Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section B / B7

A dispersive wave has phase speed depending on frequency. Its phase velocity is $v_p=\omega /k$, whereas the group velocity $v_g=d\omega /dk$ describes a narrow wave packet in a lossless single branch. Examples are deep-water gravity waves, ${\omega }^2=gk$, and electromagnetic waves in a cold plasma, ${\omega }^2={\omega }_p^2+c^2{k}^2$.

The fixed-edge membrane waveguide has transverse profiles $\sin (n\pi y/b)$ withThere is no nonzero n=0 displacement satisfying both fixed-edge conditions. Even though the unconfined membrane is nondispersive, transverse confinement makes propagation along x dispersive. The membrane-guide phase and energy speeds are, for a propagating mode,For positive $k_x$, the phase speed exceeds v while group speed is below v. At cutoff $k_x=0$ the formal phase speed diverges, but group speed vanishes; no finite longitudinal energy flux is implied.

With the supplied width and speed, $f_{c,n}=10n\mathrm{kHz}$. At15kHz only n=1 propagates:The negative value represents travel in the opposite direction; higher transverse modes are evanescent.

At5kHz every transverse mode is below cutoff, so no propagating guided mode exists. A driven boundary can produce an evanescent near field. For n=1 the decay coefficient is $\sqrt{(20\pi {)}^2-(10\pi {)}^2}=10\pi \sqrt{3}=54.4{\mathrm{m}}^{-1}$, an e-folding distance of18.4mm. This is decay along the guide, not attenuation of an otherwise propagating lossless wave.

At25kHz, the n=1 and n=2 modes can propagate, withThey have different phase and group speeds, so a launch exciting both produces multimode propagation and possible beating. Actual mode amplitudes depend on the transverse launch profile; their existence does not guarantee both are excited. Modes n≥3 remain evanescent.