Contents12 Elastodynamics 212.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Basic Equations of Elastodynamics; Waves in a Homogeneous Medium . . . 312.2.1 Equation of Motion for a Strained Elastic Medium . . . . . . . . . . . 412.2.2 Elastodynamic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.2.3 Longitudinal Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . 912.2.4 Transverse Shear Waves . . . . . . . . . . . . . . . . . . . . . . . . . 1112.2.5 Energy of Elastodynamic Waves . . . . . . . . . . . . . . . . . . . . . 1112.3 Waves in Rods, Strings and Beams . . . . . . . . . . . . . . . . . . . . . . . 1512.3.1 Compression waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512.3.2 Torsion waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512.3.3 Waves on Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1612.3.4 Flexural Waves on a Beam . . . . . . . . . . . . . . . . . . . . . . . . 1712.3.5 Bifurcation of Equilibria and Buckling (once more) .......... 1812.4 Body Waves and Surface Waves — Seismology . . . . . . . . . . . . . . . . . 2212.4.1 Body Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.4.2 Edge Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.4.3 Green’s Function for a Homogeneous Half Space . . . . . . . . . . . . 3012.4.4 Free Oscillations of Solid Bodies ..................... 3212.4.5 Seismic tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.5 The Relationship of Classical Waves to Quantum Mechanical Excitations. . . 351Chapter 12ElastodynamicsVersion 1112.2.K, 16 January 2012 . Differs from 1112.1.K solely by fixing cross referencesto earlier chapters.Please send comments, suggestions, and errata via email to [email protected], or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 12.1Reader’s Guide• This chapter is a companion to Chap. 10 (Elastostatics) and relies heavily on it.• This chapter also relies rather hea vily on geometric-optics concepts and formalism,as developed in Secs. 7.2 and 7.3, especially:phasevelocity,groupvelocity,dis-persion relation, rays and the propagation of waves, information and energy alongthem, the role of the dispersion relationasaHamiltonianfortherays,andraytracing.• The discussion of continuum-mechanics wave equations in Box 12.2 underlies thisbook’s treatment of waves in fluids (Part IV), especially in Plasmas (Part V), andin general relativity (Part VI).• The experience that the reader gains in this chapter with waves in solids will beuseful when we encounter m uch more complicated waves in plasmas in Part V.• No other portions of this chapter are of great importance for subsequent Parts ofthis book.12.1 OverviewIn the previous chapter we considered elastostatic equilibria in which the forces acting onelements of an elastic solid were balanced so that the solid remained at rest. When this23equilibrium is disturbed, the solid will undergo accelerations. This is the subject of thischapter — Elastodynamics.In Sec. 12.2, we derive the equations of motion for elastic media, paying particular at-tention to the underlying conservation laws and focusing especially on elastodynamic waves.We show that there are two distinct wave modes that propagate in a uniform, isotropicsolid, longitudinal waves and shear waves, and both are nondispersive (their phase speedsare independent of frequency).Amajoruseofelastodynamicsisinstructuralengineering,whereoneencountersvi-brations (usually standing waves) on the beams that support buildings and bridges. InSec. 12.3 we discuss the types of waves that propagate on bars, rods and beams and findthat the boundary conditions at the free transverse surfaces make the waves dispersive. Wealso return briefly to the problem of bifurcation of equilibria (treated in Sec. 11.8) and showhow, by changing the parameters controlling an equilibrium, a linear wave can be made togrow exponentially in time, thereby rendering the equilibrium unstable.Asecondapplicationofelastodynamicsistoseismology(Sec.12.4).Theearthismostlyasolidbodythroughwhichwavescanpropagate. Thewavescanbeexcitednaturallybyearthquakes or artificially using man-made explosions. Understanding how waves propagatethrough the earth is important for locating the sources of earthquakes, for diagnosing thenature of an explosion (was it an illicit nuclearbombtest?)andforanalyzingthestructureof the earth. We briefly describe some of the wave modes that propagate through theearth and some of the inferences about the earth’s structure that have been drawn fromstudying their propagation. In the process, we gain some experience in applying the tools ofgeometric optics to new types of waves, and we learn how rich can be the G reen’s functionfor elastodynamic waves, even when the medium is as simple as a homogeneous half space.Finally (Sec. 12.5), we return to physics to consider the quantum theory of elastodynamicwaves. W e compare the classical theory with the quantum theory , specializing to quantisedvibrations in an elastic solid: phonons.12.2 Basic Equations o f Elastodynamics; Waves in aHomogeneous MediumIn subsection 12.2.1 of this section, we shall derive a vectorial equation that governs the dy-namical displacement ξ(x,t)ofadynamicallydisturbedelastic medium. We shall then spe-cialize t o monochromatic plane waves in a homogeneous medium (Subsec. 12.2.2) and shallshow how the monochromatic plane-wave equation can be converted into two wave equations,one for “longitudinal” waves (Subsec. 12.2.3) and the other for “transverse” waves (Subsec.12.2.4). From those two wave equations we shall deduce the waves’ dispersion relations,whic h act as Hamiltonians for geometric-optics wave propagation through inhomogeneousmedia. Our method of analysis is a special case of a very general approach to deriving waveequations in continuum mechanics. That general approach is sketched in Bo x 12.2. We shallfollow that approach not only here, for elastic waves, but also in Part IV for waves in fluids,Part V for waves in plasmas and Part VI for general relativistic gravitational waves. W e shallconclude this section in Subsec. 12.2.5 with a discussion of the energy density and energy4flux of these waves, and in Ex. 12.4 we shall explore the relationship of this energy densityand flux to a Lagrangian for elastodynamic waves.12.2.1 Equation of Motion
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