Backward waves and quasi-resonance of shells and invariants of the time reversal operator

Abstract

Backward waves propagating on shell are guided modes with opposite phase and group velocities. For a shell in vacuum, backward modes are linked to zero group velocity modes and resonances, which have been the object of recent studies. For a shell embedded in water, the group velocity does not vanish because of the leakage into the fluid. However, the group velocity of the backward mode has a minimum associated to a quasi-resonance. These phenomena are studied on air filled steel and zircaloy hollow cylinders, using a 3 MHz linear array in pulse echo mode. The circumferential guided modes are generated and their radiation into water detected by the array. The modes are separated using the decomposition of the time reversal operator (TRO), each pair of counter-propagating modes being associated to two invariants of the TRO [Prada et al. J. Acoust. Soc. Am. (1998)]. Two resonances are revealed by the eigenvalues of the TRO, one is associated with the first longitudinal thickness resonance and the other, very high, occurring at a slightly lower frequency, corresponds to the minimum of the group velocity of the backward mode. The back-propagations of the eigenvectors of the TRO provide the phase velocities of these modes.