| course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
31 |
|
1.II.17B |
2004 |
The equation governing small amplitude waves on a string can be written as
$$\frac{\partial^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}}$$
The end points $x=0$ and $x=1$ are fixed at $y=0$. At $t=0$, the string is held stationary in the waveform,
$$y(x, 0)=x(1-x) \quad \text { in } \quad 0 \leq x \leq 1 .$$
The string is then released. Find $y(x, t)$ in the subsequent motion.
Given that the energy
$$\int_{0}^{1}\left[\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right] d x$$
is constant in time, show that
$$\sum_{\substack{n \text { odd } \ n \geqslant 1}} \frac{1}{n^{4}}=\frac{\pi^{4}}{96}$$