Past exam of the mathematical methods II course of the University of Oxford / 2013 / 10

The terms $u_{tt}$ and $c^2{u}_{xx}$ must have the same dimensions. Likewise $au$ must match $u_{tt}$. ThusOnly $c^2$ occurs in the equation, so its characteristic speed is $|c|$ when $c\ne 0$.

For a nonzero plane-wave amplitude, substituting $u=A{e}^{i(kx-\omega t+{\theta }_0)}$ gives $u_{tt}=-{\omega }^{2}u$ and $u_{xx}=-k^{2}u$. ThereforeThis dispersion relation is not proportionality between frequency and wavenumber: for $a>0$, the phase speed depends on $k$. Thus a superposition generally experiences dispersion. The negative-frequency branch is its opposite; $A=0$ is the trivial solution and imposes no constraint on the unused phase parameters.

The real part $A\cos (kx-\omega t+{\theta }_0)$ represents a monochromatic travelling wave for $k\ne 0$. The magnitude $|A|$ is its displacement or field amplitude, and $\theta =kx-\omega t+{\theta }_0$ is its phase, with ${\theta }_0$ an initial phase offset. The complex exponential is a convenient representation of that real oscillation.

Following a constant-phase point gives $kdx/dt-\omega =0$, soOn the positive-frequency, positive-$k$ branch,For negative $k$, $\omega /k$ is negative on that frequency branch and encodes the reversed direction. At $k=0$, the mode is spatially uniform with frequency $\sqrt{a}$, so a propagation phase speed is not defined.

Differentiate the positive-frequency dispersion relation with respect to $k$:Where $k\ne 0$, $c_p{c}_g=c^2$. For $a>0$ and $c\ne 0$, the magnitude of the group velocity is below $|c|$ even though the positive-$k$ phase velocity exceeds $|c|$. At $k=0$, the group velocity is zero. The two velocities describe envelope and phase transport, respectively.

A local-wave description requires an identifiable smooth phase $\Theta (x,t)$ with $\bar{k}={\Theta }_x$ and $\bar{\omega }=-{\Theta }_t$. Real propagating wavenumber and frequency must satisfy the chosen local branch of ${\bar{\omega }}^2=c^2{\bar{k}}^2+a$. In a geometrical optics or slowly modulated wavepacket approximation, these quantities and the amplitude vary on scales large compared with a local wavelength and period; for example $|{\partial }_x\bar{k}|\ll {\bar{k}}^2$ and $|{\partial }_t\bar{\omega }|\ll {\bar{\omega }}^2$ away from degenerate points.

The phase compatibility in geometrical optics means the local fields cannot be specified independently: equality of mixed phase derivatives requires ${\bar{k}}_t+{\bar{\omega }}_x=0$, derived as wave conservation below. These phase and scale conditions justify the local dispersion relation; they alone are not a guarantee of an arbitrary-amplitude exact PDE solution. Amplitude transport and higher-order envelope terms must also be satisfied.

On a locally right-moving branch with $\bar{k},\bar{\omega }>0$, a spatial phase increase of $2\pi$ counts one wave. The local number of waves per unit length and the number passing a fixed point per unit time areThe physical unsigned counts use absolute values on other branches; the signed phase convention is useful for the conservation law. A phase change around a small spacetime rectangle must close. Equivalently, ${\bar{k}}_t={\Theta }_{xt}$ and ${\bar{\omega }}_x=-{\Theta }_{tx}$, soThis wave conservation equation states that the wave count in an interval changes by the crest flux through its ends, provided the smooth phase description has no defects or creation of crests inside. It is phase compatibility, distinct from energy conservation or wave-action conservation.

For a medium with constant coefficients, $\bar{\omega }=\omega (\bar{k})$ on the selected branch. The spatial chain rule gives ${\bar{\omega }}_x=(d\omega /dk{)}_{\bar{k}}{\bar{k}}_x=c_g(\bar{k}){\bar{k}}_x$. Thus wave conservation becomesAlong a characteristic curve with $dx/dt=c_g(\bar{k})$, the total derivative $d\bar{k}/dt$ vanishes. A slowly varying wavenumber feature, and hence the corresponding modulation of a narrow-band packet, travels with the group velocity rather than the carrier's phase velocity. This interpretation assumes the modulation remains smooth on the scale required in part (e).