Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V resit / Question 5 / c

Table of contents
- Solution c

Solution

Here

f = 0

and

c t = 1

, so D'Alembert's formula gives

u (\frac{3}{2}, \frac{1}{c}) = \frac{1}{2 c} \int_{1/2}^{5/2} g (s) d s .

Only the overlap

[1/2, 1]

with the support of the initial velocity contributes. Therefore

u (\frac{3}{2}, \frac{1}{c}) = \frac{1}{2 c} \int_{1/2}^{1} (1 - s^{2}) d s = \frac{1}{2 c} [s - \frac{s ^{3}}{3}]_{1/2}^{1} = \frac{1}{2 c} (\frac{2}{3} - \frac{11}{24}) = \frac{5}{48 c} .

The relevant interval is determined by the two characteristic lines for the one-dimensional wave equation through the observation point.

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