Sound waves move at a speed that is determined by the properties of the medium. In general the sound speed is
where C is a constant describing the stiffness of the material (when there is no heat flow, which are “adiabatic” conditions) and is the mass density. In solids, this stiffness constant can depend on the direction the sound wave propagates. It equals Young’s modulus Y for the propagation of compressional waves down a rod that is much longer than it is wide. For steel,  m/s. In fluids (liquids and gases), C is the (adiabatic) bulk modulus B, which describes how much pressure is needed to achieve a given fractional decrease in volume. In gases, , where is the ratio of the specific heats at constant pressure () and volume (). (The ratio ranges from 1, for very large molecules, to 5/3, for an ideal monatomic gas, and is 1.4 for air, which is composed mostly of diatomic gases.) Consequently, the speed of sound in gases is
where R is the constant in the ideal gas law (7.​2) ( J/mol-K), and m is the molecular mass. The speed of sound in air is 343 m/s (at C), which is slower than that in steel, while in water it is 1,482 m/s (see Problems 10.4 and 10.5).

Waves are periodic in space (at one t) and repeat with a spatial periodicity called the wavelength (Fig. 10.1). They are also periodic in time (at one z) with a temporal periodicity called the period T, which corresponds to a frequency f with (Fig. 10.1). Mathematically, a disturbance moving to the right (towards larger z) can be expressed as —for any wave of functional form g, because remains constant as the wave moves to the right at speed . One example of this is . Similarly, a disturbance moving to the left (towards smaller z) can be expressed as , because remains constant as the wave moves to the left.

Frequency has units of cycles per second (cps) = Hertz (Hz). Sometimes we will use the radial frequency , which has units of rad/s or just 1/s, and which is related to the frequency by . The wavelength, frequency, and speed of a wave are interrelated by
So, low-frequency waves have long wavelengths, while high-frequency waves have short wavelengths. Using the speed of sound in air, 1,000 Hz sound waves in air have a wavelength of 0.34 m  ft. For some types of waves, depends on f, and so , but it does not for sound in air. When sound waves travel from one medium to another, the frequency stays the same, but the wavelength changes with the change in sound speed (10.3). Note that for acoustic waves it is the disturbance that is propagating; the actual molecules do not physically travel with the disturbance and move very little.

The intensity I of a sound wave is the energy carried by the wave per unit area and per unit time (in units of J/m-s or W/m). At a distance R from an isotropic source of average acoustic power , the intensity is
The acoustic intensity is also equal to the kinetic energy of the wave per unit volume, , times the wave speed, , or
where is the maximum speed of the molecules for a maximum displacement during the disturbance. Therefore, we see that
The acoustic impedance of a medium Z is given by the product of the mass density and sound speed for that medium, so
Table 10.1 lists the mass density, sound speed, and acoustic impedance for air, water, fat, muscle, and several other body materials. The acoustic impedance for water, fat, and muscle are 3,500 that for air. We will see later that this mismatch is responsible for the reflection of sound between air and these other media.

For simplicity, we will relabel the gauge pressure as P for the rest of this chapter and call it the sound pressure. Moreover, P will actually denote the maximum pressure in the cycle, . The magnitude of the maximum pressure variation in the sound wave is related to this maximum (out of phase) displacement by
We can understand this because pressure is the force per unit area and force is the change of linear momentum with time, and so pressure is the change in linear momentum per unit area per unit time. Using (10.8), the acoustic intensity is
We can present the sound intensity I in units of W/m or any other equivalent units. It is also very common to characterize the sound intensity in a more physiologically-based manner, in which I is referenced to  W/m; is a sound intensity that is barely audible at 3,000 Hz. Because sound intensities in our everyday experience can be many orders of magnitude larger than this reference intensity, we usually use a logarithmic scale—referenced to base 10—to characterize I. In units of bels, named after Alexander Graham Bell, I(in bels). It is, in fact, standard to use a finer scale in tenths of bels, called decibels or dB, with
For example, for  W/m we see that I(in dB, or 40 dB.

