Page 21 - Fall 2005

NOT YOUR ORDINARY SOUND EXPERIENCE: A NONLINEAR-ACOUSTICS PRIMER
Anthony A. Atchley
Graduate Program in Acoustics, The Pennsylvania State University University Park, PA 16802
Ordinarily, sound is a very weak
mechanical process. So weak,
in fact, that the majority of
acoustic phenomena encountered in
everyday life can be modeled to a high
degree of accuracy by assuming that
the amplitude of the sound is infini-
tesimally small. Our intuition about
sound is shaped, to a large extent, by
the consequences of this infinitesi-
mal-amplitude, or linear, theory.
However, there are many situations
where the sound amplitude is so large
that linear theory breaks down. This
high-amplitude regime is character-
ized by acoustic phenomena unantici-
pated from linear theory. Far from just an academic exer- cise, nonlinear acoustic theory has led to widespread prac- tical applications in the areas of medical imaging, biomed- ical ultrasound, nondestructive testing, and aircraft design, to name but a few.
As implied by the title, the purpose of this article is to introduce readers unfamiliar with nonlinear acoustics to a (very) few of its most fundamental concepts and to use these concepts to discuss a handful of nonlinear acoustic phenom- ena. No attempt whatsoever is made to encompass the breadth of the field. In fact we will make use of plane waves in an ideal fluid for most of this discussion. This choice serves our purposes, for it permits brevity and simplicity. However, with sincere apologies to experts, a remarkable range of phenomena will not be touched upon at all, particu- larly nonlinear acoustics in solids. References 1–3 and the works cited within them provide extensive coverage of the field.
We begin by reviewing the most basic underpinnings of linear acoustics and how these concepts are modified under high-amplitude conditions. Two examples are discussed: the propagation of an initially-single-frequency sound wave and the interaction of two such sound waves. The first example forms the basis of wave steepening, harmonic gen- eration, shock formation, and sonic booms. The second example leads to sum and difference frequency generation and the parametric array.
Four basic concepts of linear acoustics are essential to our discussion:
1) The speed v with which sound propagates is inde- pendent of the amplitude and frequency of the sound wave. As a consequence, in the absence of absorption and geomet- rical spreading, the shape of a wave, (e.g., pressure vs. time) remains unchanged as it propagates.
“Nonlinear acoustic theory has led to widespread practical applications in the areas of medical imaging, biomedical ultrasound, nondestructive testing, and aircraft design.”
2) The medium through which a sound wave propagates oscillates longi- tudinally back and forth along the direc- tion of propagation. The speed of the moving medium u is assumed to be much, much less than the speed of sound.
3) The medium returns to its initial state after the wave passes.
4) The principle of superposition holds for linear acoustic processes. This principle states that the net effect of multiple sound waves interacting at a given location and time is the sum of the effects caused by each individual wave. Much of our intuition about acoustics is
shaped by this principle.
A single cycle of a continuous sound wave is depicted in
Fig. 1. This figure shows a graph of the fluid speed u, nor- malized to its peak value, as a function of time t, relative to the acoustic period T. The graph represents, for example, the output of a sensor as a function of time as would be seen, for instance, on the display of an oscilloscope. According to linear acoustics, each point on the waveform travels with a speed equal to the infinitesimal-amplitude speed of sound c0. However in nonlinear acoustics, the speed of propaga- tion v depends on the instantaneous amplitude of the signal. In other words, every point on the waveform travels with a different speed. In general, v can be expressed as v = c0 + βu, where c0 is the infinitesimal-amplitude speed of sound, what we ordinarily call the speed of sound, and β the coefficient of nonlinearity. The amplitude dependence of the speed of propagation lies at the origin of nonlinear acoustics. As indicated on the graph in Fig. 1, v is greater than c0 during times when u is positive, and less than c0 when u is negative. When u is zero, v = c0. Imagine traveling along with this wave at speed c0. The sound in regions of positive values of u travels faster than you. Hence, it will gradually catch up with you as it propagates. In contrast, the sound in regions of negative u travels slower than you and so you tend to catch up with it. The net result of the differences in propa- gation speed is that the waveform distorts as it travels. Notice however, that because the zero-crossings travel at speed c0, the acoustic period remains constant as the wave propagates. This behavior is a property of continuous wave- forms. As we will see later, there are waveforms for which the period changes with propagation distance.
One may be tempted to recover linear acoustic behav- ior by setting β = 0, in which case, v = c0, the linear acoustic value. However, β is not zero, even for an ideal
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