Acoustic pressure

An appropriate sensor (such as a microphone or ear) placed in the sound field will detect a time-varying deviation from the equilibrium pressure found at that point within the air. The changing total pressure P measured will vary about the equilibrium pressure P₀ by a small amount called the acoustic pressure, $p\text{ =\hspace{0.5em}}P\text{-}P{}_{\text{0}}$. Because the acoustic pressure is fluctuating with time, it is customary to characterize the amplitude of a sound with the peak value or a time-averaged value, such as the root-mean-square pressure p_rms.

For sound in air, the acoustic pressure is a tiny fraction of atmospheric pressure. Standard atmospheric pressure is approximately 10⁵ Pa, while the acoustic pressure amplitude of a car horn heard 1 m away is approximately 0.1 Pa (one-millionth of atmospheric pressure). Human sensitivity to acoustic pressure ranges from 20 μPa at the frequencies at which the ear is most sensitive to about 20 Pa when the sound is painfully loud. See also: Atmosphere; Microphone; Pressure; Pressure measurement; Sound pressure

Intensity, loudness, and the decibel

The strength of a sound wave is described by its intensity I. From basic physical principles, the instantaneous rate at which energy is transmitted by a sound wave through a unit area is given by the product of acoustic pressure and the component of particle velocity perpendicular to the area. For waves that are spherical or planar (which is approximately true for any wave that is not too close to the source), intensity is proportional to the square of p_rms. If all quantities are expressed in SI units (pressure amplitude in Pa, speed of sound in m/s, and density in kg/m³), then the intensity will be in watts per square meter (W/m²). See also: Sound intensity

Because of the way the strength of a sound is perceived, it is conventional to specify the intensity of sound in terms of a logarithmic scale, known as sound intensity level (SIL), which has the dimensionless unit of the decibel (dB). To compute the SIL of sound with intensity I, it is necessary to specify a reference intensity I_ref. See Eq. (1).

(1)

The standard reference level is the threshold of hearing, which is defined as $I{}_{\mathrm{ref}}\text{ = 10}{}^{\text{-12}}$ W/m². This is the smallest sound intensity that can be perceived by an individual with unimpaired hearing for frequencies between 2 and 4 kHz, the frequency range of greatest sensitivity. Thus, the weakest sounds that can be perceived by humans have an intensity level of 0 dB. Normal conversational levels are around 60 dB, and hearing can be damaged if exposed even for short times to levels above 120 dB. See also: Decibel

Every doubling of the intensity increases the intensity level by 3 dB. Because the human perception of loudness is also logarithmic, every successive doubling of the acoustic intensity corresponds to the same incremental increase in loudness. For sounds between about 500 Hz and 4 kHz, the loudness of the sound is doubled if the intensity level increases by roughly 9 dB. Thus, a doubling of loudness corresponds to about an eightfold increase in intensity. For frequencies above 4 kHz or lower than 500 Hz, the sensitivity of the ear is appreciably lessened. Sounds at these frequency extremes must have higher threshold intensity levels before they can be perceived, and an increase in loudness requires a smaller change in the intensity. Thus, the ear behaves rather differently at high loudness levels, where sounds of equal intensities are perceived as similarly loud. It is because of this characteristic that reducing the volume of recorded music causes it to sound thin or tinny, lacking both highs and lows of frequency. See also: Hearing (human); Loudness; Psychoacoustics

Because most sound-measuring equipment detects acoustic pressure rather than intensity, it is convenient to define an equivalent scale in terms of the sound pressure level (SPL). In most applications, the intensity level and sound pressure level are identical, but this is not always true (these levels may not be equivalent for standing waves, for example). For applications that do not involve human perception, sound level is computed using Eq. (1). When the context does involve human exposure to sound (as it often does), the frequency dependence of loudness sensitivity is accounted for using a weighting function. Most commonly used is A-weighting, which is applicable when most of the sound exposure is at levels less than 90 dB. For high sound levels, C-weighting is used, which accounts for the flattening of human loudness response at higher intensities. See also: Acoustic noise

