Page 352 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

332 SURFACE ACOUSTIC WAVES IN SOLIDS

The first of these conditions leads to

exp (—a 2h) - B 2 exp (+a 2h) = 0 (10.34)
B 1
and the two other conditions lead to A = B\ + B 2 and

a 1 G 1A = a 2 G 2 (B 1 — B 2) (10.35)
This system of three linear equations has a solution different from zero (Tournois and
Lardat 1969) if
G 1a 1
tan(a 2 h = — — - (10.36)

The roots of this equation have a real value when k 1 < k < k 2, that is, when c 2 < c < c\.
Therefore, a necessary condition of existence of Love waves is that the propagation
velocity of transverse waves in the layer must be smaller than the propagation velocity
of the transverse waves in the substrate.
It is easy to deduce from Equation (10.36) that there are infinite modes owing to the
periodicity of the tangential function; when c tends to c 1, tan(a 2 h) tends toward the value
of nn (with n being 0, 1, 2, . . .), and for the first mode, the wavelength becomes infinite
compared with the thickness.
The particle displacements occurring during the propagation of Love waves are easily
obtained from Equation (10.35).

"1x2 = A
COS[(X 2(h - X 3 )] ,,ni-n
u 2x2 = A - - - exp[y(wf - kx\)] (10.37)
cos a2 n

where A is a propagation constant determined by the excitation signal.
The equations in ( 10.37) show that the displacement amplitude u 1x2 decreases exponen-
tially in the substrate. It also shows that the different modes u 2x2 correspond to 0, 1,2,...
nodal planes in the layer. Figure 10.9(a) gives the shape of particle displacements in the
layer and the substrate for the first three modes.
The displacement amplitude also depends on the frequency, and Figure 10.9(b) shows
its variation for the first mode. Therefore, it could be noticed that the energy is entirely
located in the substrate for very low frequencies and that the Love wave propagates
at a velocity c\ as if the layer does not exist. Its thickness is, in fact, negligible when
compared with the wavelength. Conversely, the acoustic energy is concentrated in the layer
for very high frequencies, and the phase velocity of the Love waves tends toward c 2, the
wavelength being very small with respect to the thickness of the layer. Between these
two limits, the energy progressively transfers from the substrate to the layer, whereas the
phase velocity varies between c\ and c 2 (Tournois and Lardat 1969).
Having obtained the nature of the displacement and the similarity between the SH-SAW
waves and Love modes, we can derive an expression for the change in the velocity and
frequency shift for a Love wave device using perturbation theory. The derivation for
the frequency shift and the corresponding change in velocity have been presented in
Appendix I using the basic equations derived in this chapter.