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.6.7; another derivationwill be given when we study fluid sound waves in Chap.15.****************************EXERCISES 11Exercise 11.1 Example: Scalar and Vector Potentials for Elastic Waves in a HomogeneousSolidJust as in electromagnetic theory, it is sometimes useful to write the displacement ¾ in termsof scalar and vector potentials,¾ = "È + " × A.(11.25)(The vector potential A is, as usual, only defined up to a gauge transformation, A ’! A+"Õ,where Õ is an arbitrary scalar field.) Show that the scalar and vector potentials satisfy thefollowing wave equations in a homogeneous solid:"2È "2A= c2 "2È , = c2 "2A.(11.26)"t2 L "t2 TThus, the scalar potential È generates longitudinal waves, while the vector potential Agenerates transverse waves.Exercise 11.2 Problem: Influence of gravity on wave speedModify the wave equation (11.12) to include the effect of gravity.Assume that the gravita-tional field is constant.By comparing the orders of magnitude of the terms involving smallperturbations to the displacement, stress and density verify that the gravitational terms canbe ignored for short wavelength elastodynamic modes.****************************11.4 Waves in Rods, Strings and BeamsLet us now illustrate some of these ideas using the type of waves that can arise in somepractical applications.In particular we discuss how the waves can be modified when themedium through which they propagate is not uniform but instead is bounded.Despite thissituation being formally global in character (cf.Box 11.3), elementary considerationsenable us to derive the relevant dispersion relations without much effort.11.4.1 Compression wavesFirst, consider a longitudinal wave propagating along a light, thin, unstressed rod.Introducea Cartesian coordinate system with the z-axis parallel to the rod.When there is a smalldisplacement ¾z independent of x and y, the restoring stress is given by Tzz = -E"¾z/"z,where E is Young s modulus (cf.end of Sec.10.4).Hence the restoring force density can beevaluated from the divergence of the stress tensor as fz = E"2¾z/"z2.The wave equationthen becomes"2¾z E "2¾z= , (11.27)"t2 Á "z2and so the sound speed for compression waves in a long straight rod is12EcC =.(11.28)Á 12
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