A method to evaluate the total infinite wavenumber spectrum of the isotropically layered pates at low frequency

A novel asymptotical and numerical tool to study the total spectrum of wavenumbers for elastic isotropically-layered plates under homogeneous boundary conditions on the faces is presented. In such a problem, for each fixed frequency, there is a finite number of real wavenumbers and an infinite (countable) set of complex-valued wavenumbers, which is the most difficult. Over the past forty years, only for two types of elastic waveguides - a homogeneous isotropic plate and a homogeneous isotropic cylindrical solid - the full spectrum has been studied more or less completely. Despite the development of many efficient numerical methods, the problem to find the entire countable set of complex-valued wavenumbers of PSV- polarized waves is still poorly studied which motivated the work. For this purpose, convenient propagator matrices are introduced, their inverse are obtained in a closed form and the matrix asymptotics at the large wavenumber moduli are deduced. The dispersion equations and their static and quasistatic limits are obtained explicitly. A method to derive the asymptotics of the wavenumbers in statics and at long waves (and low frequency) is suggested. Another iterative algorithm to refine the value and to evaluate the exact complex-valued dispersion curves is presented. Some numerical examples for spectrum of the Lamb waves in a coated plate are calculated. The numerical and asymptotic errors are estimated. The parametrical analysis is performed.