP0/ρ0 * ρ1 = 0

Simplifying, we get:

∂ρ1/∂x * a0 ± ∂ρ1/∂t - nρ1 = 0

Next, we can use the definition of a02 = n * P0/ρ0 to get:

∂ρ1/∂x * a0 ± ∂ρ1/∂t - a02ρ1 = 0

Finally, using the chain rule again, we can show that:

∂2ρ1/∂t2 = a02 * ∂2ρ1/∂x2

Substituting this into the equation above, we get:

a02 * ∂2ρ1/∂x2 - a02ρ1 = 0

which simplifies to:

∂2ρ1/∂x2 - ρ1 = 0

Therefore, we have shown that ρ1 = ρ1(x±a0t