17 Physics of Hearing

# 129 17.2 Speed of Sound, Frequency, and Wavelength

### Summary

• 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:

$\boldsymbol{v_{\textbf{w}}=f\lambda},$

where$\boldsymbol{v_{\textbf{w}}}$is the speed of sound,$\boldsymbol{f}$is its frequency, and$\boldsymbol{\lambda}$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 time.

Table 1 makes it apparent that the speed of sound varies greatly in different media. The speed of sound in a medium is determined by a combination of the medium’s rigidity (or compressibility in gases) and its density. The more rigid (or less compressible) the medium, the faster the speed of sound. This observation is analogous to the fact that the frequency of a simple harmonic motion is directly proportional to the stiffness of the oscillating object. The greater the density of a medium, the slower the speed of sound. This observation is analogous to the fact that the frequency of a simple harmonic motion is inversely proportional to the mass of the oscillating object. The speed of sound in air is low, because air is compressible. Because liquids and solids are relatively rigid and very difficult to compress, the speed of sound in such media is generally greater than in gases.

Medium vw(m/s)
Gases at 0ºC
Air 331
Carbon dioxide 259
Oxygen 316
Helium 965
Hydrogen 1290
Liquids at 20ºC
Ethanol 1160
Mercury 1450
Water, fresh 1480
Sea water 1540
Human tissue 1540
Solids (longitudinal or bulk)
Vulcanized rubber 54
Polyethylene 920
Marble 3810
Glass, Pyrex 5640
Aluminum 5120
Steel 5960
Table 1.Speed of Sound in Various Media.

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 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 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

$\boldsymbol{v_{\textbf{w}}=(331\textbf{ m/s})}$$\boldsymbol{\sqrt{\frac{T}{273\textbf{ K}}}},$

where the temperature (denoted as$\boldsymbol{T})$is in units of kelvin. The speed of sound in gases is related to the average speed of particles in the gas,$\boldsymbol{v_{\textbf{rms}}},$and that

$\boldsymbol{v_{\textbf{rms}}\:=}$$\boldsymbol{\sqrt{\frac{3kT}{m}}},$

where$\boldsymbol{k}$is the Boltzmann constant$(\boldsymbol{1.38\times10^{-23}\textbf{ J/K}})$and$\boldsymbol{m}$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$\boldsymbol{0^{\circ}\textbf{C}},$the speed of sound is 331 m/s, whereas at$\boldsymbol{20.0^{\circ}\textbf{C}}$it is 343 m/s, less than a 4% increase. Figure 3 shows a use of the speed of sound by a bat to sense distances. Echoes are also used in medical imaging.