Gallium arsenide is a piezoelectric crystalline material belonging to the cubic group (class 43¯ m).

Gallium arsenide is a piezoelectric
crystalline material belonging to the cubic group (class 43¯ m). The elastic
constants, piezoelectric constant in Coulombs per square metre and relative
permittivity constants are c11 = 11.88 × 1010N/m2 , c44 = 5.94 × 1010N/m2 , c12
= 5.38 × 1010N/m2 , e14 = 0.154C/m2 and ε11 = 12.5 × ε0. The density
of the material is ρ = 5307kg/m3 . The piezoelectric coupling coefficient
and the permittivity matrices are given by

Hence, show that the general
solution is the superposition of four-plane wave solutions, each with a unique
decay constant and each satisfying boundary conditions associated with a
piezoelectric surface. State the four boundary conditions that the solutions
must satisfy and calculate the Rayleigh SAW velocity and the electromechanical
coupling coefficient. The direction of propagation is the x1 axis (100 axis)
and the free surface is normal to the x3 axis (001 axis). Comment on your
results.