abstract

The transverse oscillations of a slightly heterogeneous, smooth string are treated using geometrical optics and first order normal mode perturbation theory. It is found that first order perturbation theory is invalid at high frequencies just as geometrical optics is invalid at low frequencies. The eigenfrequencies of the high-frequency modes are predicted accurately by first order perturbation theory, but the eigenfunctions are not. By analogy, a similar failure will presumably result if first order perturbation theory is used to treat the effect of the Earth's lateral heterogeneity on the high-frequency, surface wave-equivalent modes nSl and nTl, with n ≪ l. To gain insight into the consequences of this failure, we go on to analyze the one-dimensional analog of the earthquake excitation problem, using both normal mode, or standing wave, and traveling wave methods. We show that a simplified version of geometrical optics, in which amplitudes are only calculated to zeroth order but phases are calculated to first order, leads naturally to two methods whereby the source mechanism of a “stringquake” can be retrieved without knowing the structure of the string. Analogous methods for determining earthquake source mechanisms on a laterally heterogeneous Earth could easily be developed, if they were ever required.

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