THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

464 16. UltrasonicsTable 16.1. Value of theAdiabatic Elastic Constantsfor Quartz ×10 10 dyne/cm 2or ×10 9 N/m 2 .c 11 = 87.5c 33 = 107.7c 44 = 57.3c 12 = 7.62c 13 = 15.1c 14 = 17.2Because c mn= c nm, the number of these constants is reduced from 36 to 21. Symmetryof the axes will lessen this number even further. In the case of quartz, onlysix elastic constants are independent of one another, and the values of c mn can beexpressed as a matrix as follow:c 11 c 12 c 13 c 14 0 0c 13 c 11 c 13 −c 14 0 0c 13 c 13 c 23 0 0 0c 14 −c 14 0 c 14 0 0∣ 0 0 0 0 c 44 c 14 ∣∣0 0 0 0 c 14(c 11 − c 12)2The values of these constants for adiabatic conditions are given in Table 16.1The Dynamics of Piezoelectric TransducersA body undergoing forced vibrations can be considered analogous to an electriccircuit that is activated by an electromotive force, with the current i correspondingto body velocity u and the voltage V to the applied force F. In terms of the strains, velocity u = l(ds/dt), and the current is expressed as i =(dQ/dt) = A(dP/dt).Here l is the body length, Q is the electric charge, A is the cross-sectional area.We invoke Equation (16.26) to obtainwhich givesdPdt= e dsdti = Ae u = α T u (16.31)lwhere the transformation factor α T = Ae/l, which constitutes a characteristicconstant for a specific transducer. Because P has three components and S has sixcomponents, we can write Equation (16.31) in the general matrix formi = a T u∣