The dB scale is also used to denote the relative magnitude of intensities, such as that of relative to . With
we see that
So, a 20 dB increase in sound intensity indicates a factor of increase in I. Because the dB scale is also used to indicate relative magnitudes, this absolute acoustic unit in (10.10) is often denoted as dB SPL (sound pressure level). Table 10.2 shows the acoustic intensities of common sounds. Normal background noise is 50–60 dB SPL. Normal conversation is 60–70 dB SPL. Speech is around 70–80 dB SPL at 1 m. The threshold of pain occurs at about 120 dB SPL. Windows break at about 163 dB SPL. Shock waves and sonic booms cause levels of about 200 dB SPL at a distance of 330 m.

Apollo astronauts were exposed to very loud sounds during liftoff, over 85 dB SPL for about 80 s after liftoff—with maximum levels below 100 Hz [25, 37]. About 60 s after launch the sound in the crew area was about 123 dB over a wide frequency range, but astronauts heard much less intense sounds (because they wore helmets and suits), from 119 dB at 63 Hz to 89 dB at 2,000 Hz. On the gantry, 10 m above ground, the maximum levels were 158–168 dB SPL from 2 to 2,000 Hz and then down to 152 dB at 8,000 Hz [37, 39]. The levels were 12–31 dB lower 400 m from the gantry.

Equation (10.9) relates the sound intensity and sound pressure by . For the reference intensity , and with  W/m and  kg/m-s for air, we see that the reference pressure  N/m (Pa). Dividing these two relations for intensity and pressure gives
and using (10.10) gives
Sometimes a distinction is made between the expression for the intensity, (10.10), which gives the intensity level (IL) in units of dB IL, and that from (10.14) for pressure, which gives the sound pressure level in dB SPL, but we will not make such a distinction here and the units will be called dB SPL. Generalizing (10.12) gives
When the sound intensity increases by 60 dB, I increases by and P increases by .

There is an analog to (10.9) in electronics with power , where is the voltage drop and is the resistance. (The factor of 2 in the denominator is present for sinusoidal voltages and absent for DC voltages.) As in acoustics , the dB scale is commonly used in electronics analogous to (10.10), (10.12), and (10.15).

Within a given medium the sound wave can be attenuated by absorption and scattering. In scattering, part of the propagating beam is redirected into many directions, without being absorbed. The amplitude of the acoustic wave (A = magnitude of the pressure or displacement ) decreases exponentially with the distance the wave propagates, z. This can be written as
where is the absorption coefficient and F is the frequency dependence. For pure liquids, in Hz and for soft tissues, in Hz. Typical values of are given in Table 10.3 for body tissues and in Table 10.4 for other materials. Using (10.13), the acoustic power or intensity varies as

This variation also applies to the absorption of light in media and is called Beer’s Law. In Beer’s Law jargon the intensity absorption coefficient in (10.17) is called , and so, for instance, for tissue. Beer’s Law for light is usually expressed as:
In analyzing the measured fraction of the intensity of sound or light transmitted through a medium, all of the loss mechanisms must be considered: the reflection from interfaces (see below), absorption, and scattering. Because the wavelengths of sound waves are of the same order of magnitude as everyday objects, such as windows, pipes, heads, and so on, the diffraction and interference of sound waves is very common .

The concept of impedance is more generally used to assess the characteristics of a medium that “opposes” energy flow in a system. It is also used to characterize electrical components and circuits, where the electrical resistance is the real part of the impedance and the capacitance and inductance of a system correspond to the imaginary part of the electrical impedance. Analogous concepts are used in optics to characterize the transmission of light in media. In acoustics (and other areas), the reciprocal of the impedance Z, is called the admittance , which describes the ease of energy flow . Immittance refers to either the impedance or admittance, and is used as a general term to describe how well energy flows in a medium.

