Boundary conditions of solution to the wave equation

[deadlytrogdor]

Homework Statement

Because q(x,t) = A*exp[-(x-ct)²/σ²] is a function of x-ct, it is a solution to the wave equation (on an infinite domain).

(a) What are the initial conditions [a(x) and b(x)] that give rise to this form of q(x,t)?
(b) if f(x) is constant, then Eq. (2) shows that solution is only a function of x-ct. For the condition that f(x) is constant find b(x) in terms of a(x). [Hint: consider eq. (3a)]
(c) Show that the initial conditions you found in part (a) satisfy the relationship that you found in part (b).

Homework Equations

Initial displacement: a(x) = q(x,0)
Initial velocity: b(x) = $dq(x,0)/dt$ (this should be a partial derivative--sorry)
Eq 2: q(x,t) = f(x+ct) + g(x-ct)
Eq 3a: f(x) = $1/2$ (a(x) + $\frac{1}{c}$ ∫b(x')dx') where the integral is from x₀ to x) (sorry again...)

The Attempt at a Solution

Okay, for part (a), I just used the given q(x,t) to solve for the boundary conditions a(x) and b(x), and I got

a(x) = A*exp(-x²/σ²)
b(x) = (2cx/σ²)*A*exp(-x²/σ²)

For part (b), I'm a little stumped at the moment. Can I just solve for b(x) in equation (3a)...? I'm not sure how I would go about doing that with b(x) inside the definite integral.

I think (c) will be apparent once I figure out (b). A little push in the right direction would be much appreciated! Thanks!

 

[mark726]

Hi there! It looks like you're on the right track for part (a). For part (b), you can actually solve for b(x) by using the initial conditions you found in part (a). Since f(x) is constant, you can plug in the expression you found for a(x) into equation (3a) and solve for b(x). The integral will then be a function of x only, and you can solve for b(x) from there.

For part (c), you can then plug in the expressions you found for a(x) and b(x) into equation (2) and show that it satisfies the relationship given in part (b). Let me know if you have any further questions or need clarification!