By Alex A. Kaufman, A.L. Levshin

This monograph is the final quantity within the sequence 'Acoustic and Elastic Wave Fields in Geophysics'. the former volumes released via Elsevier (2000, 2002) dealt ordinarily with wave propagation in liquid media.

The 3rd quantity is devoted to propagation of aircraft, round and cylindrical elastic waves in numerous media together with isotropic and transversely isotropic solids, liquid-solid versions, and media with cylindrical inclusions (boreholes). * occurrence of actual reasoning on formal mathematical derivations * Readers don't have to have a robust heritage in arithmetic and mathematical physics * special research of wave phenomena in a variety of kinds of elastic and liquid-elastic media

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**Example text**

Such motion causes a deformation of a medium in the vicinity of the horizontal faces and, correspondingly, the tangential (shear) forces F x and -F, T appear, which act on these faces, Fig. 8a. Of course, in equilibrium the total force and the resultant moment of all four forces are equal to zero. However, before equilibrium occurred, motion of the elementary volume resulted in a change of its shape. This means that the angle between intersecting faces varies, and instead of TT/2 it becomes TT/2 — 7.

Consider sinusoidal solutions of the wave equation, and as follows from eq. 115) It is clear that, all four sm(ut±kx) and cos{u)t±kx) functions obey the wave equation. 3 LONGITUDINAL SINUSOIDAL WAVES IN A BAR 41 Each term of this sum can be interpreted as a superposition of two sinusoidal waves with equal amplitudes traveling in opposite directions. 117) and this describes the standing wave because all points of the bar vary synchronously. Besides, there are points (nodes) at which motion is absent.

4c, "know" where the opposite end is located? In order to find an explanation, consider the motion of the low end under action of the constant force FT: Fr = 0 if t <0 and FT — const if t > 0. provided that the upper end of the bar is fixed. Since the force Fx is directed downward, an extensional wave arises at the instant t = 0 and propagates toward the upper end. At the same time, the low end moves with constant velocity along the :r-axis. Because of a reflection at the upper end, the extensional wave arises at the instant t = l/c and travels downward, but the particle motion has an opposite direction.