1. Introduction
Elongated cylindrical structures like rods, pipes, cable strands or fibers, support the
propagation of mechanical waves at ultrasonic frequencies along their axes. This waveguide
behaviour is used in a number of scientific and engineering applications: the Non
Destructive Evaluation (NDE) of the structural health of civil engineering elements for safety
purposes (Rose, 2000), in linear displacement sensors (Seco et al., 2009) for high accuracy
absolute linear position estimation, in the evaluation of material properties of metal wires,
optical fibers or composites (Nayfeh & Nagy, 1996), and as fluid sensors in pipes transporting
liquids (Ma et al., 2007). These applications demand exact quantitative models of the
processes of wave generation, propagation and reception of the ultrasonic signals in the
waveguides.
The mathematical treatment of mechanical wave propagation in cylindrical structures was
provided by J. Pochhammer and C. Chree at the end of the XIX century, but its complexity
prevented researchers from obtaining quantitative results until the advent of computers. D.
Gazis (Gazis, 1959) reported the first exact solutions of the Pochhammer-Chree frequency
equation, as well as a complete description of propagation modes and displacement and stress
distributions for an isotropic elastic tube, found with an IBM 704 computer. Since then, the
literature on the topic has grown steadily, and references are too numerous for this book
chapter. We will only mention a few landmark developments: the study of multilayered
waveguides beginning with a composite (two-layer) cylinder by H. D. McNiven in 1963;
the extension of Gazis’ results to anisotropic waveguides, initiated by I. Mirsky in 1965; the
consideration of fluids and media with losses surrounding, or contained in the waveguides,
beginning with V. A. Del Grosso in 1968; and finally, the demonstration of ultrasonic guided
waves generated with electromagnetic transducers by W. Mohr and P. Holler in 1976, and
piezoelectrically by M. Silk and K. Bainton in 1979, for the nondestructive testing of pipes.
1.1 Modelling the response of the waveguide to external excitation
Of particular importance for transducer design is the determination of the mechanical
response of a waveguide when subjected to an external excitation. Several approaches exist
to consider this problem.
Integral transform methods (Graff, 1991) convert the differential equations that physically
model the excitation forces and the behaviour of the waveguide into a set of algebraic
Modelling the Generation and Propagation of
Ultrasonic Signals in Cylindrical Waveguides
Fernando Seco and Antonio R. Jiménez
Centro de Automática y Robótica (CAR)
Ctra. de Campo Real, Madrid
Consejo Superior de Investigaciones Científicas (CSIC)-UPM
Spain
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