Frequency and pitch

The perception of pitch corresponds to notes on a musical scale. There is a close correspondence between pitch and frequency. Sounds with a higher frequency are perceived as having a higher pitch, with small influences depending on loudness, duration, and the complexity of the waveform. For the pure tones (monofrequency sounds) encountered mainly in the laboratory, pitch and frequency are only approximately proportional. For the more complex waveforms usually encountered, however, the presence of harmonics favors a proportional relationship between pitch and frequency. See also: Pitch

Two tones generated together cannot be distinguished from each other if their frequencies are the same. If their frequencies f₁ and (slightly higher) f₂ are nearly, but not exactly, identical, the ear will perceive a slow beating, hearing a single tone that is the average of the frequencies of the two individual tones and an amplitude that varies slowly according to the difference between the two frequencies. As f₂ increases more, the beating will quicken until it first becomes coarse and unpleasant (dissonant) and then resolves into two separate tones of different pitches. With still further increase, a sense of beating and dissonance will reappear, leading into a blending or consonance that then breaks again into beats and dissonance, and the whole cycle of events repeats. These islands of consonance surrounded by beating and dissonance are attained whenever the ratio of the two frequencies becomes that of small integers, $f{}_{\text{2}}\text{/}f{}_{\text{1}}\text{ = 1/1, 2/1, 3/2, 4/3, …}$. The larger the integers in the ratio, the subtler the effects become. See also: Beat; Frequency (wave motion)

Frequency spectrum and timbre

Sound waves that consist of a single frequency have a waveform that is sinusoidal and are heard as a "pure tone." As mentioned previously, the sound of a person whistling or blowing across a bottle approximates a pure tone. Most sounds, however, are complex in that they consist of many frequency components, each with a particular amplitude. Component frequencies that are integer multiples of a base, or fundamental, frequency are called harmonics. A sound consisting primarily of harmonics will have a waveform that is periodic with the fundamental frequency and will have a perceived pitch that corresponds to the fundamental. A complex periodic waveform, together with its component harmonics, is illustrated in Fig. 2. Examples of sounds with a definite pitch, consisting of harmonics, include the human voice and those produced by many musical instruments. If the strongest components in a sound are not related harmonically, they are more likely to be perceived individually, and the overall sound will not have a definite pitch. Musicians refer to such frequency components as "overtones" or "partials." Some percussion instruments, such as bells or gongs, produce many discernible overtones when struck. Nonharmonic components that are relatively small in amplitude can contribute a complex character to a sound without interfering with a sense of pitch. Sounds that comprise a broad spectrum of component frequencies, such as footsteps, falling water, or unvoiced consonants, do not have discernible tones. See also: Bell; Harmonic (periodic phenomena)

Fig. 2 Illustration of a complex periodic waveform (black) and its five harmonic components (red is the fundamental; blue is the 2nd harmonic; green is the 3rd harmonic; violet is the 4th harmonic; orange is the 5th harmonic). The period of the full wave is the same as the period of the fundamental component. (Credit: Andrew Piacsek)

Sounds with a definite pitch, even if they have the same pitch, can be differentiated by the relative amplitudes of the harmonic components. This is perceived as a difference in timbre, which can be characterized by subjective terms such as "bright," "dark," "tinny," or "mellow." The perception of timbre provides a great deal of information about the source of a sound. It enables us to identify musical instruments, different vowel sounds, and even individual speakers. See also: Speech perception

A quantitative representation of the timbre of a sound is the spectrum plot, which shows the amplitude (or intensity) versus frequency of all the components in a sound at a given moment in time. Figure 3 shows the spectrum of the complex waveform in Fig. 2. The technique of analyzing sounds relies on the spectrum plot. Interesting sounds, such as speech and acoustical musical instruments, exhibit time-varying spectra in which the harmonic amplitudes vary with time. Time-variant spectral analysis has many applications, such as automated speech recognition, music recording, animal studies, and noise control.

