Lab. Ondes et Acoustique, Universite Paris 7, E.S.P.C.I., 10 rue Vauquelin, 75231 Paris Cedex 05, France
This work studies the acoustical field generated by a point source after reflection or transmission by a plane interface. There are two classical approaches to calculate such transient Green's functions, using either a Laplace or a Fourier transform over time. However, the second approach involves numerical integrations that are difficult to carry out because of singularities of the integrand. It is shown that it is possible to avoid this difficulty using an excitation in the form of a temporal Lorentz function of variable width, introducing several judicious variable changes and finally deforming the integration path in the complex plane. By developing the expression of the transmission/reflection coefficient into Taylor's series around suitable points, it is even possible to carry out a piecewise integration analytically, therefore resulting in an approximate expression of the acoustical field as a function of space and time. Since this expression turns out to be in a closed form, it can be evaluated fast and even be manipulated later on.