Wave Propagation in Strings with Rigid Bodies

Abstract

This paper studies wave propagation in strings with rigid bodies using the method of transfer matrices. The transmission property of a single rigid body is investigated. It is found that when the size of a rigid body is included, a symmetrically defined rigid body will transmit wave energy completely at a non-zero frequency defined by the tension, the length of the body, the mass of the string replaced by the body, and the mass of the body. Using the concept of impedance matching, the effect of a discontinuity on wave transmission in an infinite string system is revealed. The same idea is extended to the study of wave propagation in a string with multiple, equally-spaced rigid bodies (a periodic structure). The input impedance of such a system and the conditions of complete transmission are expressed in terms of the transfer matrix. The input impedance is used to identify the frequencies at which there is complete wave transmission. These frequencies are related to the natural frequencies of the corresponding finite system and constitute the so-called propagation zones. The results of this work may be applied to the propagation of vibration in complex cable systems such as oceanographic moorings.