TRANSMISSION AND REFLECTION COEFFICIENTS FOR LONGITUDINAL WAVES OBTAINED BY A COMBINATION OF REFINED ROD THEORY AND FEM

Abstract

The complex coefficients of reflection and transmission of elastic waves in rods at the location of a sudden change in cross-section are examined. Knowledge of them is useful, for example, in insulation against structure-borne sound. The values given in the literature for these coefficients are usually determined by using elementary rod and beam theory, in which the cross-sections are assumed to remain planar. In the present paper, the attention is restricted to longitudinal waves in rods. The vibrations are described by a one-dimensional linear wave equation for which the transition relations are formulated at the point of discontinuity of the cross-section. While in a previous paper simple rod theory was applied, in the present paper for small wavelengths Love's theory is used, which takes into account the kinetic energy due to transverse contraction. Although the corresponding one-dimensional wave equation can give an excellent approximation to waves obtained from the theory of three-dimensional elasticity, this is certainly not true in the immediate neighborhood of a sudden change of cross-section. The coefficients of reflection and transmission computed in this manner are therefore of questionable value. It is shown here how the coefficients of reflection and transmission can be computed by using standard finite element codes designed for the solution of eigenvalue rather than wave propagation problems. The values of the coefficients computed by using the three-dimensional theory of elasticity are then compared to the values obtained via rod theory.