Derivation of the Plessky solution to my dispersive equation

Acoustic surface transversal modes, closely similar to Bleustein-Gulyaev waves, propagating under heavy periodic metal strips on a piezoelectric substrate are considered. STW, like BG wave, has wave number close to the cut-off wave number of the bulk wave that constitutes it. The theory of ST wave propagation gives more complicated dispersive equation than the one known for Rayleigh wave. Here we show explicit solutions to the dispersive equation resulting from the spectral theory, and compare it with the known Plessky formula