 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/13e/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

The string is initially at rest, so

y (x, 0) = y_{t} (x, 0) = 0

t

converts the wave equation into

p^{2} y = c^{2} \frac{\partial ^{2} y}{\partial x ^{2}} .

For

Re p > 0

, the solution that decays as

x \to \infty

y = A (p) e^{- p x / c}

y (0, p) = f (p)

fixes

A = f

, and therefore

y (x, p) = f (p) e^{- p x / c} .

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