Introduction to the Physics of Hearing

Introduction to the Physics of Hearing

If a tree falls in the forest and no one is there to hear it, does it make a sound? The answer to this old philosophical question depends on how you define the sound. If sound only exists when someone is around to perceive it, then there was no sound. However, if we define sound in terms of physics; that is, a disturbance of the atoms in matter transmitted from its origin outward (in other words, a wave), then there was a sound, even if nobody was around to hear it.

Such a wave is the physical phenomenon we call sound. Its perception is hearing. Both the physical phenomenon and its perception are interesting and will be considered in this text. We shall explore both sound and hearing; they are related but are not the same thing. We will also explore the many practical uses of sound waves, such as in medical imaging.

Sound

By the end of this section, you will be able to:

Define sound and hearing.

Describe sound as a longitudinal wave.

Sound can be used as a familiar illustration of waves. Because hearing is one of our most important senses, it is interesting to see how the physical properties of sound correspond to our perceptions of it. Hearing is the perception of sound, just as vision is the perception of visible light. But sound has important applications beyond hearing. Ultrasound, for example, is not heard but can be employed to form medical images and is also used in treatment.

The physical phenomenon of sound is defined to be a disturbance of matter that is transmitted from its source outward. Sound is a wave. On the atomic scale, it is a disturbance of atoms that is far more ordered than their thermal motions. In many instances, the sound is a periodic wave, and the atoms undergo simple harmonic motion. In this text, we shall explore such periodic sound waves.

A vibrating string produces a sound wave as illustrated in Figure 2. As the string oscillates back and forth, it transfers energy to the air, mostly as thermal energy created by turbulence. But a small part of the string’s energy goes into compressing and expanding the surrounding air, creating slightly higher and lower local pressures. These compressions (high-pressure regions) and rarefactions (low-pressure regions) move out as longitudinal pressure waves having the same frequency as the string—they are the disturbance that is a sound wave. (Sound waves in the air and most fluids are longitudinal because fluids have almost no shear strength. In solids, sound waves can be both transverse and longitudinal.) Figure 2c shows a graph of gauge pressure versus distance from the vibrating string.

The amplitude of a sound wave decreases with distance from its source because the energy of the wave is spread over a larger and larger area. But it is also absorbed by objects, such as the eardrum in Figure 3, and converted to thermal energy by the viscosity of air. In addition, during each compression, a little heat transfers to the air, and during each rarefaction even less heat transfers from the air, so that the heat transfer reduces the organized disturbance into random thermal motions. (These processes can be viewed as a manifestation of the second law of thermodynamics presented in Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency.)

Whether the heat transfer from compression to rarefaction is significant depends on how far apart they are—that is, it depends on wavelength. Wavelength, frequency, amplitude, and speed of propagation are important for sound, as they are for all waves.

Section Summary

Sound is a disturbance of matter that is transmitted from its source outward.

Sound is one type of wave.

Hearing is the perception of sound.

Speed of Sound, Frequency, and Wavelength

LEARNING OBJECTIVES

By the end of this section, you will be able to:

Define pitch.

Describe the relationship between the speed of sound, its frequency, and its wavelength.

Describe the effects on the speed of sound as it travels through various media.

Describe the effects of temperature on the speed of sound.

Sound, like all waves, travels at a certain speed and has the properties of frequency and wavelength. You can observe direct evidence of the speed of sound while watching a fireworks display. The flash of an explosion is seen well before its sound is heard, implying both that sound travels at a finite speed and that it is much slower than light. You can also directly sense the frequency of a sound. Perception of frequency is called pitch. The wavelength of sound is not directly sensed, but indirect evidence is found in the correlation of the size of musical instruments with their pitch. Small instruments, such as a piccolo, typically make high-pitch sounds, while large instruments, such as a tuba, typically make low-pitch sounds. High pitch means small wavelength, and the size of a musical instrument is directly related to the wavelengths of sound it produces. So a small instrument creates short-wavelength sounds. Similar arguments hold that a large instrument creates long-wavelength sounds.

The relationship of the speed of sound, its frequency, and wavelength is the same as for all waves: v_w = fλ, where v_w is the speed of sound, f is its frequency, and λ is its wavelength. The wavelength of a sound is the distance between adjacent identical parts of a wave—for example, between adjacent compressions as illustrated in Figure 2. The frequency is the same as that of the source and is the number of waves that pass a point per unit of time.

Earthquakes, essentially sound waves in Earth’s crust, are an interesting example of how the speed of sound depends on the rigidity of the medium. Earthquakes have both longitudinal and transverse components and these travel at different speeds. The bulk modulus of granite is greater than its shear modulus. For that reason, the speed of longitudinal or pressure waves (P-waves) in earthquakes in granite is significantly higher than the speed of transverse or shear waves (S-waves). Both components of earthquakes travel slower in a less rigid material, such as sediments. P-waves have speeds of 4 to 7 km/s, and S-waves correspondingly range in speed from 2 to 5 km/s, both being faster in a more rigid material. The P-wave gets progressively farther ahead of the S-wave as they travel through Earth’s crust. The time between the P- and S-waves is routinely used to determine the distance to their source, the epicenter of the earthquake.

The speed of sound is affected by temperature in a given medium. For air at sea level, the speed of sound is given by

vw = (331 m/s) √ T

273 K

 

vw=(331 m/s) T

273 K

,

where the temperature (denoted as T ) is in units of kelvin. The speed of sound in gases is related to the average speed of particles in the gas, v_rms, and that

vrms = √ 3kT

m

vrms=3kT

m

,

where k is the Boltzmann constant (1.38 × 10⁻²³ J/K) and m is the mass of each (identical) particle in the gas. So, it is reasonable that the speed of sound in air and other gases should depend on the square root of temperature. While not negligible, this is not a strong dependence. At 0ºC, the speed of sound is 331 m/s, whereas at 20.0ºC it is 343 m/s, less than a 4% increase. Figure 3 shows the use of the speed of sound by a bat to sense distances. Echoes are also used in medical imaging.

One of the more important properties of sound is that its speed is nearly independent of frequency. This independence is certainly true in the open air for sounds in the audible range of 20 to 20,000 Hz. If this independence were not true, you would certainly notice it for music played by a marching band in a football stadium, for example. Suppose that high-frequency sounds traveled faster—then the farther you were from the band, the more the sound from the low-pitch instruments would lag that from the high-pitch tones. But the music from all instruments arrives in cadence independent of distance, and so all frequencies must travel at nearly the same speed. Recall that

In a given medium under fixed conditions, v_w is constant, so that there is a relationship between f and λ; the higher the frequency, the smaller the wavelength. See Figure 4 and consider the following example.

The speed of sound can change when sound travels from one medium to another. However, the frequency usually remains the same because it is like a driven oscillation and has the frequency of the original source. If v_w changes and f remains the same, then the wavelength λ must change. That is, because v_w = fλ, the higher the speed of a sound, the greater its wavelength for a given frequency.

Section Summary

The relationship of the speed of sound vw, its frequency f, and its wavelength λ is given by vwfλ, which is the same relationship given for all waves.

In the air, the speed of sound is related to air temperature T by v

w=(331m/s)√T

273 K

v_w is the same for all frequencies and wavelengths.

end of the first part!!!