Sound, Speech, and Hearing

Abstract

The basic physics of sound is first developed, including that of sound wave propagation in air and at interfaces, and of sound in resonant cavities, and these concepts are then applied to speaking and hearing. The structure and function of the relevant body components for speaking and hearing are presented, followed by a discussion of how the body produces sound and senses it. Other types of body sounds are then overviewed.

Problems

Sound Waves

10.1

Find the distance molecules in air move for 20 dB SPL and 120 dB SPL sound levels, both at 500 and 5,000 Hz.

10.2

A musician with perfect pitch can identify a 1 kHz pure tone in 4 ms. How many periods of the sound wave is this?

10.3

Use (10.6) and the mass density of air of $1.3 \times 10^{-3}$ g/cm$^{3}$ to calculate the characteristic displacement of air for 0 dB SPL and 120 dB SPL for 4 kHz.

10.4

(a) Show that the speed of sound in air $v_{\mathrm{air}}\propto \sqrt{T}$ and the mass density varies as $\rho _{\mathrm{air}}\propto 1/T$, and so the acoustic impedance of air varies as $Z_{\mathrm{air}} \propto 1/\sqrt{T}$, where T is in K.

(b) Find each of these parameters at 0, 20, and $25\,^{\circ }$C.

(c) Does this variation of $Z_{\mathrm{air}}$ with T significantly affect the reflection and transmission of sound from air to the body components? Why or why not?

10.5

Find the speed of sound in water using the bulk modulus of water of 2.26 GPa. Is this the expected result?

Sound Intensity

10.6

The scale for acoustic intensity in Fig. 10.30 ranges from $-20$ to 140 dB SPL. What pressure range does this correspond to in dyne/cm$^{2}$?

10.7

Sound with intensity 60 dB SPL in air is incident on water. How much of it is transmitted into the water (in dB SPL and W/m$^{2}$)?

10.8

Hammering on a steel plate produces sounds that two feet away reach a maximum of 115 dB SPL. If this acoustic intensity is isotropic, what is the total power of this acoustic wave in W?

10.9

What is the change in dB if the intensity of a sound wave is:

• (a) halved

• (b) doubled

• (c) tripled

• (d) quadrupled?

10.10

Why is there is a 6 dB SPL decrease in sound level for each doubling of distance from a small isotropic source?

10.11

The acoustic intensity is 60 dB SPL at a given distance from an isotropic source. What would the intensity be if this same level source were to radiate into only 1/10 of all space?

10.12

Which sound is more intense 20 m from its isotropic source: a 10 Hz sound that is 80 dB SPL a distance 4 m from its source or a 4,000 Hz sound that is 60 dB SPL a distance 3 m from its source?

Sound Intensity for Threshold Hearing Sensitivity

10.13

(a) At what distance from an isotropic $10 \; \upmu$W acoustic source is the sound at the audibility threshold for a human?

(b) The hearing threshold for dogs is $1 \times 10^{-15}$ W/m$^{2}$. At what distance can a dog hear this source?

10.14

(a) Express music’s tripleforte (very loud, $1 \times 10^{-2}$ W/m$^{2}$) and triple piano (very soft, $1 \times 10^{-8}$ W/m$^{2}$) sound levels in dB SPL.

(b) At an outdoor concert with no sound amplification, the audience sitting 4 m from the orchestra hears triple forte and triple piano sounds. What are the respective acoustic intensities (in dB SPL) for those in the audience sitting 60 m away? Treat the orchestra as a point source.

10.15

(a) Show that 1 W from an isotropic acoustic power radiator produces an intensity of 115 dB SPL a distance 0.5 m from the source.

(b) Find this acoustic intensity 1 m from the source.

(c) How far can you be from the source and still barely hear it (for a 1,000 Hz source)?

10.16

Determine $P_{\mathrm {ref}}$ and the pressure (changes) at 100 dB SPL in mmHg.

Sound Transmission, Reflection, and Ultrasound

10.17

Table 10.3 gives the absorption coefficient $\alpha$ for bone as $1.6 \times 10^{-4}$ s/m. Other sources give it as 14 dB/cm at 1 MHz frequency. Are these two values consistent? Why?

10.18

Can you talk through a person? (In other words, are people good acoustic shielding?) Estimate the dB loss for 3,000 Hz sound transmitted through your chest to help answer this question.

10.19

For each tissue in Table 10.3, determine the thickness of tissue needed to decrease the intensity of a 5 MHz ultrasound wave by half. (Assume losses are due only to absorption, and not due to reflection at interfaces.)

10.20

Often in ultrasound measurement the ultrasound transducer and detector are on the same probe, so reflected sound waves are detected. Determine the fraction of initial sound intensity that is detected for the following cases by tracking the beam that is transmitted through material X in the body, reflected at normal incidence from the interface of X with Y, and then again transmitted through material X. (Assume that all the sound from the transducer enters X and all leaving X after reflection from Y enters the detector. Consider attenuation in the medium, as well as reflection at the X–Y interface.):

• (a) X is 1 cm of muscle and Y is bone, for 1 MHz sound

• (b) X is 1 cm of muscle and Y is bone, for 10 MHz sound

• (c) X is 5 cm of fat and Y is muscle, for 1 MHz sound

• (d) X is 2 cm of blood and Y is muscle, for 5 MHz sound

• (e) X is 1 cm of bone and Y is muscle, for 1 MHz sound.

