[Fourier and PDEs 5] Wave Equation 2

1. A plucked string

The ends $x = 0$ and $x = L$ of a stretched string are fixed. The point $p$ ($0 < p < L$) is drawn aside a distance $h$ and, at the instant $t = 0$, the string is released from rest. Thus \[ u(x,0) = \begin{cases} hx/p & 0 \leq x < p \\ h(L - x)/(L-p) & p < x \leq L \end{cases} \]

and $u_t(x,0) = 0$. Use separation of variables and Fourier series to find $u(x,t)$.


2. A plucked string

Two strings of length $a$ are joined at $x = 0$ and stretched to a tension $T$. One string has constant density $\rho_1$ and is fixed at $x = -a$, the other has constant density $\rho_2$ and is fixed at $x = a$.

a) State the conditions that the transverse displacement $y(x,t)$ must satisfy at $x = -a$, $x = 0$, and $x = a$.


b) Show that in a normal mode of vibration, the transverse displacement has the form \[ y(x, t) = \begin{cases} A\sin[w(a + x)/c_1] \cos(wt + \epsilon) & -a \leq x < 0 \\ B\sin[w(a - x)/c_2] \cos(wt + \epsilon) & 0 < x \leq a \end{cases} \]

where $c_1 = \sqrt{T/\rho_1}$, $c_2 = \sqrt{T/\rho_2}$, and $A$, $B$, and $\epsilon$ are constants.


c) Deduce that the normal frequencies are $w/2\pi$, where $w$ is any positive root of the equation \[ c_1 \tan\left( \frac{wa}{c_1}\right) + c_2 \tan\left( \frac{wa}{c_2}\right) = 0. \]

Show that when $c_2 = 2c_1$ there is just one solution for $w$ in the interval $(0, c_2\pi/2a]$.


3. Energy and uniqueness

The transverse displacement $u(x,t)$ of a stretched string of unit length satisfies the boundary conditions $u(0, t) = u(1, t) = 0$ and the wave equation \[ u_{tt} = u_{xx} \]

along $0 < x < 1$ and in which the wave speed $c = 1$. Show that the energy $E$, \[ E(t) = \frac{1}{2}\int_0^1 \left[ \left(\pd{u}{t}\right)^2 + \left(\pd{u}{x}\right)^2\right] \, \de{x} = \text{constant}. \]

From this, deduce that given an initial displacement and velocity for the string, the wave equation has at most one solution that satisfies fixed boundary conditions.

Next, find $u(x,t)$ in the form of an infinite series when $u(x, 0) = f(x) = x(1-x)$ and $g(x) = 0$. Show that the fraction of the energy which is communicated to the fundamental mode of vibration is $96/\pi^4$.