An asymptotic method to analyze the vibrations of an elastic layer

The displacement components for both free and forced vibrations are sought as power series of the dimensionless wave number ε, where ε = 2π × layer thickness/wavelength. For the free vibration problem the object is to determine the frequencies, which are also sought as power series of the dimensionless wave number. The displacement and frequency expansions are substituted in the displacement equations of motion and in the boundary conditions. By collecting terms of the same order εⁿ, a system of second-order, inhomogeneous, ordinary differential equations of the Helmholtz type is obtained, with the thickness variable as independent variable, and with associated boundary conditions. For free vibrations, subsequent integration yields the coefficients of εⁿ for the displacements and the frequencies for all modes, and in the whole range of frequencies, but in a range of dimensionless wave numbers 0 < ε < ε* < 1, where ε* increases as more terms are retained in the expansions. For forced vibrations, the amplitudes are determined in an analogous manner if the external surface tractions are of sinusoidal dependence on the in-plane coordinates and on time. The response to surface tractions of more general spatial dependence is obtained by Fourier superposition.