The impedance is generally a complex parameter . The real part of Z is the resistance R. Out of phase to the resistance by (or  rad) are the mass (positive) reactance (), which is proportional to frequency f, and the stiffness (or negative) reactance , which is inversely proportional to f; and are (or rad) out of phase to each other. Overall, . The net reactance is . The magnitude of the impedance is related to these component parts by
Similarly, the admittance . The real part G is the conductance, and the imaginary parts and are the mass susceptance and stiffness (compliant) susceptance, respectively. (When the impedance has only a resistive term, the conductance is .) The net susceptance is . The components of the admittance are related to the impedance terms by , , and . The magnitude of the admittance is related to these component parts by
Problems 10.26–10.29 address these and other relations involving impedances.

This discussion can apply to the impedance in acoustics, electronics, or flow (Appendix D). For example, this terminology can be applied to the analog in mechanics of a mass attached to a spring that slides on a rough surface. The friction due to sliding on the rough surface is the resistance. The inertia due to the mass is the mass reactance and the resistance to movement due to the spring is the stiffness reactance. When used in audiology and other aspects of sound, these parameters are also characterized by the term “acoustic,” such as acoustic impedance, acoustic admittance, acoustic resistance , mass (positive) acoustic reactance, and so on. To avoid confusion we could use subscripts, such as for acoustic resistance , and so on.

The units of impedance Z is ohms (), just as in electronics. The units of admittance Y is the 1/ohm = 1 mho. R, , and all have units of ohms, and G, , and all have units of mhos. In audiology the admittance values are small and the more useful unit is the millimho or mmho.

We now examine the concept of impedance and derive (10.19) by considering the motion of a body of mass m attached to a spring of force constant k. It is subject to a viscous, frictional force, characterized by , and a driving term, . This motion is described by
The driving term has a real part , because . This model is similar to the Voigt model in Chap. 4 with (from (4.​57)). (It can be represented by the model presented later in Fig. 10.13, with the force driving term acting laterally on the mass.)

Substituting a potential solution in this equation, gives the (“particular” or steady state) solution
(See Appendix C.) Differentiating this, the speed of the body is

The ratio of the driving force, , to the speed, , is the mechanical impedance Z
This impedance is sometimes expressed as
Here, provides the resistance , is the inertial term (or inertance), and is the stiffness. In terms of the mass (positive) reactance (), which is proportional to frequency f, and the stiffness (or negative) reactance , which is inversely proportional to f, we can say
with and . Equation (10.19) follows from the magnitude of a complex number.

Sound transmission from one medium to another is very important in hearing, because sound is transmitted from the air in the auricle and ear canal into the middle ear and then into the inner ear . This can be understood by seeing what happens to a sound wave incident on the planar interface between two different semi-infinite media, such as media 1 and 2 in Fig. 10.3.

This sound wave travels in medium 1 with intensity and pressure incident on the interface. The part that is reflected back into the same medium has intensity and pressure , and the part that is transmitted into medium 2 has intensity and pressure . These pressures are related by “matching the boundary conditions” at the interface, as we will see very soon. As stated earlier, the frequency is the same in both media, but the wavelength changes with the change in sound speed by (10.3), so and . These frequencies need to be the same because the sinusoidal oscillations of pressure and matter movement on both sides of the interface must always match. The magnitudes of the pressures must match so there is no net force on the interface. This gives
The motion at the interface caused by all three waves must match. The speed of this lateral displacement is and for a wave initially moving to the right it is positive for the incident and transmitted waves and negative for the reflected wave. Using (10.8), , we know that , and so matching displacement gives
Solving these two equations gives

Using (10.9) and (10.30), the fraction of intensity that is reflected is

Using (10.31), the fraction of intensity that is transmitted is

We see that and , which means that sound energy (and intensity) is conserved. Also, the reflected fraction is large (and approximately equal to 1) when is either much larger or smaller than 1, i.e., when there is a very large acoustic impedance mismatch. We will see this means the large mismatch of acoustic impedances between air and any solid or liquid medium (Table 10.1) has important consequences in the performance of the ear. The term “to match impedance” means to minimize , and this is accomplished by making and approximately equal (10.33). Examples of this mismatch in the body are given in Table 10.5. Most sound incident on the body is reflected. Little sound is reflected between soft tissues (brain, skin, kidney, liver, and so on) and between soft tissues and blood. There is significant reflection at interfaces of soft tissues with the lungs and with bones.

