Physical Interpretation of the WKB or Eikonal Approximation for Waves and Vibrations in Inhomogeneous Beams and Plates

Abstract

The WKB or eikonal approximation is derived for flexural waves of constant frequency in inhomogeneous Euler-Bernoulli beams and plates and for waves in inhomogeneous Timoshenko beams. The derivation is based on the assumption of negligible partial reflection of waves and holds when the relative variation of parameters characterizing the medium is small over distances comparable to one wavelength. The approximate relations governing the variation of wave amplitude from point to point are derived from the requirement that no energy be lost from the wave. The extension of this technique to include evanescent waves as well as propagating waves is also discussed. As an application of the method, some simple results are given for the modal frequencies and normal modes of a cantilevered inhomogeneous Euler-Bernoulli beam. A general development of the eikonal method applicable to a wide class of mechanical systems is also given, which establishes the consistency of the energy-conservation technique with the results obtained by taking the eikonal approximation as the zeroth-order approximation to a series.