Math 054 Partial Differential Equations - HW Assignment 8

1. Use the general solution of a wave equation

u(x, t) = G(x + at) + F(x - at)

to derive D'Alembert's solution for the following initial value problem.

u_tt = a²u_xx, -∞ < x < ∞

u(x, 0) = f(x), u_t(x, 0) = g(x)

2. For an infinite string, what initial conditions would give rise to a purely forward wave? Express your answer in terms of initial displacement u(x, 0) = f(x) and initial velocity u_t(x, 0) = g(x) and their derivatives. Interpret the result intuitively. (Hint: use u(x, t) = F(x - at) + G(x + at) and the expressions found in problem 1 for F(x) and G(x).)

3. For an infinite string, suppose that u(x, 0) = f(x) and u_t(x, 0) = g(x) are zero for |x| > a, for some real number a > 0. Prove that if t + x > a and t - x > a, then the displacement u(x, t) of the string is constant. Relate this constant to g(x).

4. Solve

u_xx(x, t) - 3u_xt(x, t) - 4u_tt(x, t) = 0

u(x, 0) = x², u_t(x, 0) = e^x

5. Derive the solution of the inhomogeneous wave equation

u_tt - a²u_xx = F(x, t), -∞ < x < ∞

u(x, 0) = f(x), u_t(x, 0) = g(x)

using Green's theorem (Hint: Integrate F(x, t) over the domain of dependence).