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A direct approach to the analysis of systems with single, or commensurate delays is presented, and compared with other recently proposed methods. The finite polynomials arising in this direct method are shown to have useful sensitivity properties. Remarks are given concerning systems stable independent of delay.

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It is known that large elastic strains exist in the vicinity of a fault prior to the occurrence of an earthquake. The purpose of this paper is to study some effects of this strain on the subsequent seismic radiation. The mathematical model used is that of infinitesimal motions due to a small perturbing force superimposed on an already existing state of finite strain in an elastic medium. The suitability of this particular model is first discussed in detail in the context of the real Earth.The governing equations of motion are found to be soluble for the P-wave by means of the technique of ray expansions. The success of this approach is due to the fact that we consider a pre-strained, as opposed to a pre-stressed, Earth. In the former case, the material becomes unstrained when the stress is removed; in the latter, this is not so. In this manner, expressions for the P-wave front, the bicharacteristic rays and the nodal surface are obtained. The consequences of these results for the fault-plane method are then discussed. In general, the magnitude of the effect of the pre-strain is of the order of the magnitude of the shear strain at the source. However, in one particular region, it is found to be of higher order.One effect of pre-straining is that two quasi-S-waves appear. A study of the difference between the two corresponding characteristic equations yields an approximate expression for the difference in arrival times of these two waves.

The equations of motion for infinitesimal vibrations due to a small perturbing force superimposed on an already existing state of finite strain in an elastic medium are established. They are similar in form to the equations for infinitesimal anisotropic elasticity. For the particular case of a homogeneous strain, these may be attacked by the standard procedure of Fourier-Laplace transformation and the solution written in integral form. This is followed by a consideration of the source term using the concept of body-force equivalents. However, the formulae given by Burridge & Knopoff may not be used in the case of a pre-strained medium and their derivation must be adapted slightly to give the correct formulae.The solution to the problem of a point source of the transverse slip type in a homogeneously pre-strained medium is then obtained and in the final section, the particular example of a Hadamard-Green material is considered in detail.It is well known that the region surrounding an earthquake source is in a state of strain prior to the occurrence of the earthquake and it is the purpose of the present work to set down the mathematical foundations in order to study the effects of this pre-strain. The geophysical applications of the work will be given in a further paper. * Received in original form 1972 June 14 373 3 374 K. WaltonAny permanent (irreversible or plastic) deformation which has occurred is assumed to have been accompanied by zero permanent stress. Furthermore, stress relaxation effects are assumed to be so slow as to be negligible in the static field, and sufficiently small to neglect damping in the dynamic (infinitesimal) equations.In this paper, certain results from the general theory of waves in pre-strained perfectly elastic media are presented. Of particular interest is the study of body-force equivalents (Burridge & Knopoff 1964) in a pre-strained medium. Here comparison can be made with the results of Dahlen (1972). In one aspect, the present work is more general than that of Dahlen as the initial body force causing the pre-stress is quite general whereas Dahlen considers a body force due only to gravity and rotation. On the other hand, the present work is more restrictive since it is assumed that the pre-stress and pre-strain are related by the constitutive law for finite elasticity whereas Dahlen has no such restriction on his pre-stress.The final section deals with the particular example of a Hadamard-Green material. This material is chosen because of its particularly simple form of strain energy function.The results from this paper will be used in a further paper on the geophysical applications of the present work. Examples of which are the effect of pre-strain on the Fault-Plane method of solution and the effect of pre-straining on the dispersion of S-waves. Pre-strained mediaThe problem to be considered is that of a perfectly elastic medium which is first subjected in some manner to a static finite strain and then to small vibrations superimposed on this existing state of strain by some sm...

As a rather idealized model of an ocean sediment, we take a fluid-saturated packing of like elastic spheres and present a method for calculating the effective elastic moduli of such packings. In particular, the method is applied to the cubic packing to obtain certain new results relating to just one of the moduli, namely the one corresponding to a vertical compression. Two cases are considered; one being when the packing is subjected to a uniform compressive force and the second when the packing is at rest under gravity.

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