Page 24 - Fall 2005

common, undesirable example of this phenomenon is har- monic distortion in audio amplifiers. An ideal audio amp is linear; it amplifies the input without changing the shape or frequency content of the waveform. However, no amplifier, no matter how expensive, is truly linear. In practice, the output is never an exact replica of the input. The higher the quality of the amplifier, the more nearly linear its perform- ance, as characterized by lower harmonic distortion. Returning the discussion to the acoustics of fluids, at very low amplitudes the fluid responds approximately as a linear system. However at higher amplitudes, the nonlinear nature of the fluid becomes observable, e.g., harmonic dis- tortion
As we have seen, a high-amplitude wave becomes
increasingly distorted as it propagates. Therefore, one might expect it would follow the progression depicted in Fig. 4a, in which the cumulative distortion leads from the blue to the green to the red curves at increasingly larger propagation distances. However, the green and red wave- forms are unphysical because they predict that the fluid has three different values of u at the same instance during parts of the cycle. This cannot happen; a given volume of fluid can move at only one speed at a time. Therefore, our simple description of waveform distortion breaks down when it begins to predict unphysical results such as multi- valued physical quantities. An oversimplified, yet useful resolution of this problem is to eliminate the multi-valued region and make the waveform discontinuous. The green waveform from Fig. 4a is reproduced in Fig. 4b, with the multi-valued region shaded magenta. The discontinuity is placed according to the “equal area” rule, by which the area of the multi-valued region ahead of the discontinuity equals that behind it. The discontinuity is called a shock. In reality, a true discontinuity never exists. The abrupt change in amplitude occurs over a finite, yet short, period of time.
The propagation distance required for an initially-sinu- soidal plane wave to form a shock is termed the shock for- mation distance x– = λ/2πβM, where λ is the wavelength of the initially-sinusoidal wave. According to this equation, the longer the wavelength the farther a wave has to propa- gate before a shock is formed. This is because nonlinear effects are cumulative. Also, shocks form sooner in more nonlinear fluids and for higher amplitudes (larger values of β and M).
The three waveforms from Fig. 4a are shown again in Fig. 4c with the multi-valued regions replaced by shocks. As indicated by the red waveform, as a wave propagates it steadily evolves towards a sawtooth shape. The discussion leading to this point has assumed an initially-sinusoidal waveform. However, a somewhat non-intuitive conse- quence (linearly speaking) of nonlinear acoustics is that the fate of any waveform is a sawtooth, neglecting absorp- tion.
Table 2 (after Table 3-2 of Reference 2) contains values
of x– for a few frequencies and Mach numbers for air and
water. In addition, the absorption length La (the reciprocal
of the absorption coefficient in Np/m) is listed for the
same frequencies. This table provides two rough indica-
tors of when nonlinear effects may be important. If the
propagation distance is significantly less than x–, then
there is insufficient distance for the cumulative effects of
nonlinearites to become relevant. If the distances are com-
parable to or greater than x–, then the relevant length scale
againstwhichtocompareisL.IftheratioL /x–,also aa
known as the Gol’dberg number, is greater than unity, then a shock can form before significant absorption has occurred. In this case, nonlinear effects need to be consid- ered. If, on the other hand, the Gol’dberg number is less than unity, then absorption attenuates the waveform
Fig. 4. a) One might expect that if the waveforms depicted in Fig. 2a were allowed to propagate farther, they would take on waveforms that are multi-valued. b) However, multi-valued waveforms are unphysical. The multi-valued sections of the waveforms are eliminated by replacing them with a discontinuity, or a shock. c) The physical progression of the waveforms depicted in a) leads to shock formation. As the propagation distance increases, the waveform approaches a sawtooth shape, in the absence of attenuation.
22 Acoustics Today, October 2005