Many properties of sound waves depend on the medium the wave propagates in and on the characteristics of the enclosure. This is important in the production of sound—as in human speaking, musical instruments, megaphones, and sound produced in echo chambers—and in the collection of sound waves—as in human hearing in the outer ear . One way to analyze these properties is to consider the properties of sound waves in resonant cavities; more generally, these properties of resonant cavities can be applied to any waves in any resonant structure.

What is a resonant structure? Let us consider a wave on a string of length L. The string can be plucked so a localized wave can propagate along the length of the string. Alternatively, one can pluck the string so a (transverse wave) oscillates everywhere periodically. If the string is rigidly fixed at both ends, the lowest order periodic motion (or mode—for which the transverse displacement at any point along the string varies as ) is one with a half wave between the two ends (Fig. 10.4). Here , so the wavelength . The “boundary conditions ” for this motion are zero transverse motion at each end; such places with no motion are called nodes . Higher order modes with higher integral numbers of half wavelengths are possible. In general, for n half wavelengths , the wavelength is , and so the resonant wavelengths are
Since , the resonant or mode frequencies are
which is , , , … for ; these are also called the fundamental frequency, the first harmonic, the second harmonic, etc. The mode frequencies in this case are equally spaced by .

If one of the two ends of the massless string is not fixed, but is free to move (only transverse to string axis), this different boundary condition leads to a different set of resonant wavelengths and frequencies. At the fixed end, the transverse displacement is still zero, which is a node. At the free end, the transverse displacement now has maximum magnitude (and zero slope)—and this is called antinode an, as in Fig. 10.5. The lowest order mode now has a quarter wavelength between the two ends separated by the string length, and so and the wavelength . The next lowest order mode has an extra half wavelength cycle with , so the wavelength , etc. Now the resonant condition is , so the resonant wavelengths are
The mode frequencies are
which is , , , … for . Although the mode frequencies are different from the previous case, they are still equally spaced by . The lowest frequency is now and only odd harmonics are present. Figure 10.6 shows the mode frequencies for both boundary conditions.

Hollow tubes are good first-order models for the tubes in the body where longitudinal sound waves propagate (similar to pipes in a pipe organ) (Fig. 10.6). If the tube of length L is open at both ends, the pressure is fixed at the ambient pressure at each end and at the ends. This is formally equivalent to the string with fixed ends, so the resonant wavelengths and frequencies are given by (10.36) and (10.37). The density oscillation is also zero at each end (nodes) and the molecular displacement is of maximum magnitude at each end (antinodes). If the tube is open at one end and closed at the other, P is still zero at the open end (a node). At the closed end, the lateral displacement is fixed at zero (a node), while the pressure and density changes have maximum magnitude (an antinode). This is formally equivalent to the string fixed at one end and free at the order, so the resonant wavelengths and frequencies are given by (10.38) and (10.39). (The resonant wavelengths and frequencies for a tube closed at both ends are the same as for one that is open at both ends, but the identifications of nodes and antinodes are interchanged.) This is the reason why the tube length and boundary conditions are important in determining these resonances, and why these resonances can be changed by varying the tube conditions (length, varying cross-sectional area and shape)—as can happen in the region between the vocal cords and the mouth opening. For details about acoustic resonances see [53, 54].

