The scattering of stationary acoustic waves by a bounded elastic obstacle in an inviscid fluid is considered. The developed formalism is an extension of the transition matrix method, which has been given by Waterman for acoustic, electromagnetic, and elastic scattering. In fact the problem at hand is more complex than the elastic case, something which is apparent in the solution. It is possible to obtain some different forms for the transition matrix, depending upon whether the Rayleigh hypothesis enters or not. It is worthy of notice that the more general formalism, i.e., the one not assuming the Rayleigh hypothesis, is also the more efficient numerically. Numerical results are given for spheroids and an object resembling the number ’’8.’’