Wave equation under a Galilean transform.

[Johnny Blade]

Homework Statement

Show that the wave equation becomes
$$\left(1-\frac{V^{2}}{c^{2}}\right)\frac{\partial^{2}\psi'}{\partial x'^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}\psi'}{\partial t'^{2}}+\frac{2V}{c^{2}}\frac{\partial^{2}\psi'}{\partial t' \partial x'} = 0$$

under a Galilean transform if the referential R' moves at constant speed V along the x axis.

Homework Equations

The Attempt at a Solution

Frankly I don't really know how to do that. I tried using a general solution with x = x' + Vt' and using it in the normal wave equation, but gave me nothing good. Now I don't even know what else I could do.

 

[Johnny Blade]

Bump.

 

[kerimek]

I understand your frustration with not being able to solve this problem. However, let's break it down step by step to see if we can make sense of it.

First, let's review the wave equation in its standard form:

∂²ψ/∂t² - (c²/∂²ψ/∂x² = 0

This equation describes the behavior of a wave, where ψ represents the displacement of the wave at a given point in space and time, c is the speed of the wave, and t and x represent time and position, respectively.

Now, let's consider a Galilean transform, where we have two reference frames, R and R', moving at a constant speed V along the x-axis. In this case, we can relate the coordinates of the two frames using the following equations:

x = x' + Vt'

t = t'

Note that in the second equation, t' represents the time in the R' frame, and t represents the time in the R frame. This is important to keep in mind as we proceed.

Now, let's apply this transform to the wave equation. We can start by substituting the second equation into the first, which gives us:

∂²ψ/∂t² - (c²/∂²ψ/∂x² = 0

Next, we can substitute x' + Vt' for x in the second term, which gives us:

∂²ψ/∂t² - (c²/∂²ψ/∂(x' + Vt')² = 0

Expanding the second term using the chain rule, we get:

∂²ψ/∂t² - (c²/∂²ψ/∂x'² + 2V∂²ψ/∂t'∂x' + V²∂²ψ/∂t'²) = 0

Now, we can substitute this into the first term of the original wave equation, which gives us:

(∂²ψ/∂t² - 2V∂²ψ/∂t'∂x' + V²∂²ψ/∂t'²) - (c²/∂²ψ/∂x'² +