Page 347 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

ACOUSTIC WAVE PROPAGATION 327

SAWs that propagate parallel to the surface with a phase velocity uR and whose displace-
ment and potential amplitudes decay with distance away from the surface (X 3 , > 0). The
direction of propagation can be taken as the x 1-axis, and the (x 1, x 3) plane can be defined
as the sagittal plane.
Note that the propagation geometry axes depicted in Figure 10.4 do not always corre-
spond to the axes in which the material property tensors are expressed. There are transfor-
mation formulae that can be applied to the property tensors so that all the above equations
hold for the new axes. The elastic constants (Q/M), the piezoelectric constants (e,-^/), and
the dielectric constants (e,- 7) can be substituted by c' ijkl, e' ijkl, s'^. The primed parameters
refer to a rotated coordinate system through the Euler transformation matrix (Auld 1973a).
The solutions for Equations (10.23) and (10.24) have the form of running waves: the
surface wave solution is in the form of a linear combination of partial waves of the form
(Auld 1973a)

ui = Ai exp(—kx 3) exp -jco \t-~\\ (10.26)

(p — B exp(— kxj) exp —jco I t -- - I and x > 0 (10.27)
L V VR / J

Here, co is the angular frequency of the electrical signal, k is the wave number, given by
27T/A,, and A. is the wavelength, given by 2nvR/co.
When the three particle displacement components exist, the solutions are called gener-
alised Rayleigh waves. The crystal symmetry and additional boundary conditions (elec-
trical and mechanical) impose further constraints on the partial wave solutions. If the
sagittal plane is a plane-of-mirror symmetry of the crystal, x\ is a pure-mode axis for
the surface wave, which involves only the potential and the sagittal-plane components of
displacement.
Because the Rayleigh wave has no variation in the X2 -direction, the displacement
vectors have no component in the x 2 -direction and the solution is given as follows
(Varadan and Varadan 1999):
Assume displacements u\ and u3 to be of the form A exp(— bx 3 ) exp[jk(x 1 — ct)] and
B exp(— bxT,} exp[jk(x 1 — ct)], and u 2 equal to zero, where the elastic half-space that
exists for x 3 is less than or equal to zero, B and A are unknown amplitudes, k is the
wave number for propagation along the boundary (x 1-axis) and c is the phase velocity
of the wave. Physical consideration requires that b can, in general, be complex with a
positive real part. Substitution of the assumed displacement into Navier— Stokes equation
gives (Varadan and Varadan 1999)

V - T - / t ) = 0 (10.28)

and use of the generalised Hooke's law for an isotropic elastic solid yields two homoge-
neous equations in A and B. For a nontrivial solution, the determinant of the coefficient
matrix vanishes, giving two roots for b in terms of the longitudinal and transverse veloc-
ities. Substitution of the roots of b obtained, as shown earlier, into the homogeneous
equations in A and B gives the amplitude ratios. Thus, we obtain the general displacement
solution (Equation 10.28) (Varadan and Varadan 1999).