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

460 16. Ultrasonicsoptic axis, denoted by the z-axis. The axes joining opposite edges are designatedas x-axes, and the associated axes, which are perpendicular to these and joiningopposite faces are termed y-axes. The x- and y-axes are polar axes, and slabs cutwith their faces perpendicular to them manifest the piezoelectric effect. Crystalswhich are cut with their faces perpendicular to an x-axis or y-axis are termedx-cut and y-cut crystals, respectively. The x-cut crystals are generally utilized topropagate compression waves, and the y-cut crystals are applied to generate shearwaves.Now consider an x-cut crystal in the form of a rectangular prism shown inFigure 16.3. Applying an electric field along the x-axis produces compressionin that direction, while expansion occurs simultaneously along the y-direction.If the direction of the field is reversed, expansion occurs along the x-axis withan associated compression along the y-axis. No strain, however, occurs along thez-axis. If a pair of surfaces normal to either of the polar axes (x- and y-axes)is coated with a conductive material to form electrodes, small-amplitude oscillationswill result when an alternating voltage of frequency f is applied acrossthem. When the frequency f equals one of the natural frequencies of mechanicalvibration for a particular axis, the response amplitude jumps to a considerablyhigher value. Crystals are generally operated at resonant frequencies for either“length” or “thickness” vibrations, as denoted by the resonance occurring in thedirection parallel with or normal to the radiating surfaces, respectively. The naturalfrequency for mechanical vibrations is proportional to the inverse of the dimensionalong which they occur, so it becomes obvious the lower frequencies are generatedby “length” vibrations along the direction of the longer dimension whereasthe higher frequencies are produced by “thickness vibrations” along the directionof the smaller dimension.Maximum acoustic intensities are obviously obtained by operating at the fundamentalnatural frequencies. But material constraints in crystals may necessitate theuse of higher harmonics to obtain higher frequencies. For example, an x-cut quartzplate can be only 0.15 mm thick in order to generate a fundamental “thickness”mode for 20 MHz. Such a quartz plate is extremely brittle and it can shatter underthe impetus of a exceedingly high-applied voltage, or its dielectric properties maybreak down. To avoid this situation, it is customary to use thicker slabs of crystalswith lower resonance frequencies and operate at one of the upper harmonics. Anexample is the vibration of a 1-cm thick quartz crystal at its 191st harmonic togenerate 55 MHz ultrasound.The piezoelectric effect occurs only when opposite charges appear on the electrodes,and for that reason, only odd harmonics can be generated. At the nth harmonic,the thickness of the crystal is divided into n equal segments with compressionsand expansions alternating in adjacent sections, as illustrated in Figure 16.4.For even harmonics in the nth mode, compressions occur in n/2 segments andexpansions occur in the other n/2 segments, with the result no net strain existsin the crystal. When n is odd, the (n–1)/2 compressions offset the same numberof expansions, leaving either a compression or an expansion in the remainingsegment.