An introduction to wave propagation in anisotropic media

Summary. Wave motion in an anisotropic solid is fundamentally different from motion in an isotropic solid, although the effects are often subtle and difficult to recognize. There are such a wide range of three-dimensional variations possible in anisotropic media that it is difficult to understand the behaviour of wave motion without experimentation. Laboratory experiments are very difficult to construct and extensive numerical experiments have now given many theoretical insights so that the behaviour of waves in anisotropic media is now comparatively well understood. This introduction summarizes some of the relationships and insights required for this understanding.     

Seismic body waves in anisotropic media: propagation through a layer

Summary. The square-root energy ratios and pulse shapes are presented for P, SV and SH waves transmitted through a layer of orthorhombic olivine between two isotropic half-spaces. Off incident planes of symmetry, incident P waves generate two small amplitude SH waves (one from each interface), whose amplitudes decrease slowly with increasing period. Incident SV (or SH ) waves can generate large amplitude SH (or SV ) waves which decrease rapidly with increasing period. For incident S waves, many pulses not present in isotropic models are generated, often of large relative amplitude, with many of the transmitted S pulses showing evidence of double arrivals, either in the form of S-wave splitting, or a modification of the shape of the input waveform.     

Effective anisotropic elastic constants for wave propagation through cracked solids

Summary. Theoretical developments of Hudson demonstrate how to calculate the variations of velocity and attenuation of seismic waves propagating through solids containing aligned cracks. The analysis can handle a wide variety of crack configurations and crack geometries. Hudson associates the velocity variations with effective elastic constants. In this paper we associate the variation of attenuation with the imaginary parts of complex effective elastic constants. These complex elastic constants permit the simulation of wave propagation through two-phase materials by the calculation of wave propagation through homogeneous anisotropic solids.     

Geometrical structure of shear wave surfaces near singularity directions in anisotropic media

There are three types of surfaces which are used for studying wave propagation in anisotropic media: normal surfaces, slowness surfaces and wave surfaces. Normal surfaces and slowness surfaces have been researched in detail. Wave surfaces are the most complicated and comparatively poorly known compared with the other two. Areas of complicated geometrical structure of the wave surfaces are located in the vicinity of conical acoustic axes. There is an elliptical hole on the quick shear wave surface and complicated folds and cusps on the slow shear wave surface. Decomposition of the slow shear wave surface into smooth sheets is used for the study of its geometrical structure. Complexity of shear wave surfaces can be expressed by the number of waves corresponding to a fixed ray. An original approach to the calculation of wave normals depending on ray direction is presented.     

Guided wave propagation in laterally varying media - I. Theoretical development

Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH -waves, P - SV -waves and full anisotropy.
The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg - and Sn -waves in a multilayered model for which there are 19 coupled modes.
The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.