The propagation of a longitudinal harmonic wave in a transversely isotropic shell has been investigated in the development of ultrasonic techniques for thick hollow composite cylinders. The characteristic equation for satisfying the stress-fee inner and outer cylindrical boundaries has been obtained in an exact form in terms of the wavelength, the cylinder radii and the material constants. The phase velocity of the fundamental mode is calculated for a wide range of the wavelength for various cylinder radii for some typical sample materials. The shell wave speeds for the second mode of vibration are also presented. Comparisons are made between shell wave speeds and plate wave speeds. The spread of the wave speeds for the composite shells is shown to be much wider than that for an isotropic shell.

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