Elastic waves in Timoshenko beams: the 'lost and found' of an eigenmode

This paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler Bernoulli (EB)field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful 'branch' of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenko's solution omega((1))* similar to omega(EB)*[1+O(epsilon(2)kappa*(2))] and (ii) a singular solution omega((2))* similar to (1/2 epsilon(2)) + O(kappa*)(2); epsilon, omega* and kappa* are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of e. The second spectrum in the light of the debate on its existence, or not is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion.