Variational analysis of the reflection of surface waves by arrays of reflecting strips

Abstract

A system of approximate surface wave equations and edge conditions in one scalar variable is derived from Hamilton’s principle for linear piezoelectric media by assuming suitable depth behavior and integrating with respect to depth. The assumed behavior with depth is determined from the known surface-wave solutions of the three-dimensional equations for both the plated and unplated substrate. The influence of the inertia, stiffness, and electrical shorting of the film is included in the analysis. The approximate equations are expressed in terms of the known fundamental material constants and no measurement of model parameters is required. Bulk-wave scattering is not considered. The approximate equations, which admit of a transmission-line representation, are applied in the analysis of surface-wave reflection by both uniformly and nonuniformly spaced arrays of reflecting strips plated on various substrates. Among other things, the calculated reflection curves indicate a slight asymmetry for heavier film materials on account of the dispersion caused by the strips. Although this effect has been observed experimentally, it has not been reproduced by other analytical models.

Rayleigh wave reflection from single surface imperfections on isotropic substrates

“…[4][5][6][7][8][9][10][11][12][13][14][15][16][17] One of the fundamental characteristics allowing to estimate the influence of the surface state on the SAW propagation is the reflection coefficient from single obstacles, such as steps or grooves. Various theoretical methods have been put forward to derive this coefficient: perturbation techniques, 4-7 variational method, 8 impedance method, 9,10 and integral equation method. 11 It has been found that the absolute values of the first-order reflection coefficients from grooves and from projections of the same shape as grooves are identical but their phases differ by .…”

Singular perturbation analysis of the scattering of Rayleigh waves by thin surface layers

“…Numerous analyses of such interactions have appeared [1][2][3][4][5][6][7]. A common feature among all of these is the representation of the surface layer by a set of boundary conditions [8] to be applied to fields in the substrate.…”

“…The Wiener-Hopf technique is useful in some very special cases with semi-infinite layers [1-3] but even here usually leads to integrals which can be evaluated only numerically. A variational approach [4][5] and various related modal expansion techniques [6] are more widely applicable and have proven successful for many applications, but have not given any insight into the radiated wave, as they incorporate an assumed field consisting only of surface waves. A direct, approximate, numerical solution has been obtained for a single strip [7], but the numerical analysis is exceedingly complex.…”

“…In plane strain, the surface waves decay exponentially with depth but do not decay with distance along the surface, while the radiated wave decays as r~x/2 along any ray, and as r~3'2 along the surface. The power flux associated with the incident wave will be partitioned among the reflected, transmitted, and radiated waves.Numerous analyses of such interactions have appeared [1][2][3][4][5][6][7]. A common feature among all of these is the representation of the surface layer by a set of boundary conditions [8] to be applied to fields in the substrate.…”

“…Now let the scattered field be defined as the difference between the total field and the incident wave, and denoted with a superscript (5). If tx(x) is the shear traction between the strip and the substrate, then in view of Eq.…”