Asymptotic Theory of Wave‐Motion in Rods (Longitudinal Wave‐Motion)

Abstract

AbstractA systematic procedure is developed for constructing dynamical theories of rods, plates, and shells, based on full three‐dimensional equations. The title problem is chosen to bring out the various difficulties. The method of asymptotic expansion, used till now for static problems, is extended to time‐dependent cases. Successive approximations give classical wave equations, the effect of lateral inertia, and the effect of transverse shear. However, two additional objectives are achieved. First, it has been possible to construct a complete and well‐posed “inner problem”; this obviates the necessity of “matching” with “boundary layer” solutions at each stage. Second a modified approach is proposed to construct higher approximations; previously known methods have led to solutions where an initial singularity worsens with higher approximations. The procedure herein developed not only avoids it but gives a physically more realistic solution.