Karl Zoeppritz, 2007. "On the Reflection and Transmission of Seismic Waves at Surfaces of Discontinuity", Classics of Elastic Wave Theory, Michael A. Pelissier, Henning Hoeber, Norbert van de Coevering, Ian F. Jones
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In a recently published paper (E. Wiechert and K. Zoeppritz, “On Earthquake Waves”), Professor E. Wiechert has treated the reflection of seismic waves at the surface of the earth. Effects caused by an air or water layer were neglected in view of their negligible influence, and the reflection was treated as if it occurred at a free surface. Given today’s unavoidable uncertainty in seismic measurement methods, as can be seen from a rough estimate, this is a very good approximation. However, it seems of interest to find the formulas for the reflection and transmission of elastic waves at a boundary formed by two elastic media of arbitrary constitution. This is of interest not only with regard to the behavior at the surface of the earth but even more so with respect to interfaces in the earth’s interior. Following a suggestion by Professor Wiechert, I have treated this more general case and will present some results in the following.
I consider the planar intersection between two media, 1 and 2, with densities ρ1 and ρ2 and elastic constants a1, b1 and a2, b2. Here, a and b denote the propagation velocities of the pure dilatational and the dilatation-free shear waves. For ease of comparison, I briefly present the formulas giving the relation of these variables with Lamé’s constants λ and μ as well as with Poisson’s constant σ:
Regarding the equations of motion for isotropic, elastic media, I refer to the relevant discussion in the above-cited work by E. Wiechert. To
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Classics of Elastic Wave Theory
In this chapter, we give a brief synopsis of each of the classic papers referred to in this collection. Where relevant, we reproduce the basic equations, recast in modern notation. Supporting works also are referred to. They are listed in the “General References” section.
Table 1 is a quick outline of the key contributions of each paper reprinted in this book.
Robert Hooke, “Potentia Restitutiva, or Spring” (Oxford, 1678)
The article by Robert Hooke, “Potentia Restitutiva, or Spring,” contains the statement of the proportional relation between stress and strain universally referred to as Hooke’s law. Although the English language has evolved somewhat since 1678, the article does not require translation. Hooke describes a variety of experiments, accompanied by illustrations, confirming the stress/strain relation over a wide range of applied loads. He emphasizes the great generality of his results.
Based on his experimental work from 1660 onward, Hooke first published his law in 1676 in the form of an anagram in Latin,
which he later revealed to be “ut tensio sic vis.” Roughly translated, this means “as the force, so is the displacement” (Love, 1911; Boyce and DiPrima, 1976).
In his treatise, Hooke examined the behavior of springs, so his first casting of the equations dealt with the restoring force on a spring, for a given displacement: