Bending A wave in the scattering by a circular cylindrical shell: Its relation with the bending free modes

The steady-state problem of a plane acoustic wave scattered by an elastic circular cylindrical shell is considered. The standard resonance scattering theory (RST) procedure is used for the analysis of modal resonances. The bending waves, namely A and A0 waves, are investigated in detail. At a fixed order of modal resonance n, it is possible to obtain either the A wave resonance, or the A0 wave resonance, but never both of them together. The relation between the modal resonances of the peripheral waves A and A0 and the free modes of vibration of the shell is studied by means of the variation of the density ρ of the ambient liquid. When the density of the ambient liquid tends toward zero (at unchanged sound velocity in it), the resonance frequency approaches the one of the free vibration in two different ways. If the wave associated with the free mode has a velocity greater than the sound velocity in the fluid, the A0 wave resonance frequency tends to that of the free mode. If the wave associated with the free mode has a velocity lower than the sound velocity in the fluid, the A wave resonance frequency tends to that of the free mode. The A wave, which was up to now considered as an external wave, is rather internal at a low frequency and becomes more external when the frequency increases.