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First, we investigate the two-dimensional Lamb problem, and then move on to the three-dimensional problem. As a starting point, we take H. Lamb’s expressions of the displacements on the boundary, which we use as the boundary conditions. ∗ Tr. Seism. , 18 (1932), 41 p. Tr. Seism. Inst. is Transactions of the Seismological Institute of the USSR Academy of Sciences. – Ed. 4 S. L. Sobolev As in any method of representing a solution as a definite integral, for solving a problem we need to define so-called density of spectrum in the representation.

We replace these residues with the respectively chosen contour inteF (θ) grals. Obviously, the most convenient way to do it is to integrate in the plane of the variable of the function Qt . For this purpose, instead of θ we introduce the new variable H via the formula H = θx + a2 − θ2 y. (27) For the sake of definiteness, we assume that x > 0. Let us make a cut in the plane θ along the real axis between the points ±a. First, we consider the transformation of the real axis of θ. We take as the first sheet of the Riemann surface the one such that H = θx + a2 − θ 2 y Application of the Theory of Plane Waves to the Lamb Problem 21 on the lower lip of the cut and H = θx − a2 − θ 2 y on the upper lip.

On the real axis. These isolated linear discontinuities, sliding on the surface and not related to the inner surfaces 1 of the discontinuities, have to move with the velocity , as proved. c As we have already noted, our physical idea can be justified by summing the Fourier integrals used by H. Lamb in his memoir, and therefore, it is not new in principle. However, we think this idea was not explicitly presented yet. 3. For the sake of convenience of the further presentation, we need to give a somewhat different form of H.

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