Sound Scattering by Elastic Cylinders

The problem of steady‐state sound scattering by an infinite elastic circular cylinder is treated by performing a Sommerfeld‐Watson transformation on the normal‐mode series. The complex velocities of the ensuing circumferential waves are found by obtaining zeroes of a 3 × 3 determinant in the complex plane, identical to that used by Goodman and Grace in the theory of free vibrations of an elastic cylinder. We find numerically two kinds of zeroes: (α) Franz‐type zeroes, similar to (and for aluminum cylinders, almost identical with) those appearing in scattering from rigid cylinders; (β) Rayleigh‐type zeroes, as found by Goodman and Grace, which for large cylinders tend to the Rayleigh and Stoneley wave velocities, and which enter the cylinder surface at a certain critical angle. These two types correspond to the “diffracted” and the ordinary surface waves conjectured by Keller and Karal. We also consider the causality relations of sound pulses and show mathematically that they lead to arrival times in accord with the (complex) ray paths of Keller's theory. Finally, the trajectories of the zeroes in the complex plane for variations of ka and the group velocities of circumferential sound pulses have been obtained.