Normal modes and waves (PS2): The wave equation and dispersive waves

1. Sketching d'Alembert's solution

At time $t = 0$, the displacement of an infinitely long string is defined as: \[ y(x,t) = \begin{cases} \sin(\pi x/a), & -a \leq x \leq a \\ 0 & \text{otherwise} \end{cases} \]

and the string is initially at rest. Using d'Alembert's solution, and assuming that the waves may move along the string with speed $c$, sketch the displacement of the string at $t = 0$, $t = a/2c$, and $t = a/c$.


2. Energy for the wave equation

Two transverse waves are on the same piece of string. The first and second have displacements \begin{align*} y_1 &= A\sin(kx + \omega t) & kx + \omega t \in [\pi, 2\pi] \\ y_2 &= A\sin(kx + \omega t) & kx + \omega t \in [-2\pi, -\pi], \end{align*}

and zero outside their intervals. Calculate the energy of the two waves at $t = 0$. What is the displacement of the string at $t = 3\pi/(2\omega)$? What is the energy here?


3. Moving and stationary waves

What is the difference between a moving wave and a stationary wave?


4. Dispersive, non-dispersive, phase velocity, and group velocity

What is meant by a) a dispersive medium, and b) phase velocity, $c_p$? Explain the relevance of group velocity $c_g$ for the transmission of signals in a dispersive medium. Justify the equation \[ c_g = \dd{w}{k}. \]


5. Relativistic motion

In quantum mechanics, a particle of momentum $p$ and energy $E$ has associated with it a wave of wavelength $\lambda$ and frequency $f$ given by \[ \lambda = \frac{h}{p}, \qquad f = \frac{E}{h}, \]

where $h$ is Planck's constant. Find the phase and group velocities of these waves given that \[ p = \frac{m_0 v}{\sqrt{1-(v/c)^2}}, \qquad E = \frac{m_0c^2}{\sqrt{1-(v/c)^2}}, \]

where the particle's rest mass is $m_0$ and its speed is $v$, and where $c$ is the speed of light.


Show that \begin{align} c_g &= c_p + k \dd{c_p}{k} \notag \\ c_g &= c_p - \lambda \dd{c_p}{\lambda} \label{cg2} \\ c_g &= \frac{c}{\mu}\left(1 + \frac{\lambda}{\mu} \dd{\mu}{\lambda}\right). \notag \end{align}

Assuming that $\mu$ is the refractive index for waves of wavelength $\lambda$ and wavenumber $k$ (in the medium), show that \[ c_g = c_p \left[ 1 - \frac{1}{1+\frac{c_p}{\lambda'}\dd{\lambda'}{c_p}}\right], \]

where $\lambda'$ is the wavelength in vacuum.


7. Kinetic and potential energy of a string

Show that the kinetic energy $U$ and the potential energy $V$ for string of density $\rho$, tension $T$, and length $L = 2\pi/k$ subjected to transversal movement are given by \[ U = \int_0^L \frac{\rho}{2} \left(\pd{y}{t}\right)^2 \, \de{x}, \qquad V = \int_0^L \frac{T}{2}\left( \dd{y}{x}\right)^2 \, \de{x}. \]

Evaluate these for the wave $y = A\cos(kx + \omega t + \phi)$ and show that $U = V$.