By David Stephen Ballantine; et al
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Extra info for Acoustic wave sensors : theory, design, and physico-chemical applications
9). Shear Mode Resonator I -"' > 55 56 3. Acoustic Wave Sensors and Responses The decay length, c5, is [ 16] "2~\1/2 . 30 is  Ov~ ] - rlVx° (1 + j). 32 yields the surface mechanical impedance due to a semi-infinite liquid : Zs = ( . ~ r / ) l / 2 (1 + j ) . 34b) where we note that R2 --- tosL2 for loading by a Newtonian fluid . 34 were derived for one-sided liquid contact; for two-sided, L2 and R2 are doubled. The motional inductance L2. 23 in agreement with the prediction of Kanazawa and Gordon : ~f,---- L2fs _ 2LI f3/2 .
33) 8Note that the source term is non-zero only in regions where it varies spatially. This is an important consideration in the mechanism by which acoustic waves are launched in the thickness-shear mode resonators that we consider later. 28 2. Fundamentals of Acoustic Waves The stiffness parameter c55 has, in effect, been increased by the factor (1 + K 2) - - a n effect known as piezoelectric stiffening. 34) C55~11 Piezoelectric stiffening increases the wave velocity from that obtained in the non-piezoelectric case.
Reprinted with permission. See Ref. . 1 43 TSM RESONATOR MASS SENSITIVITY The presence of displacement maxima at the crystal surfaces makes the thickness-shear mode very sensitive to surface mass accumulation. Mass that is rigidly bound moves synchronously with the crystal surface, perturbing the TSM resonant frequency. The relation between surface mass accumulation and resonant frequency change can be derived from a simple variational principle proposed by Rayleigh : resonance in a mechanical system occurs at frequencies at which the peak kinetic energy Uk exactly balances the peak potential energy Up.