Seismic Waves

Seismic Waves 

oscillations propagating in the earth from the foci of earthquakes, explosions, and other sources. Near the foci of strong earthquakes, seismic waves have destructive force over a period of tenths of a second. At significant distances from the epicenters, seismic waves are elastic waves.

Figure 1. Block diagram of oscillations in a compressional wave (a) and a shear wave (b)

Compressional waves (P waves) transmit changes of volume in a medium; that is, they transmit compression and extension. Oscillations in such waves occur in the direction of propagation (Figure l, a). Shear waves (S waves) do not cause volumetric changes in the medium; they are oscillations of particles in a direction perpendicular to the direction of wave propagation (Figure l, b). At each moment and at each point in the medium, seismic oscillations satisfy the wave equations (for P and S waves). In a homogeneous, isotropic, elastic medium, the propagation velocities of compressional (a) and shear (b) waves are determined by the formulas

Here, k + (4/3)μ = λ + 2μ, k is the bulk modulus, and λ and μ are Lame constants; μ is referred to as the shear modulus. The velocity of compressional waves is greater than that of shear waves.

The following phenomenon is characteristic of the propagation of seismic waves (elastic waves in a solid medium): when such waves obliquely strike the boundary between media with different parameters (such as propagation velocities and densities) for a given type of wave (such as a compressional wave), reflected and refracted shear waves arise, in addition to reflected and refracted compressional waves (Figure 2). Surface waves arise in the earth near the surface of a boundary. When an inhomogeneous SH wave is propagated along a horizontal layer a Love wave arises. When a P wave strikes the plane of a boundary, reflected P and SV waves may arise in the layer. In this case, if α₂> β₂ > α₁ > β₁, where α₁ and β₁ are velocities in the layer and α₂ and β₂ are velocities in a nonadjacent medium, then the reflected SV wave for a small e₁ possesses the property of total internal reflection, just as the reflected P wave does. As a result, Rayleigh waves form in the layer. Like Love waves, Rayleigh waves are dispersive of velocity. They arise in a half-space without stratification. Rayleigh waves do not disperse and their velocity c ≈ 0.9 β.

Figure 2. Reflection and refraction of P waves at a boundary

P and S waves are propagated from the source through the body of the earth and are called body waves. For a homogeneous and isotropic medium, their amplitude diminishes in inverse proportion to the distance. Surface waves propagate along the surface and possess an amplitude that diminishes in inverse proportion to the square root of the distance. For this reason, surface waves are dominant in terms of amplitude for oscillations from remote earthquakes.

As a result of changes in the properties of the earth with increasing depth, the propagation velocities of body waves also change; this brings about refraction of the waves in the earth’s interior.

Figure 3. Velocity of compressional (P) and shear (S) waves as a function of depth

Observations of the propagation of seismic waves made on the earth’s surface make it possible to study the structure of the earth. The relation between the propagation velocity of P and S waves and depth (Figure 3) has made it possible to identify a series of structural shells in the earth.

REFERENCES

Savarenskii, E. F., and D. P. Kirnos. Elementy seismologii seisomometrii, 2nd ed. Moscow, 1955.
Bullen, K. E. Vvedenie v teoreticheskuiu seismologiiu. Moscow, 1966. (Translated from English.)
Savarenskii, E. F. Seismicheskie volny. Moscow, 1972.
Brekhovskikh, L. M. Volny v sloistykh sredakh, 2nd ed. Moscow, 1973.

E. F. SAVARENSKII