The mathematics of resonant excitation of an acoustic medium are the same as those used in Chap. 8 to analyze (8.​48) and (8.​50), and are due to acoustic losses. This damping leads to the finite widths of the resonances for acoustic energy, which are , in Hz, between the points with half the maximum response. Figure 10.7 compares the unrealistically narrow resonances to the more realistic, broader, damped resonances. The number of cycles that an excitation lasts during its decay (to 1 / e of the initial energy) is , where the quality factor , where and f are the resonant frequencies. (See Appendix D for more details.)

Human speech is made by air from the lungs as it passes through the trachea (windpipe), larynx (which houses the vocal folds or cords ), and pharynx (throat), through the mouth and nasal cavities , and then out of the mouth and nose (Fig. 9.​1). For general sources about speech see [6, 7, 16, 17, 23, 29, 31, 43, 66, 69, 76].

The vocal folds (or vocal cords) are folds of ligament extending on either side of the larynx , with the space in between them called the glottis . The structure above the larynx is called the vocal tract . In adult females, the average length of the pharynx is 6.3 cm and that of the oral cavity is 7.8 cm, for a total vocal tract length of 14.1 cm. In adult males, the average length of the pharynx is 8.9 cm and that of the oral cavity is 8.1 cm, for a total vocal tract length of 16.9 cm.

Why can we vocalize distinct sounds well? One reason is that we are able to modify our very round tongues and the other parts of our vocal tract rapidly and precisely [43, 44]. Another is that our horizontal vocal tracts (above the tongue) and vertical tracts (below the tongue) are about equally long (Fig. 9.​1), unlike those tracts in primates. Our horizontal tracts are shorter than in primates because our faces and jaws protrude less and our vertical tubes are longer because our larynxes are lower in the neck because they are suspended from the hyoid, a small floating bone at the base of the tongue. On the downside, with the larynx so low, there is an open region where food and air travel before entering the esophagus or trachea, so with this design we are susceptible to choking on food. (In other mammals, the epiglottis and soft palate touch and there are two tubes, an inner one for air flow and an outer one for food and fluid transport.)

The shape of the vocal tract can be varied by moving the soft palate, tongue, lips, and jaw, and these are the articulators. Adjusting the vocal tract to produce speech sounds is called articulation . The basic elements of speech are classified as the (1) phonemes (the basic sounds), (2) phonetic features (how the sounds are made), and (3) the acoustic signal (the acoustic nature of the sounds). We will first discuss the phonemes and the acoustic signal, and then describe how we make sound.

Phonemes are the shortest segments of speech, which in General American English are the 14 vowel sounds and 24 consonant sounds. Each phoneme is produced by distinctive movements of the vocal tract , which are the phonetic features of speech. Vowels are produced with the vocal tract relatively open, with different shapes of the opening. Consonants are produced by a constriction or closing of the vocal tract. The production of phonemes is characterized by three phonetic features (a) voicing, (b) place of articulation , and (c) manner of articulation.

Sounds produced as air rushes though the vibrating vocal folds in the larynx (Fig. 9.​1) are called voiced sounds. All vowels are voiced sounds. (Touch the middle of your throat as you speak them.) Many consonants are also voiced, such as the “d,” “m,” “w,” and “v” sounds. (We will not use the standard formal notation for phonemes [6, 17, 69].) Sound produced without the vocal folds vibrating and only involving air flow through constrictions or past edges produced by the tongue, teeth , lips, and palate are called unvoiced sounds, such as for the “t” and “f” sounds.

The obstructions needed to produce consonants are formed in different places of articulation , including (in order from the front to the back of the mouth) bilabial (both lips, or labial) for the “p,” “b,” “m,” and “w” sounds, labiodental (the bottom lip and upper front teeth ) for the “f” and “v” sounds, dental or interdental (teeth) for both “th” sounds (as in “thin” and “them”), the alveolar ridge or alveolar (upper gums near the teeth) for the “d,” “t,” “s,” “z,” “n,” “r,” and “l” sounds, palatal or alveopalatal (hard palate, which is behind the upper gums) for the “sh,” “zh” (as in “vision”), “ch,” “j,” and “y” sounds, velar (soft palate, which is behind the hard palate) for the “k,” “g,” “ng,” and “w” sounds, and glottal for the “h” sound.

