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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-3/14a/e/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 14A / e / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Here the D'Alembert formula with initial velocity becomes

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

When

t > 1

and

1 - t \leq x \leq t - 1,

the integration interval contains the whole support

[- 1, 1]

. Since

g

is odd,

u (x, t) = \frac{1}{2} \int_{- 1}^{1} ξ d ξ = 0.

Thus the requested region containing the line

x = 0

is

t > 1, ∣ x ∣ \leq t - 1.

This is the central zero region for odd compactly supported initial velocity. The initial disturbance splits into left-moving and right-moving waves; after they pass, their opposite signed impulses cancel and leave an expanding undisturbed region around the origin.

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