Solutions for problems involving wave propagation in a semi-infinite half space are of interest for interpreting measurements of radiation fields at locations near or on the free surface. Solutions to these problems as derived for elastic media have formed the basis for the initial interpretation of seismograms and resultant inferences concerning the internal structure of the Earth.

Analytic solutions and corresponding numerical examples for problems involving general SI, P, and SII waves incident on the free surface of a viscoelastic half space are presented in this chapter (Borcherdt, 1971, 1988; Borcherdt and Glassmoyer, 1989; Borcherdt et al., 1989). Closed-form expressions for displacement and volumetric strain are included to facilitate understanding and interpretation of measurements as might be detected on seismometers and volumetric strain meters at or near the free surface of a viscoelastic half space.

The procedures to solve the reflection–refraction problems for a general SI, P, or SII wave incident on a free surface are analogous to those for the corresponding problems for a welded boundary. For brevity, many of the expressions and results in medium V for a welded boundary applicable to the free-surface problems will be referred to here, but not rewritten.

Boundary-Condition Equations

Solutions of the equations of motion for problems of general P, SI, and SII waves incident and reflected from the surface of a viscoelastic half space are specified by (4.2.1) through (4.2.45) with respect to the coordinate system illustrated in Figure (4.1.3), where medium V′ is assumed to be a vacuum.

The response of a stack of multiple layers of viscoelastic media to waves incident at the base of the stack is of special interest in seismology. Solutions of the problem for elastic media with incident homogeneous waves have proven useful in understanding the response of the Earth's crust and near-surface soil and rock layers to earthquake-induced ground shaking. Solutions are provided here for general (homogeneous or inhomogeneous) SII waves incident at the base of a stack of viscoelastic layers. The derivations of solutions for the problems of incident general P and SI waves are similar, but more cumbersome. The method for deducing solutions of the incident P and SI wave problems will be illustrated by those developed here. The results provided here for viscoelastic media include those derived for elastic media (Haskell, 1953, 1960). The method used here to derive the solutions for viscoelastic waves uses a matrix formulation similar to that initially used by Thompson (1950) and implemented with the correct boundary condition for elastic media by Haskell (1953).

To set up the mathematical framework for multilayered media consider a stack of n – 1 parallel viscoelastic layers in welded contact underlain by a viscoelastic half space. Spatial reference for the layers is provided by a rectangular coordinate system designated by (x1, x2, x3) or (x, y, z) as shown in Figure (4.1.3) with the plane x3 = z = 0 chosen to correspond to the boundary at the free surface. The layers are indexed sequentially with the index of each layer corresponding to that of its lower boundary as indicated in Figure (9.1.1).

Theoretical results in the previous chapter predict that plane harmonic waves reflected and refracted at plane anelastic boundaries are in general inhomogeneous with the degree of inhomogeneity dependent on the angle of incidence, the degree of inhomogeneity of the incident wave, and properties of the viscoelastic media. As a result physical characteristics of the waves such as phase velocity, energy velocity, phase shifts, attenuation, particle motion, fractional energy loss, direction and amplitude of maximum energy flow, and energy flow due to wave interaction vary with angle of incidence. Consequently, these physical characteristics of inhomogeneous waves propagating in a stack of anelastic layers will not be unique at each point in the stack as they are for homogeneous waves propagating in elastic media. Instead these physical characteristics of the waves will depend on the angle at which the wave entered the stack and hence the travel path of the wave through previous layers. Towards understanding the significance of these dependences of the physical characteristics on angle of incidence and inhomogeneity of the incident wave, numerical models for general SII and P waves incident on single viscoelastic boundaries are presented in this chapter. Study of this chapter, especially the first three sections, provides additional insight into the effects of a viscoelastic boundary on resultant reflected and refracted waves.

A computer code (WAVES) is used to calculate reflection–refraction coefficients and the physical characteristics of reflected and refracted general waves for the problems of general (homogeneous or inhomogeneous) plane P, SI, and SII waves incident on a plane boundary between viscoelastic media (Borcherdt et al., 1986).

Analytic closed-form solutions for problems of body- and surface-wave propagation in layered viscoelastic media will show that the waves in anelastic media are predominantly inhomogeneous with the degree of inhomogeneity dependent on angle of incidence and intrinsic absorption. Hence, it is reasonable to expect from results in the previous chapter that the physical characteristics of refracted waves in layered anelastic media also will vary with the angle of incidence and be dependent on the previous travel path of the wave. These concepts are not encountered for waves in elastic media, because the waves traveling through a stack of layers are homogeneous with their physical characteristics such as phase speed not dependent on the angle of incidence. This chapter will provide the framework and solutions for each of the waves needed to derive analytic solutions for various reflection–refraction and surface-wave problems in subsequent chapters.

Specification of Boundary

To set up the mathematical framework for considering reflection–refraction problems at a single viscoelastic boundary and surface-wave problems, consider two infinite HILV media denoted by V and V′ with a common plane boundary in welded contact (Figure (4.1.3)). For reference, the locations of the media are described by a rectangular coordinate system specified by coordinates (x1, x2, x3) or (x, y, z) with the space occupied by V described by x3 > 0, the space occupied by V′ by x3 > 0, and the plane boundary by x3 = 0. For problems involving a viscoelastic half space V′ will be considered a vacuum.

The behavior of many materials under an applied load may be approximated by specifying a relationship between the applied load or stress and the resultant deformation or strain. In the case of elastic materials this relationship, identified as Hooke's Law, states that the strain is proportional to the applied stress, with the resultant strain occurring instantaneously. In the case of viscous materials, the relationship states that the stress is proportional to the strain rate, with the resultant displacement dependent on the entire past history of loading. Boltzmann (1874) proposed a general relationship between stress and strain that could be used to characterize elastic as well as viscous material behavior. He proposed a general constitutive law that could be used to describe an infinite number of elastic and linear anelastic material behaviors derivable from various configurations of elastic and viscous elements. His formulation, as later rigorously formulated in terms of an integral equation between stress and strain, characterizes all linear material behavior. The formulation, termed linear viscoelasticity, is used herein as a general framework for the derivation of solutions for various wave-propagation problems valid for elastic as well as for an infinite number of linear anelastic media.

Consideration of material behavior in one dimension in this chapter, as might occur when a tensile force is applied at one end of a rod, will provide an introduction to some of the well-known concepts associated with linear viscoelastic behavior. It will provide a general stress–strain relation from which stored and dissipated energies associated with harmonic behavior can be inferred as well as the response of an infinite number of viscoelastic models.

Preamble

In the twentieth century, there were a number of innovations in applied mathematical techniques. Some had their origins in the years before. Rayleigh seems to have begun many of these innovations that were taken up by others in the twentieth century. Rayleigh's criterion establishes the instability of a rotating fluid. Based on elementary physical arguments, he was able to conclude (loosely) that the flow configuration is stable when the square of the circulation increases outward. In the early years of the twentieth century, work was done to put many of these physical arguments on a stronger mathematical footing. One of the techniques that has proven to be useful is the study of eigenvalue problems for differential equations. This technique relates particularly to the method of normal modes of vibration of a physical system and to discrete–mode instabilities of fluid flows. Many examples of this latter application, which is given some limited discussion in this chapter, may be found in (Chandrasekhar) and (Drazin and Reid), for example.

The general mathematical ideas were first developed by Sturm and Liouville, but Fourier had laid much of the groundwork in his theory of heat conduction. This “continuous” treatment, by means of differential equations, has discrete analogs as well, in the theories of matrices and of particle systems.