 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2005/calculus-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/4/b/i/solution

For the wave equation with wave speed one, D'Alembert's formula with arbitrary twice continuously differentiable initial displacement

h (x) = u (x, 0)

u (x, t) = \frac{h ( x + t ) + h ( x - t )}{2} + \frac{1}{2} \int_{x - t}^{x + t} g (s) d s .

The arbitrary function

h

is essential: only the initial velocity is prescribed. For

g = 0

, this gives

u (x, t) = \frac{h ( x + t ) + h ( x - t )}{2} .

Both travelling terms solve the wave equation, and their time derivatives cancel at

t = 0

. Conversely every classical solution has this form for its initial displacement

h

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