Subsonic Free Surface Waves in Linear Elasticity

Abstract

For general anisotropic linear elastic solids with smooth boundaries, Rayleigh-type surface waves are studied. Using spectral factorizations of matrix polynomials, a self-contained exposition of the case of a homogeneous half-space is given first. The main result is about inhomogeneous anisotropic bodies with curved surfaces. The existence of subsonic free surface waves is shown by giving ray series asymptotic expansions, including formulas for the transport equation. Introduction. Rayleigh[29] discovered waves which propagate along a plane traction-free surface of an isotropic and homogeneous elastic solid. The surface wave speeds are subsonic, that is, they are strictly less than the wave speeds of interior body waves. Furthermore, the amplitudes attenuate exponentially with distance to the surface. Synge [31] raised doubts about the existence of Rayleigh-type waves in anisotropic solids. Stroh [30] pointed out that these doubts were unfounded, and he introduced a sextic eigenvalue problem which became useful in the theory of free surface waves in anisotropic solids. In the early 1970s, the existence and uniqueness problem of free surface homogeneous plane waves in a semi-infinite half-space was settled by Barnett, Lothe, and coworkers. For any given horizontal propagation direction they showed that there is at most one free surface wave speed, and they gave criteria for the existence of such waves. Lothe and Barnett [24] rederived their results by the surface impedance tensor method. The surface impedance tensor relates the surface displacement to the surface traction required to sustain it. This tensor was introduced by Ingebrigtsen and Tonning [19]. Detailed presentations of the existence and uniqueness results were given by Chadwick and Smith [7] and Barnett and Lothe [6]. Much later, Mielke and Fu [25] simplified some proofs of the Barnett-Lothe theory by using a Ricatti equation satisfied by the impedance tensor. A crucial property of the tensor, the positive definiteness of its real part, follows from an integral identity which, in the original treatments, arises somewhat magically by averaging over rotations in the plane spanned by the normal to the surface and the propagation direction. Existence of subsonic free surface waves was shown by Kamotskiȋ and Kiselev [22] with a completely different approach based on the variational principle.Concerning Rayleigh-type waves in inhomogeneous elastic solids with curved boundaries, Petrowsky [28] exhibited the following locality principle: If a surface wave exists, the velocity of its discontinuity at a given point must be equal to the velocity in the homogeneous elastic half-space which is obtained by freezing the elastic parameters at that point. This locality principle is efficiently implemented by asymptotic ray methods, which substitute, in the high-frequency regime, standard plane waves by geometrical-optical "plane waves." For inhomogeneous, isotropic elastic solids with

Rayleigh and Stoneley Waves in Linear Elasticity

“…Roughly speaking, Rayleigh waves can be regarded as a special (limit) case of Stoneley waves. Both geophysical and mathematical works have been done for these two kinds of waves, see [15,16,2,20,1,8,13,14,24,28,7,9,2,3,26,5,18,17] and their references. Most geophysical works on them are considering specific situations, for example, the case of flat boundaries, plane waves, or homogeneous media.…”

“…Kazuhiro Yamamoto in [28] shows the existence of Stoneley waves as the propagation of singularities in two isotopic media with smooth arbitrary interfaces. Sönke Hansen in [7] derives the Rayleigh quasimodes by the spectral factorization methods for inhomogeneous anisotropic media with curved boundary and then in [9] shows the existence of Rayleigh waves by giving ray series asymptotic expansions in the same setting. In particular, the author derives the transport equation satisfied by the leading amplitude which represents the term of highest frequency.…”