free oscillations

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free oscillations After a large earthquake the Earth rings like a bell. The earthquake sets up standing waves that can persist for weeks, each with a characteristic oscillation period measured in minutes. These are called free oscillations because they continue without any forcing, unlike tides for example, which are continuously driven by the Moon's gravitational attraction. Since the discovery of free oscillations in 1960, the periods of many hundreds have been measured precisely. They depend on details of the Earth's internal structure, and consequently have told us much about the Earth's interior.

Plucking a guitar string sets up standing waves on the string with an integral number of wavelengths between the fixed ends. The standing wave with the longest wavelength is called the fundamental; the overtone number keeps count of the number of half-wavelengths. Points where the string remains stationary are called nodes. The frequency (pitch) is determined by the wavelength, which is altered by changing the length of the string using the frets. Free oscillations of a sphere are three-dimensional standing waves and require three overtone numbers. The nodes are surfaces lying on spheres (constant radius), cones (constant latitude), and meridians (constant longitude): see Fig. 1. The radial overtone number (n) counts the number of spherical nodal surfaces and the angular order number (l) counts the rest. The azimuthal order number (m) counts those on meridians, leaving l–m on cones of constant latitude. The ‘wavelength’, and therefore the frequency, depends on n and l; m merely keeps track of the distribution of nodal surfaces with respect to geographic north. Many free oscillations therefore have the same frequency and are said to be degenerate. The theory for a uniform sphere was worked out around the end of the nineteenth century. In 1882 H. Lamb showed there were two types of free oscillation, called spheroidal and torsional, the latter having purely horizontal motion with no compression. In 1911 A. E. E. Love determined a period of 60 minutes for the Earth's slowest oscillation.

Observation of the waves was held up by the technical difficulty of recording long periods: rapid vibrations are much easier to detect (they make cups rattle on shelves, for example), but periods of minutes to an hour require a very stable instrument. By 1952 H. Benioff was detecting long-period oscillations with his strainmeter, but positive identification had to await the very large Chilean earthquake of May 1960. Several research groups announced measurements at the 1960 meeting of the International Association of Seismology and Physics of the Earth's Interior (IASPEI) in Helsinki, heralding a new era of long-period seismology. Some of the frequencies observed with strainmeters were missing from gravimeter records: these were the torsional oscillations, which do not register on a gravimeter because their motion is purely horizontal.

Over a thousand modes, with periods ranging from 56 minutes down to less than 40 seconds, have been identified using the present network of long-period seismometers and gravimeters. Any record of a moderate-sized earthquake may be transformed from time to frequency to reveal a comb of peaks at each free oscillation frequency (Fig. 2). The frequencies are in close agreement with theoretical predictions calculated using detailed models of the Earth's internal structure.

In 1975, F. Gilbert and A. M. Dziewonski used frequencies of 1044 free oscillations to refine our understanding of the internal structure still further, including a demonstration of the solidity of the Earth's inner core. The Earth is not quite a perfect sphere, and departures from spherical symmetry split the degeneracy: each single frequency peak becomes a ‘fine structure’ of several closely spaced peaks. The fine structure can be explored using high-resolution seismograms. Splitting studies have revealed a large-scale departure from spherical symmetry in the transition zone of the mantle.

The peaks in Fig. 2 are fundamentals: they have no nodes in radius. Fundamentals are set off by shallow earthquakes; radial overtones require a deep earthquake source and are relatively rare. On 9 June 1994 the largest deep earthquake in recorded history occurred beneath Bolivia. Although it caused little damage, it was felt as far away as Canada. It registered on the entire global network of modern instruments, including many temporary field stations, and has provided data for the identification of new free oscillations and their fine structure.

David Gubbins

Bibliography

Dahlen, F. A. and and Tromp, J. (1998) Free oscillations of the Earth. Princeton University Press.

free oscillations The harmonics at which any body, e.g. the Earth, tends to vibrate most freely, i.e. resonates. There are two fundamental types: torsional (vibration with motions perpendicular to the Earth's radius); and spheroidal (vibrations that are both radial and tangential to the Earth's surface). The study of such resonances, e.g. those induced by major earthquakes, provides information on the internal nature of the Earth. A major earthquake can make the entire globe vibrate or ring like a bell, and some earthquakes have been so large that sensitive seismometers have continued to record the oscillations for weeks after the event. The decay of the vibrations gives valuable information about the elastic layering of the Earth, and especially of the low velocity zone. Moonquakes produce similar phenomena.