## Osk

Acoustic Terminology

Acoustics has a language all its own. Following is a short primer describing some of the terms that will be useful in this chapter.

Sound is a periodic oscillation of pressure about a mean that propagates through a medium in the form of a traveling wave. The medium can be fluid, like air or water, or it can be solid. The simplest type of sound manifests itself as a sinusoidal oscillation of pressure (Fig 10A.1), p (Pa), with respect to time, t (s), described by the equation:

This equation is a periodic function. Two important numbers describe it. The amplitude, A, measures how far pressure is elevated above (or depressed below) the mean pressure. Variations of amplitude are perceived by us as variations in loudness, or intensity, of a sound: higher amplitudes correspond to louder sounds. The frequency, f, describes how frequently the pressure oscillation repeats itself in a second. Its units are cycles per second, sometimes abbreviated as s-1 ("per second") or hertz (Hz), after the German physicist, Heinrich Hertz. Variations of frequency are perceived by us as variations of pitch, or tone: high frequency sounds correspond to high-pitch tones. The range of frequencies perceptible by humans is from about 500 Hz to 5,000 Hz. Two other numbers are sometimes used to describe sound waves. One, the period, P is simply the inverse of frequency: it describes the time required for a sound wave to complete one oscillation. The wavelength, 2, measures how far apart in space two similar points on a sound wave are from one another. (If the horizontal axis in Fig. 10A.1 were distance, not time, the length labeled P would be the wavelength, 2.) This distance will depend upon how rapidly the oscillation takes place and how rapidly sound waves propagate through the medium, also known as the speed of sound, c (m s-1). Speed of sound varies with the medium: in air, c « 330 m s-1, but sound travels much faster in water (c « 1,500 m s-1). Sound travels fastest in solid media; in steel, for example, c « 6,000 m s-1. For pressure pressure

Figure 10A.1 A sine wave of pressure, indicating amplitude (A) and period (P).

the range of frequencies perceptible to humans, wavelengths range from about 66 cm at the low end (f = 500 Hz) to about 6.6 cm at the high end (f = 5,000 Hz). It is important to keep these numbers in mind, because many aspects of the ability to generate and emit sounds are related to the wavelength of the sound emitted.

Sound intensity, I, refers to the energy being carried in sound. Sound pressure, p, described above, is a measure of the potential energy in a sound wave. As sound travels, this potential energy does work moving air molecules in a back-and-forth motion as the pressure wave passes by. These movements are very small, on the order of a hydrogen atom's diameter, but they still represent work. This work is the product of the movement and the force doing the moving. In sound, work rate, or power, is the product of the sound pressure and the velocity of the particles of air it moves. The power in sound is the sound intensity, which has units of watts per square meter (W m-2). Sound intensity is related to pressure by the formula:

where p is the density of medium (in the case of air, roughly 1 kg m-3).

Sound intensity can be measured directly. A pressure amplitude of zero in a sound therefore corresponds to no sound. It is frequently the case that sound intensities are measured in comparison to some standard, or in comparison to some other sound. The standard of comparison in this case is the decibel, abbreviated dB. The decibel level of a sound is calculated from a ratio of intensities in two sounds:

Two sounds of equal intensity will differ from each other by 0 dB (/1//2 = 1, and logarithm of 1 is 0). A sound ten times louder than the other (/1//2 = 10) will be 10 dB louder (log1010 = 1). A sound a hundred times louder (/1//2 = 100) will be 20 dB louder (log10100 = 2), and so forth. Conversely, a sound a hundred times softer than another (y/2 = 0.01) will differ from it by -20 dB (log10 0.01 = -2).

