Abstract
This paper presents various numerical techniques for the analysis of the one dimensional wave equation with variable coefficients, for governing materials with spatially varying thermal, inertial and elastic properties. Approximate time domain Finite Elements (FE) and frequency domain Spectral Finite Elements (SFE) are formulated, being computationally efficient, accurate and fast convergent compared to any other existing tool. The Sturm-Liouville problem corresponding to this wave equation is discussed and numerically solved using Prufer transformation, and the effect of inhomogeneity on the natural frequencies and mode shapes is elicited. As an application, Functionally Graded Materials (FGM), where the material properties are tailored to vary in spatial directions, are chosen, and the longitudinal wave propagation in an FGM rod due to high frequency impact load is studied. As a possible application of FGM, attenuation of waves reflection at a joint of dissimilar materials is investigated. The notion of inhomogeneous waves, used in dissipative media of more than one dimension, is retained in this one-dimensional approximation.
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