Consider the Love-wave model of this chapter: a layer of thickness and shear speed…

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Problem

Consider the Love-wave model of this chapter: a layer of thickness

H

and shear speed

v_{S1}

welded to a half-space of shear speed

v_{S2} > v_{S1}

, with the Love-wave dispersion relation

\tan\!\left(\omega H\sqrt{\frac{1}{v_{S1}^2}-\frac{1}{c^2}}\right) = \frac{\mu_2\sqrt{1/c^2-1/v_{S2}^2}}{\mu_1\sqrt{1/v_{S1}^2-1/c^2}}\,,\qquad v_{S1}<c<v_{S2}.

The chapter solves this for the lowest root only, the fundamental mode, and states that a family of higher roots exists at shorter periods without finding them. ((a)) Write

x=\omega H\sqrt{1/v_{S1}^2-1/c^2}

. The fundamental branch has

0<x<\pi/2

. Explain why a second branch,

\pi<x<3\pi/2

, also solves the dispersion relation (the right-hand side is always positive for

v_{S1}<c<v_{S2}

), and why this branch describes the first higher (overtone) Love mode. ((b)) Show that this first higher mode exists only above a cutoff angular frequency

\omega_1

, found by setting

x=\pi

at the point where

c\to v_{S2}

(the right-hand side vanishes there), and that the corresponding cutoff period is

T_1 = 2H\sqrt{\frac{1}{v_{S1}^2}-\frac{1}{v_{S2}^2}}.

((c)) Evaluate

T_1

for the continental model of this chapter,

H=35\,\mathrm{km}

v_{S1}=3.5\,\mathrm{km\,s^{-1}}

v_{S2}=4.5\,\mathrm{km\,s^{-1}}

, and say whether the

T=20\,\mathrm{s}

wave computed in the text for this model could be mistaken for the first higher mode instead of the fundamental.

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