Bleustein-Gulyaev waves in some functionally graded materials

Functionally Graded Materials are inhomogeneous elastic bodies whose properties vary continuously with space. Hence consider a half-space (x_2>0) occupied by a special Functionally Graded Material made of an hexagonal (6 mm) piezoelectric crystal for which the elastic stiffness c44, the piezoelectric constant e15, the dielectric constant epsilon11, and the mass density, all vary proportionally to the same "inhomogeneity function" f(x_2), say. Then consider the problem of a piezoacoustic shear-horizontal surface wave which leaves the interface (x_2=0) free of mechanical tractions and vanishes as x_2 goes to infinity (the Bleustein-Gulyaev wave). It turns out that for some choices of the function f, this problem can be solved exactly for the usual boundary conditions, such as metalized surface or free surface. Several such functions f(x_2) are derived here, such as exp($\pm 2\beta x_2) (\beta is a constant) which is often encountered in geophysics, or other functions which are periodic or which vanish as x_2 tends to infinity; one final example presents the advantage of describing a layered half-space which becomes asymptotically homogeneous away from the interface. Special attention is given to the influence of the different inhomogeneity functions upon the characteristics of the Bleustein-Gulyaev wave (speed, dispersion, attenuation factors, depth profiles, electromechanical coupling factor, etc.)