Elastic Surface Waves Guided by Thin Films

An isotropic, elastic solid permits the propagation of one straight-crested, nondispersive surface wave, called a Rayleigh wave. If a thin film of a different material is deposited on the substrate, an infinite number of straight-crested surface waves are possible, all of which are dispersive, including the one corresponding to the Rayleigh wave. Since the solution of the three-dimensional equations of elasticity is extremely unwieldy and the films may be considered thin in the frequency range of interest, the approximate equations of low-frequency extension and flexure of thin plates are employed in the solution to account for the motion of the platings. These approximate equations enable the entire effect of the plating to be treated as a boundary condition at the surface of the substrate. The accuracy of the approximation is shown to be excellent in the frequency range of interest. It is shown that platings which load the substrate reduce the velocity of straight-crested surface waves, while platings which stiffen the substrate increase the velocity. A condition is derived which tells whether a given combination of materials will lower or raise the surface-wave velocity. The solution for straight-crested waves in isotropic solids is extended to include waves with trigonometric and hyperbolic crests. An approximation technique is introduced, which enables the use of the dispersion curves for straight-crested waves in connection with the extended solution for waves with variable crests, in order to obtain approximate solutions for surface waves guided by thin films of finite width. The results of the calculation are shown to compare favorably with experiment. The analysis shows that strips of finite width may be used to guide surface waves only when the film loads the substrate and that slots of finite width in the film must be used when the film stiffens the substrate. It is also shown that while the strip configuration has a highly dispersive surface-wave velocity, the slot configuration has a large frequency range around which the surface wave velocity is essentially nondispersive. A formula for the coupling length between two weakly coupled, adjacent strips (or slots) is obtained by means of a perturbation technique, and shows that the coupling length increases rapidly with decreasing wavelength.