The decibel comparison is used in a variety of ways. Commonly, the comparison is made against an agreed-upon standard sound. Unfortunately, the standard sound is anything but standard: engineers and audiophiles have, for their own good reasons, adopted different conventions for the standard sound. For physics, the standard is a pressure fluctuation of 200 mPa, which corresponds to an intensity of about 6 x 10-11 W m-2. Audiophiles, because they are designing things to be heard by humans, build their standard around the acoustical behavior of the ear, which varies with frequency. More frequently, though, decibel comparisons are made between two nonstandard sounds, say the peaks in a power spectrum. If a sound is composed of a dominant frequency and its harmonics, for example, the harmonics may be described as being so many decibels relative to the dominant. Alternatively, a sound may be described as being so many decibels louder than the noise that surrounds it.

disk being driven back and forth in an oscillatory motion. We assume that the disk is surrounded by a medium (let us assume it is air) that can carry sound. We need not concern ourselves for the moment with what the engine is, only that it is doing mechanical work on the disk at a rate Qm. We know from the First Law that the work done by the engine must be conserved: energy transmitted to the disk from the engine must equal the energy transmitted from the disk to the medium surrounding it. Whether that energy emerges as sound depends upon what type of work the disk does on the medium. The type of work done, in turn, is influenced by the properties of the medium itself—its density, viscosity, and rigidity. The heart of the matter turns on two questions. First, what kinds of work does a vibrating disk do on the air surrounding it? Second, how do these different kinds of work translate into sound?

Regarding the first question, the answer depends in large measure on how fast the vibration is. Imagine first that the disk is oscillating back and forth very slowly. As the disk moves back and forth, it will drive synchronous motions of the air surrounding it. At the disk's leading edge—the one that is moving forward— air will be pushed ahead, while air behind the disk's trailing edge will be pulled along toward it. At the same time, air surrounding the edge of the disk will be moved in the opposite direction, flowing from the leading surface of the disk and around its edge to the trailing surface. This air has mass, and setting it in motion means accelerating it. Therefore, work must be done in overcoming the air's inertia, which we shall designate inertial work. The rate at which inertial work is done is I.

Now imagine that we speed up the disk's oscillation rate. At low vibration rates, nearly all the work done by the disk will be inertial work. As oscillation rates increase, though, appreciable pressures will appear around the disk, just as they do around a car speeding down the highway. Specifically, pressure will be elevated at the disk's leading surface, where the air is compressed, and will be reduced at the disk's trailing surface, where the air is temporarily rarefied. The faster the oscillation rate, the greater these pressures will be. We remind ourselves now that pressure is potential energy, and work must be done to impart it to the air. Consequently, the work done by the driver on the disk does not now go completely into doing inertial work: some portion of it goes into imparting potential energy to the air.

The secret of sound production lies in what happens to this potential energy. At the top or bottom of the disk's stroke, there will be a pressure difference between the disk's leading and trailing edges. Some of this pressure will help drive the disk back during its return stroke. In other words, some of the potential energy stored in air pressure is recovered to help drive the oscillation of the disk. This work we designate as capacitative work: the rate at which capacitative work is done is symbolized as C.

If the rate of oscillation is fast enough, not all the potential energy does capacitative work. Some of the compressed air at the disk's leading surface helps compress air ahead of it, setting in motion a wave of elastic recoil that propagates away as a wave of high pressure. As the disk draws back, it pulls the air back with it, again launching a wave of elastic recoil that propagates away, this time as a trough of low pressure. This propagating wave of up-and-down pressure, of course, is a sound wave. The portion of energy that powers this elastic recoil we designate as dissipative work, which is done at a rate D.

You are now in a position to grasp one of the essential facts of acoustics, be it physical or physiological: it is an energy balance problem (Fig. 10.2). The energy balance on the disk involves three components of work: inertial work, I; capacitative work, C, and dissipative work, D. The energy flow through the disk can be expressed as a straightforward energy balance equation, where the work done by the disk on the air must equal the work done on the disk by the engine, Qm:

Figure 10.2 The energy budget at a sound emitter, in this case a disk shown from the side (at left) and from above (at right). a: Work done on the emitter by a driver, Qm, must emerge from the emitter as inertial work, I, capacitative work, C, or dissipative work, D. b: Placing a baffle around the emitter enables a larger proportion of energy to emerge as sound.

Figure 10.2 The energy budget at a sound emitter, in this case a disk shown from the side (at left) and from above (at right). a: Work done on the emitter by a driver, Qm, must emerge from the emitter as inertial work, I, capacitative work, C, or dissipative work, D. b: Placing a baffle around the emitter enables a larger proportion of energy to emerge as sound.

