A mathematical model for the propagation of elastic disturbances through a cylindrically wound thick composite specimen is presented. The primary purpose of the modelling effort is to provide a means by which the elastic constant tensor corresponding to the specimen can be determined by in-situ measurements, in which case the effect of the curved fibre geometry must be addressed in the theoretical model. Hence the material is modelled as a cylindrically anisotropic medium, which is of orthotropic symmetry. Ray theory techniques give rise to the identification of curved ray paths along which a disturbance propagates in any one of three normal modes with a constant velocity of propagation. Thus the determination of the elastic constants from in-situ time-of-flight measurements has been reduced to the level of simplicity involved in ascertaining elastic constants from measurements made on a media possessing the usual cartesian symmetry. Finally, it will be shown that the eikonal equations for the cylindrically orthotropic media give rise to the following simple description of the ray geometry: The rays propagate so as to maintain a constant angle of attack with respect to the surfaces which describe the symmetry of the medium.