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D'Alembert's solution

  1. Nov 7, 2006 #1
    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!
     
    Last edited: Nov 7, 2006
  2. jcsd
  3. Nov 7, 2006 #2
    Use the relativistic D'Alembert's solution then..
     
  4. Nov 7, 2006 #3
    Solution to what?
     
  5. Nov 7, 2006 #4

    Galileo

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

    [tex]\frac{\partial^2 f}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 f}{\partial t^2}[/tex]

    Could be the speed of sound or whatever. It depends on what the speed 'c' in your wave equation is.
     
  6. Nov 7, 2006 #5
    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 travelling 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 travelling at the speed of ... of ...
     
  7. Nov 7, 2006 #6

    Galileo

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    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.
     
  8. Nov 7, 2006 #7
    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?
     
  9. Nov 7, 2006 #8

    Galileo

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    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.
     
  10. Nov 7, 2006 #9
    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.:confused:
     
  11. Nov 13, 2006 #10
    ...light, because you're solving the wave equation for light. The solution is a light wave moving with the speed of light "c".
     
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