Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 5 / a / ii

Table of contents
- Solution ii

Solution

At

x = L /2

, the integration interval first moves through positive arguments, then crosses zero, and finally contains all of

[- L, L]

. The result is

y (\frac{L}{2}, t) = ⎩ ⎨ ⎧ L t /2, \frac{L ^{2} + ( c t - L /2 ) ^{2}}{4 c}, \frac{L ^{2}}{2 c}, 0 \leq c t \leq L /2, L /2 \leq c t \leq 3 L /2, c t \geq 3 L /2.

The final plateau is the integral of the entire initial velocity profile divided by

2 c

.

Figure 1.
Two traces of the D'Alembert solution for the compactly supported initial velocity
. The spatial profile has two peaks at half the initial support radius. The time trace at that location eventually reaches a constant plateau. Axes use dimensionless variables; plotted curves use $L = c = 1$ . Details.

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