Fig. 3 Spectrum of a sound consisting of the first five harmonics; the fundamental frequency is 440 Hz. The horizontal axis represents frequency (in kHz); the vertical axis is sound intensity (in dB, relative to total intensity). With a linear frequency axis, harmonics are clearly evident as evenly spaced peaks. (Credit: Andrew Piacsek; plot produced using Raven Lite software)

As noted, the relative amplitude of spectral components, and thus the timbre, changes over time. When a chime is struck, many overtones are produced, providing the initial "clang." However, the highest-frequency motions of the bell decay more rapidly than lower ones, leaving a tone with a clear sensation of pitch. Plucking a guitar string near the bridge with a fingernail yields a similar effect. The initial sound of a wind instrument or a bowed string instrument, known as the attack of a note, is indicative of the mechanism by which vibrations are initiated. The brief squeak of the reed in a clarinet, the rasp of a bow on a violin string, or the buzzing of lips on a trumpet is produced before these sources quickly adjust to the natural frequency of the medium. A gong gives a very complex impression throughout its sound because many inharmonic and nonconsonant overtones are present, which die away at different rates, so that the sound seems to shift in pitch and timbre. It is the abundance of harmonics and overtones, the distribution of intensity among them, and how they die away preferentially in time that provide the subjective evaluation of the nature or timbre of the sound. See also: Musical acoustics

Speed of sound

The speed of sound in a medium, such as air, is temperature-dependent. For instance, sound moves faster through hot, less dense air than through cold, denser air because the molecules in hot air move faster than in cold air, and thus transmit vibrations more effectively. Expressing this relationship mathematically, acoustic pressure and the density variation are related by Eq. (4),

(4)

where the condensation s is $\text{(ρ – ρ}{}_{\text{0}}\text{)/ρ}{}_{\text{0}}$ and is a constant of proportionality. This is a three-dimensional Hooke's law stating that stress and strain are proportional. The propagation of sound requires such rapid density changes that there is not enough time for thermal energy to flow from compressed (higher-temperature) elements to decompressed (lower-temperature) elements. Because there is little heat flow, the process is nearly adiabatic and is the adiabatic bulk modulus (rather than the more familiar isothermal modulus relevant to constant-temperature processes). A derivation of the wave equation for a fluid reveals that the speed of sound is given by Eq. (5). Equations (4) and (5) can be combined to give Eq. (6). See also: Adiabatic process

(5)

(6)

In most fluids, the equation of state is not known, and the speed of sound must be measured directly or inferred from experimental determinations of the bulk modulus. In distilled water, the measured speed of sound in meters per second can be represented by a simplified Eq. (7),

(7)

where T is the temperature in degrees Celsius and ₀ is the equilibrium pressure in bars. For most gases at reasonable pressures and temperatures, however, the equation of state is known to be very close to the perfect gas law, so a theoretical expression for the speed of sound in an ideal gas, is Eq. (8).

(8)

Thus, the speed of sound in an ideal gas depends on temperature, increasing as the square root of the absolute temperature. Because many applications of Eq. (8) are for gases around room temperature, it is convenient to re-express the formula in terms of degrees Celsius and the speed of sound c₀ at $T\text{ = 0°C}$ (32°F), as in Eq. (9).

(9)

This equation can be used to calculate the speed of sound for a perfect gas at any temperature in terms of the speed of sound in that gas at 0°C (32°F). In air, for example, $\text{c}{}_{\text{0}}\text{ =}$ 331 m/s (1086 ft/s), and at 20°C (68°F) the speed of sound is calculated to be 343 m/s (1125 ft/s). See also: Gas; Heat capacity; Underwater sound

In solids, the sound speed depends on the transverse extent of the solid with respect to the wave. If the solid is rodlike, with transverse dimensions less than the acoustic wavelength, the solid can swell and contract transversely with relative ease, which tends to slow down the wave. For this case, the speed of sound is given by Eq. (10),

(10)

where is Young's modulus. If, at the other extreme, the solid has transverse dimensions much larger than the acoustic wavelength, then a longitudinal wave will travel with a greater speed of sound given by Eq. (11),

(11)

where is the shear modulus. These two different propagation speeds are often called bar and bulk sound speeds, respectively. See also: Elasticity; Young's modulus

Andrew Piacsek