(The minus sign in front of Qm is needed to differentiate between work done on the disk from work done by the disk.)

### No Loud Crickets

Equation 10.1 is at the heart of the reason cricket songs should not be loud. Sound is produced by only one type of work, dissipative work. Efficient sound production involves manipulating the disk's energy budget so that energy fed into it by the motor does mostly dissipative work, because capacitative and iner-tial work do not produce sound.2 Unfortunately, most sound-producing insects are small, and small sound

2. This is not strictly true for insects that transmit sounds over very short distances, on the order of fractions of the wavelength of the sound.

emitters generally are poor sound emitters—too much of the energy fed into them goes to capacitative and in-ertial work. A few calculations illustrate the difficulty.

It is an acoustic rule of thumb that a sound emitter should be large with respect to the wavelength of the sound it produces. Wavelength is the distance (not the time) between the high points of pressure in adjacent sound waves. Specifically, the diameter, d, r of the emitter should exceed the ratio of the sound's wavelength, A, and n:

This is bad news for crickets. Consider the harp of a common cricket, Oecanthus, as illustration. These crickets produce chirps with a carrier frequency of about 2 kHz. The wavelength A (in meters) of any sound wave is the ratio of the speed of sound, c, which in air is about 330 m s-1, and its frequency f (Hz):

A sound wave with f = 2,000 Hz therefore has a wavelength in air of about 165 mm. The diameter of the Oecanthus harp is only about 3.2 mm, though, fifty times smaller. The rule of thumb indicates the harp could emit sound pretty well at wavelengths of (3.2 mm X n) ~ 10 mm, which correspond to frequencies in air of roughly 33 kilohertz (kHz) or higher. But it does not—it emits sound at 2 kHz. Therefore, the cricket harp should be a very poor emitter of sound. Let me hasten to add that this is not necessarily a bad thing—even a very inefficient emitter of sound can, with some clever manipulation of various properties of sound, be useful in communication. (For some examples, see Box 10B.)

The Baffle Leaf of Oecanthus burmeisteri Even with an inefficient harp, however, crickets must still compete with other crickets for the attention of mates. All other things being equal, benefits should accrue to crickets that manage even modest increases in efficiency of sound production. One simple solution is to mount the sound emitter in a baffle—a wall with a hole cut in it to accommodate the oscillating disk. Baffles work by presenting a physical obstacle to the movement of air around the disk's margins (Fig. 10.2). Energy that would have powered this movement is now available to increase pressure across the face of the disk, some of which may now produce sound. This property of baffles is familiar to anyone who has ever worked with loudspeakers or high-fidelity sound reproduction: you simply must baffle a loudspeaker, or the sound quality is dismal.

Some insects have learned this trick: a remarkable example is found in a South African cricket, Oecanthus burmeisteri. The Oecanthus harp oscillates at a frequency of about 2 kHz and, as we have seen, its production of sound should be very inefficient. Baffling the harp seems an obvious thing to do. All crickets baffle their harps to some extent by situating the harp membrane in the center of the wings, where the surrounding membranes will act as baffles. The wing's effectiveness as a baffle is itself limited, however, by the wing's small size. A baffle is most effective when its diameter is one-third of the sound's wavelength or larger: for a 2 kHz tone with a wavelength of 165 mm, the baffle should have a diameter of at least 56 mm. Baffles smaller than this will help, but not as much as a 56 mm baffle would. The cricket's own wings measure about 10 mm by 5 mm, which means they should not be very effective baffles.

Oecanthus burmeisteri gets around this problem by building itself a larger baffle (Fig. 10.3). These crickets normally feed on leaves, chewing out small holes or portions from the edge of the leaf. Prior to calling, however, the male cricket chews out of the center of a leaf a pear-shaped hole, measuring roughly 8 mm by 14 mm. The leaves they call from usually measure from about 70 by 80 mm at the smallest to as large as 170 by 300 mm. During calling, the cricket positions itself so that its wings are located in the center of the pear-shaped hole. The dimensions of the leaf and hole and the positioning of the cricket are consistent with

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