Solve Wave Equation with D'Alembert's Solution

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In summary, the conversation revolves around the use of coordinates and variables to solve the wave equation and the concept of changing frames in physics. It is clarified that mathematically, the value of "c" in the equation does not necessarily represent the actual speed of light in a physical scenario. The D'Alembert solution to the Schrodinger equation is also briefly mentioned.
  • #1
actionintegral
305
5
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!
 
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  • #2
Use the relativistic D'Alembert's solution then..
 
  • #3
Solution to what?
 
  • #4
actionintegral said:
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:

[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.
 
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  • #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 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 ...
 
  • #6
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.
 
  • #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?
 
  • #8
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.
 
  • #9
Galileo said:
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.:confused:
 
  • #10
actionintegral said:
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".
 

FAQ: Solve Wave Equation with D'Alembert's Solution

1. What is D'Alembert's solution?

D'Alembert's solution is a mathematical formula used to solve the wave equation, which describes the propagation of waves in a medium. It is named after French mathematician Jean le Rond d'Alembert, who first proposed the solution in 1746.

2. How is D'Alembert's solution used to solve the wave equation?

D'Alembert's solution involves breaking down the wave equation into two separate equations for the forward and backward propagating waves. These equations can then be solved using the initial conditions and boundary conditions to find the general solution for the wave at any point in time and space.

3. What are the assumptions made in D'Alembert's solution?

D'Alembert's solution assumes that the medium in which the wave is propagating is linear and homogeneous. This means that the properties of the medium, such as density and elasticity, do not change over time or space.

4. Can D'Alembert's solution be used for any type of wave?

D'Alembert's solution is most commonly used for solving the wave equation for one-dimensional waves, such as sound or transverse waves on a string. It can also be used for other types of waves, but may require modifications for more complex scenarios.

5. Are there any limitations to D'Alembert's solution?

While D'Alembert's solution is a powerful tool for solving the wave equation, it does have some limitations. It is based on simplifying assumptions and may not accurately model more complex wave phenomena. Additionally, it is only valid for linear and homogeneous media, and may not be applicable to nonlinear or inhomogeneous systems.

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