Longitudinal Modes of Elastic Waves in Isotropic Cylinders and Slabs

HE classical exact treatments of the modes of propagation of elastic waves in isotropic media having stress-free surfaces b u t extending indefinitely in at least one dimension are those of Rayleigh 1 for semiinfinite media bounded by one plane, of Lamb 2 for slabs bounded by two parallel planes, and of Pochhammcr* for solid cylinders. Rayleigh showed t h a t a wave could be propagated without attenuation parallel to the surface, in which the displacement amplitude of the medium decreased exponentially with distance from the surface, at a velocity independent of frequency and somewhat lower than t h a t of either the plane longitudinal or plane transverse waves in the infinite medium. Such "Rayleigh surface waves" have received application in earthquake theory. For slabs or cylinders the t-eatments lead to a transcendental secular equation, establishing a relation (the "geometrical dispersion") between the frequency and the phase velocity, which for some time received only asymptotic application in justifying simpler approximate treatments. T h e past decade, however, has seen a revival of interest in the exact results 4 5 stimulated by experimental application of ultrasonic techniques to rods®- * and slabs, 7 by the use of rods and the like as acoustic transmission media, and perhaps by curiosity as to what qualitative correspondence may exist between such waves and the mor? intensively studied electromagnetic waves in wave guides. T h a t this correspondence might not be close could be anticipated by observing t h a t an attempt to build up modes by the superposition of plane waves in the medium reflected from boundaries would encounter an essential difference between the two cases: the elastic medium supports plane waves of two types (longitudinal and transverse) with different velocities, and reflection from a boundary transforms a wave of either type into a mixture of both.