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Table 2
before nonlinear effects have become significant.
Once a shock forms, propagation is never the same, as illustrated by the following example. According to linear acoustics, if the amplitude of a source is doubled, the ampli- tude of the sound is doubled. According to linear theory, which of course eventually breaks down, there is no limit to the amplitude of sound. This is not true in the presence of a shock. Once a shock is formed, there is an upper limit to the amplitude that the sawtooth can reach. In a process known as acoustic saturation, any additional power delivered by a source is dissipated in the shock. It does not go into increas-
ing the amplitude of the wave.
Referring back to Fig. 3b, one may wonder about the
significance of the n = 0 term in the output of a nonlinear system. Consider the nonlinear system to be a small volume of fluid exposed to a passing high-amplitude sound wave. At low acoustic amplitudes, the response of this volume of gas will be very nearly linear. For instance, the displacement of the gas in the volume will be a linear function of the pres- sure amplitude. As the pressure undergoes a complete acoustic cycle, the displacement of the volume undergoes a complete acoustic cycle, returning to its starting point after that cycle.
However, at high amplitudes the volume of gas no longer behaves as a linear system; it behaves nonlinearly. According to Fig. 3b, the response to a single-frequency sinusoid pres- sure, will be a displacement that is described by a combina- tion of sinusoids of frequencies nf. The n = 0 term corre- sponds to a net displacement of the gas volume after one cycle. In other words, the volume of gas does not return to its starting point after a complete cycle. This behavior carries over to other variables as well, the general conclusion being that the gas volume does not return to its original state after the passage of a high-amplitude acoustic wave.
The amplitude-dependent speed of sound also gives the space shuttle its characteristic double-boom, heard during reentry to the atmosphere. The waveform in the near vicin- ity of a supersonic vehicle is composed on a sequence of shocks radiated from various parts of its surface, the nose, wings, tail, etc., as depicted in Fig. 5. As this sequence prop- agates towards the ground, the higher-amplitude, faster shocks overtake and “consume” the lower amplitude, slow- er ones, b. Given sufficient propagation distance, i.e., suffi- cient time, eventually only two shocks remain, c, resulting in the characteristic N-wave. Notice that the duration of the sequence of shocks lengthens as it propagates. The reason for this stretching is that the front shock travels faster than the back shock. As a result the time interval between the
two steadily increases with propagation distance. By the time the wave reaches a listener on the ground, the front and back shocks are sufficiently separated in time to be per- ceived as two distinct events, boom-boom. Because these booms come without warning, they can be startling and their impact annoying. So annoying, in fact, that overland supersonic flight is currently banned.
Advances in computer modeling technology have, how- ever, led recently to improved aircraft designs that could, perhaps, significantly reduce the annoyance caused by supersonic flight. By carefully shaping the aircraft surface, the shape of the sonic boom can be modified, hence reduc- ing its perceived annoyance. This concept was demonstrat- ed during the August 2003 test flights of Northrop Grumman’s Shaped Sonic Boom Demonstrator, a modified U.S. Navy F-5E. At least two companies, Aerion and Supersonic Aerospace International (working with Lockheed Martin), have publicly announced plans to build supersonic business jets.
Up to this point we have considered the nonlinear response to a single input. Now suppose that two initially- sinusoidal sound waves propagate together. Linear theory predicts that the resulting action is simply the sum, or superposition, of the two individual signals acting alone. That is, the sum of a sound wave at f1 and one at f2. However, nonlinearity causes a much more complicated output as depicted in Fig. 6. Based on the previous discus- sion, one would expect the output to consist of harmonics of the two individual signals, nf1 and mf2. In addition, how- ever, the output contains the sum and difference of all the possible harmonics, 􏰁nf1 + 􏰁mf2. For instance suppose
Fig. 5. A notional diagram depicting the propagation of a waveform initially con- taining numerous shocks, such as might be generated near a supersonic aircraft. As the propagation distance increases (a – d), the duration of the event increases and the shocks coalesce, forming an N-wave (c).
Not Your Ordinary Sound Experience 23