Direct Sturm-Liouville problem for surface Love waves propagating in layered viscoelastic waveguides

This paper presents theoretical model for shear-horizontal (SH) surface acoustic waves of the Love type propagating in lossy waveguides consisting of a lossy viscoelastic layer deposited on a lossless elastic half-space. To this end, a direct Sturm-Liouville problem that describes Love waves propagation in the considered viscoelastic waveguides was formulated and solved, what constitutes a novel approach to the state-of-the-art. To facilitate the solution of the complex dispersion equation, the Author employed an original approach that relies on the separation of its real and imaginary part. By separating the real and imaginary parts of the resulting complex dispersion equation for a complex wave vector k = k0 + jα of the Love wave, a system of two real nonlinear transcendental algebraic equations for k0 and α has been derived. The resulting set of two algebraic transcendental equations was then solved numerically. Phase velocity vp and coefficient of attenuation α were calculated as a function of the wave frequency f, thickness of the surface layer h and its viscosity η44. Dispersion curves for Love waves propagating in lossy waveguides, with a lossy surface layer deposited on a lossless substrate, were compared to those corresponding to Love surface waves propagating in lossless waveguides, i.e., with a lossless surface layer deposited on a lossless substrate. The results obtained in this paper are original and to some extent unexpected. Namely, it was found that: 1) the phase velocity vp of Love surface waves increases as a function of viscosity η44 of the lossy surface layer, and 2) the coefficient of attenuation α has a maximum as a function of thickness h of the lossy surface layer. The results obtained in this paper are novel and can be applied in geophysics, seismology and in the optimal design and development of viscosity sensors, bio and chemosensors.