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P. Borejko:
"Exact Pressure Response for the Impulse-Excited Liquid-Solid Interface Problem";
Talk: 1st International Conference "Inverse Problems: Modeling ans Simulation", Fethiye, Turkey (invited); 2002-07-14 - 2002-07-21; in: "Abstracts of the First International Conference "Inverse Problems: Modeling and Simulation"", S. Cohn, A. Hasanoglu, S. Kabanikhin, A. Tolstoy (ed.); (2002), ISBN: -975-8047-12-4; 36 - 38.

The purpose of this paper is to obtain exact and complete time records of the pressure due to a point source in the vicinity of a fluid-solid interface in order to indicate (1) the relative importance of the various wave-forms contributing to these pressure responses and (2) the possibility of utilizing in situ measurements of the arrival times of the interface waves to determine the bottom rigidity in fluid-covered areas. Both an impulse-excited source and a point receiver are placed in a liquid medium overlying a solid bottom. This type of problem was studied experimentally by Roever and Vining [1] and theoretically by Strick [2], who demonstrated close pressure-response-form agreement between experiment and theory for the particular parameters (such as the characteristic wave speeds in the adjacent media and the source-receiver distance) assumed in the model experiments and the theoretical investigation. Using the Cagniard method [3], Strick [2] obtained the rigorous solution for the pressure response first for the case of the mathematically simpler line rather than point source. Then he derived the asymptotic solution for the point source and showed that at source-receiver distances selected in the experiments conducted by Roever and Vining [1] it closely approximated to the exact line source solution, thus concluding that at these distances the geometric difference of the two sources would not appreciably affect the form of the pressure response. His conclusion was confirmed by good agreement between the theoretical and experimental results that were obtained when the fluid-solid interface was excited by an impulsive point source located close to the interface.. A new theoretical point source solution for the problem under consideration is presented in this paper. This exact solution is in the form of a sum of two partial wave-motions, where the first is radiated from the source (the direct wave) and the second is excited in the fluid by the interaction of the former with the bottom, the latter including the contributions from the critically refracted longitudinal and shear waves (the head or conical waves), the pseudo-Rayleigh and Stoneley interface waves, and the totally reflected wave. In the first instance each Laplace-transformed partial wave-motion appears as a double integral over an infinite domain, synthesized by superposition of respective Laplace-transformed plane wave components. Each integral is then transformed into the time domain (without introducing any approximation) by applying the Cagniard formalism [3], where it emerges as a single definite integral taken over a fixed finite range. This integral is further transformed into a form amenable to numerical integration executed by the Gaussian quadrature. With this accomplished the numerical values of the integral are convoluted with the time history of an impulsive source to obtain complete partial response at the receiver. Opposite to Strick's approximate point source solution (Ref. [2]), obtained by means of the asymptotic expansion method and thus applicable to large source-receiver separations, the present point source solution is accurate for any separation. In particular, it is also accurate for small separations where the asymptotic solution is apparently invalid and the normal mode solution (for the case of a point source in a liquid layer with a solid bottom) becomes impracticable due to poor convergence. Similarly as in Ref. [2] the theoretical pressure response curves are obtained for two fluid-solid systems: one where the shear wave speed in the solid is lower than the sound speed in the fluid and the other where the shear wave speed is higher. For the case of a low shear wave speed bottom, there are four distinct wave-form arrivals: (1) the critically refracted longitudinal wave, (2) the direct wave, (3) the totally reflected wave, and (4) the Stoneley wave. Both the critically refracted longitudinal wave and the Stoneley wave appear to be the most prominent wave-forms (large in amplitude and wide in time) on the response curve. For the case of a high shear wave speed bottom, besides the four arrivals occurring in the former case, there are two additional arrivals: the critically refracted shear wave and the pseudo-Rayleigh wave, and the most significant of these six arrivals are the Stoneley and pseudo-Rayleigh waves. The existence of the Stoneley wave along the fluid-solid interface is re-examined, using a modern matrix formulation of the secular equation governing the speed of the Stoneley wave given in Ref. [4]. As has already been pointed out, the arrival times of the pseudo-Rayleigh and Stoneley waves can be used to solve the inverse problem of in situ determination of the bottom rigidity in water-covered areas.

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