Numerical results are obtained for a finite circular elastic cylinder with spherical end caps using Waterman’sT‐matrix method. In addition to the important practical applications that this geometry has in underwater acoustics, for the first time this method is applied to elastic scatterers that have a discontinuity in the first derivative of the normal to the surface. This makes the problem numerically difficult and is a good test of the effectiveness of theT‐matrix method. The frequency dependence of the backscattering cross section is presented for a cylinder whose overall length is twice its diameter. Our results are compared with experiments showing excellent agreement.