This paper deals with the elastic waves propagated along an interface between two solid elastic half-spaces (Cagniard's problem). Classically it has been shown that interface (Stoneley) waves should exist only for those limited values of the elastic parameters of the two solids for which the Stoneley pole is real and lies on the sheet of integration. Solutions for the similar, but algebraically simpler, Lamb's problem indicate that interface waves may also be associated with complex poles not on the sheet of integration. Exact solutions are presented for Cagniard's problem for a large number of materials, lying both inside and outside the classical existence diagram. These seismograms support the conclusion that attenuated Stoneley waves can be propagated at the interface of almost any two solid materials. Additional information on critical refraction phenomena is also presented.

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