Solve Wave Equation with D'Alembert's Solution

[actionintegral]

Friends,

I have seen the wave equation solved by changing the coordinates to
u=x+ct and v=x-ct.

This is preposterous! Solve the wave equation by moving at the speed of light! Outlandish!

 

[Reshma]

Use the relativistic D'Alembert's solution then..

 

[actionintegral]

Solution to what?

 

[Galileo]

Friends,

I have seen the wave equation solved by changing the coordinates to
u=x+ct and v=x-ct.

This is preposterous! Solve the wave equation by moving at the speed of light! Outlandish!

Um, the 'c' is just the speed appearing in the wave equation:

$$\frac{\partial^2 f}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 f}{\partial t^2}$$

Could be the speed of sound or whatever. It depends on what the speed 'c' in your wave equation is.

 

[actionintegral]

Yes. Suppose c is the speed of sound. What is the effect of setting
u=x-ct? You are changing to a frame that is traveling at the speed of sound. Suppose c is the speed of light. What is the effect of setting u=x-ct? You are changing to a frame that is traveling at the speed of ... of ...

 

[Galileo]

But so what? It's just a change of variable that will help you solve the equation. It has nothing to do with violating relativity if that's what you're thinking of.

 

[actionintegral]

Fair enough. Let me counter. Suppose I solve a physics problem by changing to a frame where
x= .99ct Can I do so with a clear conscience?

 

[Galileo]

You're not 'changing to a frame'. You introduce new variables u and v to help solve the differential equation. It's mathematics, not physics. Mathematically you can make c 20 times the speed of light and you'll still get a solution to the differential equation in the form F(x+ct)+G(x-ct). In a relativistic physical theory you wouldn't find something that obeys that wave equation with c greater than lightspeed so that's not an issue.

 

[actionintegral]

You're not 'changing to a frame'.

I understand perfectly now. Saying "my friend is at x=5" and then saying "let u=x-5. My friend is at u=0" is distinct from saying "I am where my friend is".

That said, maybe someone can help me understand the d'alembert solution to the schroedinger equation. I did this with u=x+ct, v=x-ct.
Then I got nervous and changed the "c" to a "v".

Now I don't know what the h******ell to think.

 

[ObsessiveMathsFreak]

Yes. Suppose c is the speed of sound. What is the effect of setting
u=x-ct? You are changing to a frame that is traveling at the speed of sound. Suppose c is the speed of light. What is the effect of setting u=x-ct? You are changing to a frame that is traveling at the speed of ... of ...

...light, because you're solving the wave equation for light. The solution is a light wave moving with the speed of light "c".