∑∑∑=L ∑∑s,L1 Fom:Φ=2 (48.8) When this combination is substituted into the first three equations of Eq. (48.6)and terms gathered, they become identical to Eq (48.1)but with each ci replaced by e LL E=c(1+K2) (489) Using these so-called stiffened elastic constants, we obtain the same third-order eigenvalue equation, Eq (48.3),and hence the velocities of each of the three modes and the corresponding mechanical displacement components. The potential is obtained from Eq (48.8). The velocities obtained for the piezoelectric material are usually at most a few percent higher than would be obtained with the piezoelectricity ignored. The parameter Kin Eq. (48.9)is called the electromechanical coupling constant. 48.4 One-Dimensional Propagation If an acoustic plane wave as in Eq(48.2)propagating within one medium strikes an interface with another boundary conditions at the interface, it will be necessary in gencral esponding case in optics. To satisfy the medium there will be reflection and transmission much as in the cor enerate three transmitted modes and three reflected modes. Thus, the concepts of reflection and transmission coefficients for planar interfaces tween anisotropic media are complicated In many propagation and excitation geometries, however, one can consider only one independent pure mode with energy flow parallel to k and particle displacement polarized along k or perpendicular to it. This mode (plane wave) then propagates along the axis or its negative in Eq (48.2). Discussion of the generation, propagation, and reflection of this wave is greatly assisted by consid- ering analogies to the one-dimensional electrical transmission line. With the transmission line model operating in the sinusoidal steady state, the particle displacement u; of Eq (48. 2)is represented by a phasor, u. The time derivative of the particle displacement is the particle velocity and is represented by a phasor, v= jou, which is taken as analogous to the current on the one-dimensional electrical transmission line. The negative of the stress, or the force per unit area, caused by the particle displacement is represented by a phasor, (n)=jkcu, which is taken as analogous to the voltage on the transmission line. Here cis the appropriate stiffened elastic constant for the mode in question in Eq (48.3). with these definitions, the general impedance, the characteristic impedance, the phase velocity, and the wave vector, respectively, of the equivalent line are given by (48.10) Vp Some typical values of the characteristic impedance of acoustic media are given in Table 48. 1. The characteristic impedance cor ing to a mode is given by the product of the density and the phase velocity, pV, even in the anisotropic case where the effective stiffness c in Eq.(48.10)is difficult to determine. As an example of the use of the transmission line model, conside wave propagating in n isotropic solid and incident normally on the interface with a second isotropic solid. There would be one reflected wave and one transmitted wave, both longitudinally polarized. The relative amplitudes of the stresses in these waves would be given, with direct use transmission line concepts, by the voltage reflection and transmission coefficients c 2000 by CRC Press LLC© 2000 by CRC Press LLC (48.8) When this combination is substituted into the first three equations of Eq. (48.6) and terms gathered, they become identical to Eq. (48.1) but with each cijkl replaced by (48.9) Using these so-called stiffened elastic constants, we obtain the same third-order eigenvalue equation, Eq. (48.3), and hence the velocities of each of the three modes and the corresponding mechanical displacement components. The potential is obtained from Eq. (48.8). The velocities obtained for the piezoelectric material are usually at most a few percent higher than would be obtained with the piezoelectricity ignored. The parameter K in Eq. (48.9) is called the electromechanical coupling constant. 48.4 One-Dimensional Propagation If an acoustic plane wave as in Eq. (48.2) propagating within one medium strikes an interface with another medium, there will be reflection and transmission, much as in the corresponding case in optics. To satisfy the boundary conditions at the interface, it will be necessary in general to generate three transmitted modes and three reflected modes. Thus, the concepts of reflection and transmission coefficients for planar interfaces between anisotropic media are complicated. In many propagation and excitation geometries, however, one can consider only one independent pure mode with energy flow parallel to k and particle displacement polarized along k or perpendicular to it. This mode (plane wave) then propagates along the axis or its negative in Eq. (48.2). Discussion of the generation, propagation, and reflection of this wave is greatly assisted by consid￾ering analogies to the one-dimensional electrical transmission line. With the transmission line model operating in the sinusoidal steady state, the particle displacement ui of Eq. (48.2) is represented by a phasor, u. The time derivative of the particle displacement is the particle velocity and is represented by a phasor, v = jvu, which is taken as analogous to the current on the one-dimensional electrical transmission line. The negative of the stress, or the force per unit area, caused by the particle displacement is represented by a phasor, (–T) = jkcu, which is taken as analogous to the voltage on the transmission line. Here c is the appropriate stiffened elastic constant for the mode in question in Eq. (48.3). With these definitions, the general impedance, the characteristic impedance, the phase velocity, and the wave vector, respectively, of the equivalent line are given by (48.10) Some typical values of the characteristic impedance of acoustic media are given in Table 48.1. The characteristic impedance corresponding to a mode is given by the product of the density and the phase velocity, rV, even in the anisotropic case where the effective stiffness c in Eq. (48.10) is difficult to determine. As an example of the use of the transmission line model, consider a pure longitudinal wave propagating in an isotropic solid and incident normally on the interface with a second isotropic solid. There would be one reflected wave and one transmitted wave, both longitudinally polarized. The relative amplitudes of the stresses in these waves would be given, with direct use transmission line concepts, by the voltage reflection and transmission coefficients F F = = ÂÂÂ ÂÂ e LLU L L e U ijk i k j kji ij i j ji e e Form: c c e e LL L L cc K K e c ijkl ijkl mij nkl m n nm mn m n nm =+ = ÂÂ ÂÂ e e Form: = (1 + ) with 2 2 2 Z T v Z cV c k V = = == (– ) 0 r r w