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An acoustic ray-wave synthesis is developed by combining the wave theory methods of Kirchoff and Helmholtz with the ray theory methods of Sommerfeld and Runge. The result is a generalization of the Kirchoff-Helmholtz integral formulation of the wave equation in which the kernels are ray theory expressions and in which an additional term appears. The ray theory kernels represent the gross properties of the fields while the extra term represents scattering by the medium and is the additional field required for an exact solution. The acoustic ray-wave synthesis is applied to the problem of signal variation in a random medium. The problem is found to separate naturally into three regions which are designated near field, intermediate field and far field.

The reflectivity method for the calculation of theoretical body-wave seismograms is extended to include a double-couple point source. Theoretical seismograms of SV waves from this type of source are presented for models of the Earth's crust and the crust-mantle boundary, and for models of the whole Earth. In the models of the crust-mantle boundary, there are up to four SV head waves, depending on the sharpness of the transition. The most remarkable one is slightly slower and later than Sn and has unusually low frequencies. Theoretical SV-wave seismograms for models of the whole Earth for periods from 15 s to 60 s and in the epicentral distance range from 10° to 160° show as prominent phases S, ScS, SKS and SKKS. SKS, and SKKS are different in wave form, in agreement with observations from long-period WWNSS stations. A diffracted wave SPdiffKS + SKPdiffS is found in the theoretical seismograms whose travel-time curve is tangential to that of SKS at a distance of 107 °. The resulting interference causes the wave form of SKS to change markedly around 120°. Since the theoretical seismograms are the complete response of the Earth models from the crust-mantle boundary down to the inner core, they also include many multiple and converted phases. Besides those associated with the crust-mantle boundary there are phases related to the transition zones in the upper mantle. Their amplitudes depend strongly on the sharpness of these zones.           ARK: https://n2t.net/ark:/88439/y089618 Permalink: https://geophysicsjournal.com/article/249 &nbsp

The behaviour of internal waves in vertically inhomogeneous fluids has been studied theoretically by various authors, from the point of view of general hydrodynamics as well as of meteorology and of physical oceanography (see the list of references, which, however, is not meant to be complete). For mathematical reasons most of them assumed discontinuities at certain levels, either in the density or in its first derivative with respect to the vertical coordinate, z. Any transition layer was often assumed to be thin in comparison with the wave length. FJELDSTAD [7], on the other hand, by using numerical integration, succeeded in giving an.approximate method of solving the problem for certain general density-distributions, such as may actually occur in the sea, a method, however, which is only applicable for very long waves. Besides this restriction to long waves only, FJELDSTAD's method has one other disadvantage, viz. of not directly yielding general rules or relations between the properties of the internal waves and certain parameters of the density-distribution. It is therefore, that we have gone back to somewhat more special density distributions, which are perfectly continuous with respect to the density and its first derivative to z (as are FJELDSTAD's density-distributions), but which appear to be capable of an analytical treatment; furthermore, the results are also valid for small wavelengths. In the present paper we deal only with fluids extending to infinity both upwards and downwards. At first sight this seems rather unrealistic. We know, however, that the wave-motions are always confined to a certain layer, above and below which they are negligibly small, so that, if only the boundaries of the fluid fall without this layer, they will not interfere essentially with the solutions we shall find here. The thickness of this layer depends on the wave length (see section 5). . For the rest, it is quite possible to introduce a free surface and a rigid bottom, if n

The classical problem of describing the characteristics of deep water waves of finite amplitude is considered. The method of analysis initially follows that of Nekrasov, but differs in that a non-linear algebraic equation is derived. This equation is solved to the third, fifth and fifteenth order by means of a digital computer and the data is presented in tabular form. Expressions for the wave speed and wave shape, predicted by the analysis, are compared with the results obtained by Stokes. The highest wave in water is also discussed.