EXERCISES
4.2 Physics of Sound
4.2.1 Basics
The generation of sound involves the vibration of particles in a medium. In speech production, air particles are perturbed near the lips, and this perturbation moves as a chain reaction through free space to the listener; the perturbation of air molecules ends in the listener’s ear canal, vibrating the ear drum and initiating a series of transductions of this mechanical vibration to neural firing patterns that are ultimately perceived by the brain. The mechanism of the chain reaction in sound generation can be illustrated by a simple analogy. Consider a set of pool balls configured in a row on a pool table. The striking of the cue ball sets up a chain reaction whereby each ball in the series is struck until the last ball reaches the intended pool-table pocket.
Each ball has a “particle velocity” (m/s) and there is a “pressure” (newtons/m2) felt by each ball due to the interaction between balls. After the first ball is struck, a pressure increase is felt successively by each ball, as well as a change in velocity. As the balls move closer together, we say they are in acompressionstate which occurs locally in space, i.e., in the vicinity of each ball. On the other hand, if the pool stick were to move fast enough away from the first ball, we can stretch our imagination to envision a vacuum set up behind the first ball, thus moving each ball successively in the other direction, creating a local decrease in pressure. For this case, the balls move farther apart which is referred to as ararefactionstate near each ball. If we move the pool stick sinusoidally, and fast enough, we can imagine creating atraveling waveof local compression and rarefaction fluctuations.
A deficiency in this pool-ball analogy is that there is no springiness between the balls as there is between air particles. Air particles stick together and forces exist between them. To refine our analogy, we thus connect the pool balls with springs. Again we have two states: (1) compression (pushing in the springs between balls) and (2) rarefaction (pulling out the springs between balls). Also, as before, a traveling wave of compression and rarefaction fluctuations is created by pushing and pulling the springs. Observe that the pressure and velocity changes are not permanent; i.e., the balls are not permanently displaced, and this is why we refer to the changes in pressure and velocity aslocalfluctuations in compression and rarefaction.
In describing sound propagation, we replace the pool balls with air molecules and the springs with forces between the air molecules; in Figure 4.1 we depict these forces with little springs. As with the pool balls with connecting springs, air molecules are locally displaced, they
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y
x
(a)
(b)
(c)
Infinitely Large Wall
Figure 4.1 Compression and rarefaction of air particles in front of an infinitely large wall: (a) illustration of springiness among air particles; (b) compression; (c) rarefaction.
have a particle velocity, and there is pressure built up among the molecules due to the springiness and collisions that occur with themselves.1
Consider now placing a moving wall in front of the air molecules, and suppose that the wall is infinitely large in the y, z direction (z coming out of the page). As with the balls attached by springs, by moving the wall to the right or to the left, we create a chain reaction of compression or rarefaction, respectively, and, with a sinusoidal movement of the wall, we create a traveling wave of compression and rarefaction fluctuations. When the wall is pushed to the right, the molecules in front of the wall are compressed and collide with one another and build up pressure. The pressure increase in front of the wall moves as a chain reaction to the right. When pulled back, the piston creates a region of rarefaction in front of the wall. The local decrease in pressure travels as a chain reaction to the left. When moved sinusoidally, the wall generates
1Pressure = force/unit area = time rate of change of momentum/unit area [2].
a traveling wave of rarefaction and compression fluctuations characterized by local sinusoidal changes in particle pressure. In addition, local changes in particle velocity and displacement accompany pressure changes. Here the sound travels as aplane wavelongitudinally along the xaxis of Figure 4.1 [2]. This is in contrast to perturbations of air particles in other forms, i.e., other than the longitudinal flow wave of the compression-rarefaction type, such as rotational or jet flow [14].
We are now in a position to give a formal definition of a sound wave.
Asound waveis the propagation of a disturbance of particles through an air medium, or, more generally, any medium,2 without the permanent displacement of the particles themselves. Associated with the disturbance are local changes in particle pressure, dis- placement, and velocity. The termacousticis used in describing properties or quantities associated with sound waves, e.g., the “acoustic medium.”
A pressure disturbance will reside about some ambient (atmospheric) pressure. The pressure variation at the ear causes the ear’s diaphragm to vibrate, thus allowing the listener to “perceive”
the sound wave. For a sinusoidal disturbance, with the sinusoidally-varying wall, there are a number of important properties that characterize the sound wave (Figure 4.2).
Thewavelengthis the distance between two consecutive peak compressions (or rarefac- tions) in space (not in time), and is denoted byλ. If you take a snapshot at a particular time instant, you will see a sinusoidally-varying pressure or velocity pattern whose crests are sep- arated by the wavelength. Alternately, the wavelength is the distance the wave travels in one cycle of the vibration of air particles. Thefrequencyof the sound wave, denoted byf, is the number of cycles of compression (or rarefaction) of air particle vibration per second. If we were to place a probe in the air medium, we would see a change in pressure or in particle velocity of f cycles in one second. Therefore, the wave travels a distance off wavelengths in one second.
The velocity of sound, denoted byc, is therefore given byc=f λ. Because the speed of sound c=f λand observing that the radian frequency= 2πf, then/c= 2π/λ. We denote 2π/λbyk, which we call thewavenumber. At sea level and 70◦F,c =344 m/s; this value is approximate because the velocity of sound varies with temperature. The velocity of sound is distinctly different from particle velocity; the former describes the movement of the traveling compression and rarefaction through the medium, and the latter describes the local movement of the particles which, on the average, go nowhere (as we saw earlier in the pool-ball analogy).
λ 1
0 –1
x Figure 4.2 Traveling sinusoidal wave characterized by wavelengthλand frequencyf.
2Sound waves can also propagate in, for example, solid and liquid media.
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EXAMPLE4.1 Suppose the frequency of a sound wavef = 50 cycles/s (or Hertz, denoted by Hz) and assume that the velocity of sound at sea level isc = 344 m/s. Then the wavelength of the sound wave λ = 6.88 m. For f = 1000 Hz, λ = 0.344 m, while for f = 10000 cycles/s, λ = 0.0344 m. This wide range of wavelengths occurs in the propagation of speech
sounds. 䉱
In describing sound propagation, it is important to distinguish pressure change due to variations that occur rapidly or slowly [2]. Anisothermalprocess is a slow variation that stays at constant temperature. Isothermal compression of a gas results in an increase in pressure because a given number of molecules are forced into a smaller volume and will necessarily collide with each other (and the boundaries of an enclosure) more often, thus increasing the time rate of change of momentum. However, because the variation is slow, there is time for the heat generated by collisions to flow to other parts of the air medium; thus the temperature remains constant. On the other hand, anadiabaticprocess is a fast variation in which there is no time for heat to flow away and thus there occurs a temperature change in the medium. The heat generated causes an even greater number of collisions. The molecules get hotter faster and collide with each other (or the boundaries of an enclosure) more frequently. They have greater momentum (they are moving faster) and thus transfer more momentum to each other during collisions. It follows that an adiabatic gas is “stiffer” (than an isothermal gas) since it takes more force to expand or compress it.3For most of the audible range, a sound wave is an adiabatic process,4a property that we use in deriving the wave equation.