Mathematics 373 – Applied Mathematics Individual Problem Set 2 – D’Alembert’s Solution of the Wave Equation Due: Tuesday February 6, 2001

A) Solve u_{tt} = c^{2}u_{xx} for x ∈ R, u(x, 0) = e^{x}, u_{t}(x, 0) = sin(x). Sketch (or use Maple to plot) the solution (u versus x, with t fixed) for t = 0, 1, 2, 3. What are the two travelling waves that are superposed to get the solution u(x, t)?

B) A string originally at rest with u(x, 0) = f(x) = 0 is struck with a hammer at time t = 0. The hammer blow imparts an initial velocity

u_{t}(x, 0) = g(x) = \{ \begin{matrix}1 & if |x| ≤ a \\ 0 & if |x| > a. \end{matrix}

1) Show that the solution of u_{tt} = c^{2}u_{xx} with these initial conditions is

u(x, t) = \frac{1}{2c} ∫_{x−ct}^{x+ct} g(s) ds,

and explain why this says u(x, t) is \frac{1}{2c} times the length of the interval (x − ct, x + ct) ∩ (−a, a) for all t, x.

2) Plot u versus x (t fixed) for t = \frac{a}{2c}, \frac{a}{c}, \frac{3a}{2c}, \frac{2a}{c}, \frac{5a}{2c}, \frac{5a}{c}, and describe what the string is doing in words.

3) As a function of t, what is the maximum displacement max_{x∈R} u(x, t)?

C) Recall that we derived D’Alembert’s solution of the wave equation by factoring the operator \frac{∂^{2}}{∂t^{2}} − c^{2} \frac{∂^{2}}{∂x^{2}} . Apply the same method to solve

1) u_{xx} − 3u_{xt} − 4u_{tt} = 0, u(x, 0) = x^{2}, u_{t}(x, 0) = e^{x}.

2) u_{xx} + u_{xt} − 20u_{tt} = 0, u(x, 0) = f(x), u_{t}(x, 0) = g(x), f, g arbitrary.