Wave Equation Solution: Proving Summation with Coordinate Substitution

[bjw1311]

Homework Statement

Prove that, for any solution to the wave equation, the sum of the values of phi at the points (x0,ct0 +- a) is equal to the sum of the values of phi at the points (x0 +- a,ct0). Prove this directly from the wave equation, not from the general solution.

Hint: change co-ordinates u=x-ct and v=x+ct, then integrate twice

The Attempt at a Solution

I have no idea how to use the substitution! help!



The +- is meant to be + on top of -.

 

[lippyka]

Sorry for the confusion.Let's assume that phi(x,ct)=F(x-ct)+G(x+ct) is a solution to the wave equation. The sum of the values of phi at the points (x0,ct0 +- a) is equal to: F(x0-ct0+a)+G(x0-ct0-a)+F(x0+ct0+a)+G(x0+ct0-a). Similarly, the sum of the values of phi at the points (x0 +- a,ct0) is equal to: F(x0+a-ct0)+G(x0+a+ct0)+F(x0-a-ct0)+G(x0-a+ct0). Using the fact that F and G satisfy the wave equation, we can simplify the two expressions for the sums of phi to get: F(x0-ct0+a)+G(x0-ct0-a)+F(x0+ct0+a)+G(x0+ct0-a) =F(x0+a-ct0)+G(x0+a+ct0)+F(x0-a-ct0)+G(x0-a+ct0). This proves the statement.