Analysis of elastic wave propagation in long beam using Fourier transformation

This paper presents a novel method for modeling elastic wave propagation in long beams. The proposed method derives a solution for the transient transverse displacement of the beam's neutral axis without assuming the separation of variables (SV). By mapping the governing equation from the space domain to the frequency domain using Fourier transformation (FT), the transverse displacement function is determined as a convolution integral of external loading functions and a combination of trigonometric and Fresnel functions. This method determines the beam's response to general loading conditions as a linear combination of the analytical response of a beam subjected to an abrupt localized loading. The proposed solution method is verified through finite element analysis (FEA) and wave propagation patterns are derived for tone burst loading with specific frequency contents. The results demonstrate that the proposed solution method accurately models wave dispersion, reduces computational cost, and yields accurate results even for high-frequency loading.