# Boundary conditions of solution to the wave equation

1. Feb 16, 2012

1. The problem statement, all variables and given/known data

Because q(x,t) = A*exp[-(x-ct)22] 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).

2. Relevant 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...)

3. 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(-x22)
b(x) = (2cx/σ2)*A*exp(-x22)

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!!!