 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Mathematics course of the University of Oxford

Past exam of the mathematics course of the University of Oxford

2017

Prelims

Mathematics V resit

Question 5

{c}

Here $f=0$ and $ct=1$, so <D'Alembert's formula> gives
$$
u\left(\frac32,\frac1c\right)
=\frac1{2c}\int_{1/2}^{5/2}g(s)\,ds.
$$
Only the overlap $[1/2,1]$ with the <support> of the initial velocity contributes. Therefore
$$
\begin{aligned}
u\left(\frac32,\frac1c\right)
&=\frac1{2c}\int_{1/2}^{1}(1-s^2)\,ds\\
&=\frac1{2c}\left[s-\frac{s^3}{3}\right]_{1/2}^{1}
=\frac1{2c}\left(\frac23-\frac{11}{24}\right)
=\boxed{\frac5{48c}}.
\end{aligned}
$$
The relevant interval is determined by the two <characteristic lines for the one-dimensional wave equation> through the observation point.

Solution

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

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

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)