Past exam of the mathematical methods II course of the University of Oxford / 2019 / 9

Assume small transverse slopes, a flexible string, constant tension $T$ and uniform linear mass density $\rho$. For an element of equilibrium length $dx$, the vertical force is the difference of the vertical tension components at its ends. To first order in slope,Its mass is $\rho dx$ and its transverse acceleration is $y_{tt}$. Newton's second law therefore gives $\rho dx{y}_{tt}=Tdx{y}_{xx}$, orThis is the one-dimensional wave equation for small-amplitude transverse waves. Changes in string length and horizontal acceleration are second order in slope and are neglected in this linear approximation.

At distance $x$ below the support, the cable must carry the weight of the remaining length $L-x$. Thus its equilibrium tension is $T(x)=\mu g(L-x)$, and the local transverse wave speed isThe linear mass density cancels. More precisely the small-displacement equation with variable tension is $\mu y_{tt}={\partial }_x[T(x)y_x]$; its local characteristic speed is the expression above, even though a first-spatial-derivative term is also present.

The wave travel time on a hanging uniform cable is obtained from $dt=dx/c(x)$:Although the local speed tends to zero at the free end, the endpoint singularity in the travel-time integral is integrable. This finite result belongs to the ideal flexible-cable model, with bending stiffness neglected.

A dispersive medium has frequency-dependent phase velocity, so different Fourier components generally propagate with different phases. A dispersion relation specifies $\omega$ as a function of the wave number $k$, or an equivalent relation between them. For a monochromatic factor $e^{i(kx-\omega t)}$, a point of fixed phase moves atFor a narrow wave packet centered at $k_{\ast }$, expand $\omega (k)\simeq \omega (k_{\ast })+(k-k_{\ast }){\omega }^{\prime }(k_{\ast })$. The envelope then translates at the group velocityThe phase velocity tracks crests of a monochromatic wave. The group velocity tracks a slowly varying envelope when the narrow-band approximation is valid; in the ideal lossless guide here it also equals the longitudinal energy-transport speed. It is not a universal formula for signal-front speed in every possible dispersive medium.

For the propagating branch of the supplied electromagnetic waveguide relation, take ${\omega }_0>0$, $k>0$ and $\omega =\sqrt{{\omega }_0^2+c^2{k}^2}$. Differentiation givesThe product alone would also allow $v_p=v_g=c$; the strictly positive cutoff is what makes the inequalities strict. At $k=0$ the phase-speed expression is undefined, and these formulas are interpreted through the propagating limit.

A moving monochromatic crest does not carry a new message. An information-bearing modulation is made from a range of frequencies; here its packet and energy move below $c$, and the high-frequency limit approaches $c$. Causal propagation of a newly switched-on disturbance is governed by the electromagnetic equations, so the superluminal phase velocity does not permit faster-than-light signalling. These are the waveguide phase and group velocities for a mode with nonzero cutoff, rather than a zero-cutoff mode with both velocities equal to $c$.