Past exam of the mathematical methods II course of the University of Oxford / 2021 / 11

Wave dispersion means that different frequencies propagate with different phase velocities, so a wave packet generally changes shape. Equivalently the dispersion relation is not proportional to $k$ over the relevant band. Three natural examples are the colour separation in a rainbow, where the refractive index of water depends on wavelength; deep-water surface gravity waves, for which ${\omega }^2=gk$ and long waves have larger phase velocity; and capillary waves, whose short-wave limit is ${\omega }^2=(\gamma /\rho )k^3$ and whose shorter waves have larger phase velocity. Here $g$ is gravitational acceleration, $\gamma$ is surface tension, and $\rho$ is fluid mass density, with positive $k$ and negligible viscosity.

A distinct application is an optical prism spectrometer: the wavelength-dependent refractive index produces different refraction angles, separating spectral components for measurement. This exploits wave dispersion deliberately.

Take constant complex amplitude $A$ and real wavenumber $k$ and angular frequency $\omega$. Substitution of the complex exponential givesThus every nonzero plane wave of the given form solves the equation precisely whenThe wavenumber specifies spatial wave phase change per unit length, with wavelength $\lambda =2\pi /|k|$. The angular frequency specifies temporal wave phase change per unit time, with period $2\pi /|\omega |$ and ordinary frequency $|\omega |/(2\pi )$. The constant $m$ in this equation has dimensions time divided by length squared; without an additional dimensional coefficient, it is not an ordinary physical mass.

For propagating oscillatory solutions assume real $m\ne 0$. If $m=0$, the equation becomes $y_{xx}=0$ and only $k=0$ among these spatial plane waves is possible; the usual dispersion relation branch and velocity discussion would degenerate.

A fixed wave phase satisfies $kx-\omega t=\text{constant}$, so the signed phase velocity is $v_p=\omega /k$. The group velocity $v_g=d\omega /dk$ gives the leading translation of the wave envelope of a narrow-band wave packet on a differentiable dispersion relation branch.

Here $v_p=k/m$. For the conventional positive branch $m>0$, $\omega >0$ and $k>0$,On the opposite propagation branch $k<0$ it is $-\sqrt{\omega /m}$, so $\sqrt{\omega /m}$ is the phase speed for either sign of $k$. More generally, for real $m\ne 0$ use $v_p=k/m$ with $k^2=m\omega$, allowing the sign of $\omega$ to follow that of $m$. The $k=0$ constant solution has no distinguished phase velocity.

For equal complex amplitudes, the superposition principle and the complex exponential formula for the cosine give the exact identityThe cosine is a real periodic modulation of the rapidly varying carrier. It varies slowly when $|\delta k|\ll |k_0|$ and $|\delta \omega |\ll |{\omega }_0|$. Its signed wave envelope travels with velocity $\delta \omega /\delta k$; its magnitude gives the visible amplitude envelope.

Equal magnitudes need not mean equal initial wave phases. If the two complex amplitudes are $a{e}^{i{\varphi }_{+}}$ and $a{e}^{i{\varphi }_{-}}$, write $\bar{\varphi }=({\varphi }_{+}+{\varphi }_{-})/2$ and $\delta \varphi =({\varphi }_{+}-{\varphi }_{-})/2$. The same result becomes $2a{e}^{i\bar{\varphi }}\cos (\delta kx-\delta \omega t+\delta \varphi )e^{i(k_{0}x-{\omega }_{0}t)}$, still with a real slowly varying modulation.

Apply the dispersion relation separately to the two components and subtract:In the narrow-band limit this is the group velocity, since $\omega (k)=k^2/m$ givesEquivalently on the positive branch $k=\sqrt{m\omega }$, $dk/d\omega =m/(2k)$ and its reciprocal is $2k/m$. The factor of two applies to signed velocities on either nonzero propagation branch.

There is a small but useful finite-band qualification. Adding the two component relations gives $m{\omega }_0=k_0^2+(\delta k{)}^2$. Thus the carrier formed by their exact midpoints has phase velocity ${\omega }_0/k_0=k_0/m+(\delta k{)}^2/(m{k}_0)$, and its finite-band envelope-to-carrier velocity ratio isThe exact derivative relation $v_g=2v_p$ refers to a single central point on the dispersion relation; the two-tone midpoint ratio approaches two as $\delta k/k_0\to 0$.

The result depends on the component amplitudes and wave phases; adding many frequencies does not by itself guarantee one localized wave packet. For a definite sketch, take many equally spaced wavenumbers $k_j=k_0+j\Delta k$ in the same fixed band, with equal, aligned complex amplitudes $B/N$. Linearize the dispersion relation as $\omega (k_j)\simeq {\omega }_c+v_g(k_j-k_0)$, where ${\omega }_c=k_0^2/m$. Then with $\xi =x-v_{g}t$ the wave envelope isfor odd $N$. This is a normalized Dirichlet kernel: constructive interference occurs near $\xi =0$, while most other positions undergo partial cancellation. At finite $N$ there are repeated groups separated by $2\pi /\Delta k$.

As $N$ grows with the half-bandwidth $K$ fixed, $\Delta k$ decreases and the repetition spacing grows. The continuous flat-band limit isThus a sinc function envelope has a central group between its first zeros at $\xi =\pm \pi /K$, smaller side lobes and rapidly oscillating crests. At fixed bandwidth the limiting central width is $2\pi /K$; it does not shrink indefinitely just because more components are added. Smoother spectral weights can reduce the side lobes, and random wave phases would produce a different shape.

For the exact quadratic dispersion relation,The additional quadratic wave phase makes the wave packet spread and change shape. The translating sinc function approximation is valid while $|t|K^2/|m|\ll 1$; the preceding figure is the exact $t=0$ comparison, where that approximation introduces no error.