Page 348 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

328 SURFACE ACOUSTIC WAVES IN SOLIDS

Unstressed Rayleigh wave

Figure 10.5 Particle displacement on the sagittal plane for the Rayleigh wave

These displacements are as shown in Figure 10.5. It is seen that as u$ is in phase
quadrature with MI, the motion of each particle is an ellipse. Because of the change
in sign in u 1 at a depth of about 0.2 wavelengths, the ellipse is described in different
directions above and below this point. At the surface, the motion is retrograde, whereas
lower down it is prograde.

u 1 = [A 1 exp(-£i.*3) + A 2exp(-b 2*3)]exp[jk:(.xi - x 1)]

"3 = (- - ct)] (10.29)
2
2 1 / 2
2
2
where b 1 = k(1 - c /v )1/2 and b 2 = k(1 - c /v )
The longitudinal and transverse velocities, v 1 and v,, are given by

where the Lames' constants G is given by E m /2(l + v) and A is given by vE m/[(l + v)
(1 — 2v)] with v being Poisson's ratio and E m being Young's modulus.

10.5.2 Shear Horizontal or Acoustic Plate Modes

Acoustic plate modes (APW) or shear horizontal (SH) waves in a half-space utilise single-
crystal quartz substrates. These act as an acoustic waveguide by confining the acoustic
energy between the upper and lower surfaces of the plate. Such a mechanism is used to
confine waves traveling between an input and output IDT. SH modes may be thought
of as those waves with a superposition of SH plane waves, which are multiply reflected
at some angle between the upper and lower surfaces of the quartz plate. These upper
and lower faces impose a transverse resonance condition, which results in each SH mode
having the displacement maxima at the surfaces, with sinusoidal variation between the
surfaces.
The solution is simply a plane shear wave propagating parallel to the surface, with its
amplitude independent of X3, within the material. The phase velocity is equal to v,. The
particle displacement associated with the nth order SH plate mode (propagating in the