Use of Wavelet transform in the Study of Propagation of Transient Acoustic Signals Across a Plane Interface Between Two Homogeneous Media

Wavelets, Time-Frequency Methods and Phase Space, Proceeding of the International Conference, Marseille, France,December 14-18, 1887. Eds JM Combes, A. Grossmann, P. Tchamitchian, SPRINGER-VERLAG, Berlin, 1989International audienceThe problem we study can be defined as follows: In the 3D space, we consider two homogeneous media-"air" and "water"-separated by plane interface. There is a point source of sound in "air" at height h above the interface. Its emission is given by a function F(t) of time . We are interessed in the behavior of pressure in water, at time t, depth z, and radial distance r from a vertical line going through the source. The classical methods of resolution, while well adapted to the case of monochromatic sources, are less suited to the description of transients. This is due to the fact that waves of different frequencies both follow different paths and undergo different attenuations. in such time-ans-frequency dependent situations, it is natural to apply wavelet transform techniques. As a mater of fact, the main motivation for the introduction of wavelet methods. In J. Morlet's work was the need, in geophysics, to study frequency-dependent propagation phenomena. In the first part , we shall briefly describe the time behavior of the wavelet transform of the acoustic field at a fixed point under water. (This is not the same as attemping to use wavelet techniques to solve the partial differential equations of the propagation problem). It is straightforward to write down an expression for the wavelet transform of the propagator for our problem. This expression is much smoother that the propagator itself, and allows selective reconstitutions with arbitrary precision. It has a natural decomposition into three contributions corresponding to branch-points of the integrand. In a second part, we use the different contributions to obtain a formula for the reconstruction of the time-dependent source.