## Acoustic and Elastic Wave Fields in Geophysics, III by Alex A. Kaufman, A.L. Levshin

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 by means of 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 several 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 should not have to have a robust historical past in arithmetic and mathematical physics * exact research of wave phenomena in a number of forms of elastic and liquid-elastic media

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3a. Suppose that the wave propagates along the :r-axis and reaches the cross-section S(x - dx/2). From this moment we begin to observe deformation of this bar element, dV = Sdx. With some time delay, the wave arrives at the front face of the volume, S(x+ dx/2). Force F(x, t), which accompanies the wave, has the same direction at both crosssections, but may differ in magnitude. 2 LONGITUDINAL WAVES IN A THIN BAR 17 takes place. The force at points of the section S(x + dx/2) acts on a medium located in front of the bar element.

107) and the influence of E vanishes. Consider, as in example four, a transition from an elastic bar to an ideally rigid one. As we already know, with an increase of the Young modulus the wave velocity increases, but the particle velocity becomes smaller. In other words, with an increase of E, both the particle velocity VQ and the time interval T — l/ci, during which the velocity v(t) remains constant, decrease. At the same time the product QVO is preserved. Respectively, in the limiting case of an ideally rigid body Q —> oo and VQ —> 0 the velocity becomes a continuous function V(t), and it describes the motion of all points of the bar.

111 both frequencies are related to each other. If the wave velocity Q is frequency-independent, then there is a linear relationship between k and UJ. In a dispersive medium, where the wave velocity is a function of UJ, this relation becomes more complicated. Next let us describe fields that accompany the sinusoidal waves. First of all, they do not have a beginning or an end. 113) Xx(x, t) = —A kEcos(ojt - kx) Similar formulas describe the incoming sinusoidal wave. Of course, as in a general case of nonstationary waves, eqs.