## Treatise on materials science and technology, Volume 3 |

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Page 35

These authors propagated 25 MHz ultrasonic beams in a quartz crystal with

dimensions 16 mm along the x

beam was directed along the x

beam associated with longitudinal wave propagation along the z

symmetry associated with this

Fig. 12. Schlieren photograph of longitudinal ultrasonic wave propagating along

pure mode z ...

These authors propagated 25 MHz ultrasonic beams in a quartz crystal with

dimensions 16 mm along the x

**axis**and 32 mm along the y and z axes. The lightbeam was directed along the x

**axis**in all cases. Figure 12 shows the ultrasonicbeam associated with longitudinal wave propagation along the z

**axis**. Thesymmetry associated with this

**axis**is such that longitudinal wave propagation is.Fig. 12. Schlieren photograph of longitudinal ultrasonic wave propagating along

pure mode z ...

Page 36

with this

mode wave propagates with no deviation of the energy flux from the wave normal

(specimen

quasilongitudinal wave propagation along a nonpure mode

generated by an X-cut piezoelectric transducer attached to the y face of the

quartz crystal specimen, i.e., the y

case the energy flux ...

with this

**axis**is such that longitudinal wave propagation is istropic, i.e., a puremode wave propagates with no deviation of the energy flux from the wave normal

(specimen

**axis**). Figure 13 shows the ultrasonic beam associated withquasilongitudinal wave propagation along a nonpure mode

**axis**. This beam wasgenerated by an X-cut piezoelectric transducer attached to the y face of the

quartz crystal specimen, i.e., the y

**axis**is vertical in the figure. Note that in thiscase the energy flux ...

Page 92

If we substitute the expressions from Eqs. (249), properly differentiated, into Eq. (

245) we obtain p0 Vt, = (2ai + X) + a (5A + 1 On + 2/ + 4m) + 2p (A + 21), (253)

where again we have neglected terms of order a2 or greater, and where we have

set VLa = co/A:. (254) The subscripts L and a denote longitudinal wave along the

a

substitute the expressions for a and /? from Eqs. (252) into Eq. (253) and

rearrange we ...

If we substitute the expressions from Eqs. (249), properly differentiated, into Eq. (

245) we obtain p0 Vt, = (2ai + X) + a (5A + 1 On + 2/ + 4m) + 2p (A + 21), (253)

where again we have neglected terms of order a2 or greater, and where we have

set VLa = co/A:. (254) The subscripts L and a denote longitudinal wave along the

a

**axis**and compressive stress along the a**axis**, respectively. Finally, if wesubstitute the expressions for a and /? from Eqs. (252) into Eq. (253) and

rearrange we ...

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