Circumferential waves in thin-walled cylindrical shells

Abstract

Three types of multiple echo returns confirming the existence of circumferential waves traveling within shells were observed from the backscattering of short ultrasonic pulses(1 μsec for main peak) by air-filled or water-filled nuclear fuel cladding tubes in water. The contribution of the circumferential waves in the backscattering patterns was found to be more significant at high frequencies(2.25 and 5Mc) than at low frequency(1Mc). By the simplified geometrical approach of Fekih and Quentin, the values for velocity and attenuation of the circumferential waves traveling along the effective mean circle with radius $r_{eff}$=(outer radius a + inner radius b)/2 were obtained. The measured velocities and attenuations were plotted as a function of the product of frequency f and shell thickness d (the region studied is $0.22\times10^6mm/sec<f \cdot d<3.35\times10^6mm/sec$) and the ratio of radii (the region studied is 0.88<b/a<0.97). The observed three modes of circumferential waves were then identified by the trends of the velocity measurements versus f · d as the first and second transverse modes in air-filled tubes and the Rayleigh mode in water-filled tubes. As b/a increased, the velocity of the transverse mode increased and approached to that of the zero order symmetric($S_o$) Lamb mode in a plate of the same material and thickness, while that of the Rayleigh mode decreased. The attenuation measurements of the first transverse and the Rayleigh modes showed minimal values in the region 0.4<f·d<0.7 and increased almost linearly as f·d or b/a increased in the region f·d>0.7. The increasing rate of attenuation of the first transverse mode, however, was larger than that of the Rayleigh mode. Contrary to the $S_o$ Lamb plate mode, the first transverse mode had cut-off frequency, which was given by $f \cdot d = (1-b/a) C_T/2cos^{-1}(b/a)$. The cut-off frequency thus increased as the curvature of a shell increased.