Page 22 - Fall 2005

location changes continuously. Consequently, the instanta- neous speed of sound changes continuously as well.
The second cause of nonlinearity in a fluid is that the speed of sound depends upon the local thermodynamic con- ditions of the fluid. In the case of an ideal gas the adiabatic speed of sound depends on temperature according to c=􏰀γRT/M, where R is the universal gas constant, T the absolute temperature, and M the molar mass of the gas. Just as u changes continuously during the cycle, so does the tem- perature. The combination of convection and the depend- ence on the thermodynamic state are both captured in the parameter β.
The evolution of a high-amplitude wave as it propagates is illustrated in Fig. 2. The upper graph, Fig. 2a, shows the normalized acoustic amplitude of a plane wave as a function of time at three propagation distances. The initially-sinu- soidal waveform is depicted in blue. The green and red curves show the waveform at successively greater propaga- tion distances, respectively. Because the positive-amplitude regions of the wave travel faster than c0, they will arrive suc- cessively earlier during the cycle. In contrast, the negative- amplitude regions travel slower than c0 and arrive succes-
Fig. 1. Graph of fluid speed u, normalized to its maximum value, as a function of time t, relative to the acoustic period T for one cycle of a continuous wave. The speed of sound v is greater than c0 at times when u is positive, and less than c0 when u is negative. The zero-crossings travel at speed c0. The net result of the amplitude dependence of the speed of sound is that the waveform distorts as it propagates, while maintaining a constant acoustic period.
fluid. It is a physical property of the fluid. For example, in a fluid, β = 1 + B/2A. Table 1 contains representative values of the parameter B/A, referred to as “b-over-a.” For an ideal gas, B/A = (γ - 1), and so β = (1 + γ)/2, where γ is the ratio of specific heats. γ = 1.4 for air. Extensive tables of the parame- ter B/A for gases, fluids, and tissues, including the values list- ed in Table 1, can be found in Reference 4 (Chapter 2 of Reference 1) and Reference 2.
Table 1
Values of the parameter B/A for selected fluids. Values are from Tables I, II, and III of Ref. 4 and
assume an ambient pressure of 1 atm.
Fluid B/A Acetone 9.2 Distilled Water 5.2 Human Liver 7.6±0.8 Liquid Nitrogen 7.7 Mercury 7.8 Seawater 5.25
Linear acoustics is instead recovered, not by setting β = 0, but rather by setting u = 0. Of course, u is zero only if the sound amplitude is zero. However, from a practical point of view, linear-acoustic propagation occurs if the ratio u/c0, also known as the acoustic Mach number M, is much less than unity. How much less? A few basic principles of nonlinear acoustics need to be introduced, before this essential ques- tion can be answered.
There are two physical causes for nonlinearity in a fluid, i.e., two reasons why β is non-zero. One reason is that the speed of sound in a fluid moving with speed u is c0 + u. A, perhaps, familiar consequence of this result is that sound traveling up wind is slower than that traveling down wind. In our case, the fluid motion is not caused by wind, but by the passage of the sound wave itself. During the passage of a sound wave, the speed of the fluid medium u at a particular
Fig. 2. a) Graph of normalized acoustic amplitude as a function of t/T at three propagation distances. The initially-sinusoidal waveform is depicted in blue. The green and red curves show the waveform at successively greater propagation dis- tances, respectively. The waveform becomes more and more distorted as it propa- gates. b) Graph of the frequency spectrum of the waveforms shown in a). Increased waveform distortion is accompanied by increased harmonic content.
20 Acoustics Today, October 2005