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sively later in the cycle. As the wave propagates, it experi- ences more and more distortion. The continuously increas- ing level of distortion demonstrates one of the important properties of nonlinear acoustic processes—the effect is cumulative. This is why nonlinear processes may need to be considered, even for seemingly small acoustic Mach num- bers.
Figure 2b shows that the distortion is accompanied by a spread of energy from the original frequency f to its harmon- ics. In other words, nonlinear distortion is accompanied by harmonic generation. The two effects are not independent. In fact, they imply one another. To see this, consider a con- tinuous initially-sinusoidal sound wave of frequency f. As we have learned, the waveform distorts as it propagates, owing to the amplitude-dependence of v. Nevertheless, the waveform is a periodic function of time, the period T being 1/f. Fourier’s theorem tells us that any periodic waveform can be represented in terms of an infinite series of harmonically related sinusoids. The number of harmonics needed to repre- sent the waveform depends on the amount of distortion. As indicated by the blue line in Fig. 2b, only one term in the series is required to represent the initially-sinusoidal wave. As the waveform becomes distorted, however, a single term is not sufficient. Increasingly more harmonics are necessary to reproduce the waveform.
A practical application of harmonic generation can be found in medical imaging using ultrasound. A typical imag- ing system uses a transducer to generate a pulse of ultra- sound. As this pulse propagates through tissue, it reflects off interfaces between different types of tissue, bone, organs, tumors, blood vessels, etc. The reflected signals travel back to the transducer where they are processed to produce an image. The resolution of the image depends on the frequen- cy of the ultrasound. Higher frequencies give higher resolu- tion. Because body tissue is nonlinear, harmonics are gen- erated as the ultrasonic pulse propagates. If these harmon- ics are used to process the image, rather than the funda- mental frequency, the resolution of the image is greatly enhanced. This is the basis for a technique known as tissue harmonic imaging. A number of remarkable images can be found by searching for “tissue harmonic imaging” on the Web.
We deviate here slightly to introduce the concepts of lin- ear and nonlinear systems. In this context, a “system” is any- thing that responds to a stimulus, i.e., the proverbial black box. The term “system” could refer to a piece of sound recording or sound-reproducing equipment (or both), a microphone, the ear, or, as in our case, a small volume of fluid exposed to sound. One consequence of linear theory is that, given sufficient time, a system excited by a stimulus of fre- quency f, responds at and only at frequency f, as depicted in Fig. 3a. This concept lies at the heart of a branch of study known as linear systems theory. Linear acoustics is governed by this property. An example of a (nearly) linear system is a very well designed microphone operating well within its specified input range. The output of an ideal (i.e., linear) microphone is an electrical signal that faithfully captures the amplitude and phase of the sound to which it is exposed. If
Fig. 3. a) A stimulus of frequency f applied to a linear system results in an output at, and only at, the same frequency f. b) In contrast, a nonlinear system responds to an input of frequency f, with an output consisting of harmonics of f.
the input is a single-frequency, constant-amplitude, constant- phase sinusoid, the output is also a single-frequency, con- stant-amplitude, constant-phase sinusoid.
A nonlinear system responds differently. As before, sup- pose that the input is a sinusoid of single-frequency f. If the system is nonlinear, then in general the output will consist of a combination of sinusoids of frequencies nf, where n = 0, 1... A
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