Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2013 / Paper 1 / 3

For group velocity from a period-dependent phase velocity, start with $k=\omega /c_n(\omega )$ rather than equating group velocity to phase velocity. Differentiating,For the stated inverse-frequency term, $d{c}_n/d\omega =-a/{\omega }^2$, soThe inequality assumes positive frequency and physical positive phase velocity, with $a>0$. The symbol $a$ here has units of acceleration and is unrelated to the plate thickness in Question 1.

Write the oscillation period as $P=2\pi /\omega$, so locally $c_n=c_o+(a/2\pi )P$. The first overtone is the lowest curve above the fundamental, not the fundamental mode itself. Near $P=50\mathrm{s}$, Figure 1 gives approximately $c_1=5.13\mathrm{km}{\mathrm{s}}^{-1}$. Representative nearby readings are about $4.96\mathrm{km}{\mathrm{s}}^{-1}$ at $40\mathrm{s}$ and $5.30\mathrm{km}{\mathrm{s}}^{-1}$ at $60\mathrm{s}$. ThusThe local group-velocity estimate isThese are graphical estimates, roughly $a=0.11\pm 0.02\mathrm{km}{\mathrm{s}}^{-2}$ and $u_1=4.4\pm 0.1\mathrm{km}{\mathrm{s}}^{-1}$, rather than high-precision fitted constants. The nearby straight-line approximation need not describe that mode over the whole plotted period range. No source curve is reproduced here.

The phase velocity in Figure 1 is measured along the Earth's surface. Spherical geometry makes the local horizontal phase speed at radius $r$ equal to $cr/R_E$. In the ray approximation, squared radial shear wavenumber isThe spherical surface-wave evanescence threshold immediately below the discontinuity is thereforeEquality is the grazing limit. Using $5.08\mathrm{km}{\mathrm{s}}^{-1}$ directly would omit the requested spherical correction. Immediately above the discontinuity the corresponding boundary is $4.87(6371/5961)\simeq 5.20\mathrm{km}{\mathrm{s}}^{-1}$. Hence the interval $5.20<c<5.43\mathrm{km}{\mathrm{s}}^{-1}$ specifically describes propagation just above the jump and evanescence just below it. Slower modes can already be evanescent above $410\mathrm{km}$.

Intersecting the plotted branches with $5.43\mathrm{km}{\mathrm{s}}^{-1}$ gives the following approximate upper periods. Below each upper period, that branch is evanescent immediately beneath the discontinuity:

For these overtones the displayed ranges thus start near $10\mathrm{s}$ and end at their respective crossings. Any still higher curve already above the threshold at the lower plotted period has no evanescent interval within the displayed range. Readings of the closely spaced higher curves are only accurate to about a second, and the plot does not establish their continuation to unshown periods. Nor does the local inequality prove that a mode remains evanescent at every greater depth in an unspecified Earth model.

Use the flat two-layer model, with thickness $h=30\mathrm{km}$, crustal shear speed ${\beta }_1=3.5\mathrm{km}{\mathrm{s}}^{-1}$ and mantle speed ${\beta }_2=4.5\mathrm{km}{\mathrm{s}}^{-1}$. For SH-wave travel times in a crustal waveguide, unfolding each reflected path gives all reflection orders at once:with $x$ in kilometres and times in seconds. The direct wave begins at the origin; reflected orders have intercepts $60m/3.5=17.14m\mathrm{s}$ and approach the direct-wave slope at large distance.

The critical angle, measured from the vertical, is $i_c=\arcsin (3.5/4.5)\simeq 51.1^{\circ }$. Associated critical refraction and its multiples giveEach head-wave line is tangent to its reflection curve at the critical distance. In particular the primary head wave exists before it becomes the first arrival; it overtakes the direct wave only around $170\mathrm{km}$.

The slope of a reflected travel-time curve is $p=d{t}_m/dx=\sin i/{\beta }_1$. For $x<x_{c,m}$, $p<1/{\beta }_2$, so the mantle supports outgoing shear radiation and successive reverberations leak energy. For $x>x_{c,m}$, $1/{\beta }_2<p<1/{\beta }_1$: the mantle field is evanescent and ideal interface reflection is total. These leaking and trapped SH-wave reverberations explain the annotations in the original travel-time drawing below. A reflected ray is not itself a separate Love mode: constructive interference of the trapped contributions, including reflection phase, selects the normal modes.

For a horizontally propagating mode with ${\beta }_1<c<{\beta }_2$, define vertical wavenumbersThe free upper surface and welded interface impose the Love-wave dispersion conditionwhere ${\mu }_j={\rho }_j{\beta }_j^2$ are shear moduli. Each phase-velocity branch falls towards $3.5\mathrm{km}{\mathrm{s}}^{-1}$ with increasing frequency. Higher branches join $4.5\mathrm{km}{\mathrm{s}}^{-1}$ at their low-frequency cutoffs. At cutoff $s\to 0$, soThus the first three overtones have maximum periods about $10.77$, $5.39$ and $3.59\mathrm{s}$. The fundamental mode has $n=0$ and no positive-frequency cutoff in this flat layer-over-half-space model: it approaches $c={\beta }_2$ as $f\to 0$, with ever-deeper penetration. The phase-velocity sketch uses an explicitly illustrative equal-density choice, because the question gives speeds but no density or rigidity ratio. That missing ratio changes detailed branch curvature, not the cutoff values or the requested branch topology.

For the actual finite spherical Earth, the very-long-period continuation must instead be described by discrete toroidal normal modes, not a flat infinite waveguide at arbitrary wavelength. The flat-model answer therefore does not assert a continuum of arbitrarily low-frequency global nonrigid oscillations. A global lowest oscillation frequency cannot be computed from the supplied crust-and-half-space data alone.^[ref]