The mechanical means by which consonants are formed, including the way air is pushed through the opening, is the manner of articulation . Plosive or stop sounds such as “d,” “b,” “p,” “g,” “t,” and “k” have a staccato nature because they are formed by blocking air flow and then letting a slight rush of air. Fricative sounds such as the voiceless “f,” “th” (as in “thin”), “s,” and “sh,” and the voiced “v,” “th” (as in “them”), “z,” and “zh” (each set in order from the front to the back of the mouth) have a hissing nature because air flow is constricted at the place of articulation , making the air flow turbulent . The “m” sound is nasal because the soft palate is lowered to couple the nasal cavities to the pharynx and air is suddenly released to flow through the nose . Some unvoiced sounds have a combination of initially plosive and then fricative character (and are affricative), such the “ch” and “j” sounds and other gutterals. The approximants are produced by moving one articulator to another without creating a closed constriction, such as for the “w,” “y,” “r,” and “l” sounds. The “w” sound is also said to be formed in a semivowel manner.

The acoustic signal is the set of acoustic frequencies and intensities as a function of time. For example, Fig. 10.8a shows a sound spectrogram—a plot of the frequency components of sound versus time—for the expression “She had her dark suit in …” Distinctively different signals are seen for the different vowels and consonants. This spectrogram clearly has more useful information than the time-domain waveform for the same expression in Fig. 10.8b (even though they have the same information content).

These different contributions of vowels and consonants are more clearly seen in Fig. 10.9 [5]. In part (a) the “sh” and “ch” consonant spectra are seen to be different and have different durations; also there is an interval of silence before the “ch.” They are both very different from the “s” spectrum. Each vowel has characteristic bands. For example, the long a in “say” in (a) has three frequency bands, near 500, 1,700, and 2,500 Hz. These can be, respectively, labeled as F, F, and F. Such multiple bands are characteristic of vowels and are called formants , the first-formant (F), the second-formant (F), and so on. These formant frequencies differ for the different vowels, as is seen for this long a, the “ah” in “father” and “pot” in part (a) and /ba/ in part (c), and the long i in “mite” in part (b). As we will see later, these formants define the vowels and the characteristic feature in the perception of vowels. For all three syllables in part (c), the lowest part of the first spectrum of the consonant increases into the F of the vowel. However, this variation is different for the second and third formants and this helps define these different sounds. For /ba/, F and F rise into the vowel; for /da/, F and F fall into the vowel; and for /ga/, F falls, while F rises into the vowel.

There are three sequential neuromuscular systems involved in speech production (Fig. 10.10). (a) The lungs are the airstream mechanism in which muscle force is used to produce a stream of compressed air. (b) The larynx is the phonation mechanism, which takes the compressed air and turns it into an acoustic buzz , hiss, or explosion. (c) The vocal tube track is the articulation mechanism , which takes the larynx sounds and turns them into speech sounds. These three systems, respectively, function as a power supply of compressed air, a buzzer , and a filter resonating system. We have described the properties of the lungs in Chap. 9. We will examine the phonation and articulation mechanisms now.

The separation of the vocal folds (vocal cords, glottis opening) varies with our state of speaking (Fig. 10.11). The open glottis is V-shaped because the vocal folds are held together in the front of the larynx and move apart in the back. When we are voiceless, the vocal folds are totally open for normal breathing. During whispering the folds are closer together. Air moves through the glottal constriction (picks up speed) and rotates in turbulent eddies to give it its distinctive sound. As detailed later, our voice during the normal speaking of voiced sounds is created by the folds periodically opening and closing, to give periodic bursts of air. No air flows when there is a glottal stop.