10.21

Calculate the relative delay times between sound reflecting at the beginning of the X medium and that reflecting at the X–Y interface, for each case in Problem 10.20.

10.22

Estimate the relative delay times for 1 MHz sound reflecting normally from the first outer surface of the aorta, the first inner surface of the aorta, the second inner surface of the aorta, and second outer surface of the aorta.

10.23

When an ambulance siren blaring at a frequency f approaches you at speed v, you hear the frequency upshifted to $f^{\prime }=f(1+v/v_{\mathrm {s} })$, while when it is distancing itself from you, you hear the frequency downshifted to $f^{\prime }=f(1-v/v_{\mathrm {s}})$. This is the Doppler effect.

(a) Show that this is consistent with $f^{\prime }=f(1-(v/v_{\mathrm {s}})\cos \theta )$, where $\theta$ is the angle between the velocity vector of the moving object and the position vector from you to it.

(b) Show that Doppler ultrasonography echocardiography (Fig. 10.56) can be used to determine the blood flow speed v to be

$$\begin{aligned} v=\frac{(\delta f)v_{\mathrm {s}}}{2f\cos \theta }, \end{aligned}$$

(10.66)

where $\delta f$ is the measured Doppler shift ($f^{\prime }-f$) and $v_{\mathrm {s}}$ is the speed of sound in body tissue.

(c) Calculate the maximum Doppler shift for blood flowing in the aorta, using 1 MHz ultrasound.

Fig. 10.56

Doppler examination of blood flow in a vessel at a speed u, with scanning shown (Reprinted from [51]. Used with permission of Elsevier)

10.24

A generous dab of gel is put on the ultrasound probe head before it is placed on the skin (Fig. 10.56):

• (a) Why?

• (b) What must be the desired acoustic properties of this gel?

10.25

Express the unit of acoustic admittance, the mmho, in SI units.

10.26

(advanced problem) With $Z=R+\mathrm{i}X_{\mathrm {m}}+X_{\mathrm {s}}/\mathrm{i}$, derive (10.19).

10.27

(advanced problem) With $Z=R+\mathrm{i}(X_{\mathrm {m}}-X_{\mathrm {s}})$ and $Y=G+ \mathrm{i} (B_{\mathrm {m}}-B_{\mathrm {s}})$, derive:

• (a) Equation (10.20)

• (b) G, $B_{\mathrm {m}}$, and $B_{\mathrm {s}}$ in terms of R, $X_{\mathrm {m}}$, and $X_{\mathrm {s}}$

• (c) R, $X_{\mathrm {m}}$, and $X_{\mathrm {s}}$ in terms of G, $B_{\mathrm {m}}$, and $B_{\mathrm {s}}$.

10.28

(advanced problem) With $Z=R+\mathrm{i}\omega M+S/\mathrm{i}\omega$, where $\omega$ is the radial frequency in rad/s and $\omega =2\pi f$ where f is the frequency in Hz or cycles per second, show that

$$\begin{aligned} {\mid Z\mid } =\sqrt{R^{2}+(2\pi fM+S/2\pi f)^{2}}. \end{aligned}$$

(10.67)

In electronics problems, R is the resistance, M corresponds to the inductance L, and S corresponds to the reciprocal of the capacitance C ($S=1/C$).

10.29

(advanced problem) Use (10.24) to show that the resistive term of the speed varies as a cosine wave and the inertial and stiffness terms both vary as sine waves, but with opposite signs.

10.30

The speeds of sound in the brain and skull bone are 1,550 and 4,090 m/s, respectively. What fraction of sound is lost in reflection from air to the skull bone and then from the skull bone to the brain? Assume the densities of the brain and skull bone are 1 g/cm$^{3}$.

10.31

We are usually concerned with light entering the eye, but what happens when sound enters the eye? Calculate the reflection coefficient at each interface between the air/cornea/aqueous humor/eye lens/vitreous humor. The speeds of sound in the aqueous humor, eye lens (crystalline lens), and vitreous humor are 1,510, 1,630, and 1,540 m/s, respectively. Assume that the density of each medium in the eye is 1 g/cm$^{3}$ and that the cornea and eye lens have the same properties.

10.32

The speed of sound in collagen is 3,640 m/s along the fiber axis and 2,940 m/s across this axis. What is the reflection coefficient between blood and collagen, for sound traveling in both directions in the collagen?

Speaking

10.33

The oral cavity of a child is 8 cm long, as measured from her lips to vocal folds. What is the fundamental oscillation frequency of this cavity? (Treat the oral cavity as a cylinder open at one end and closed on the other.) Does this make sense in light of the differences of the voices of children and adults?

10.34

Plot on the same set of axes all of the resonant frequencies below 5 kHz for an 18-cm long cylinder that is open on both ends and a 16-cm long cylinder that is open on one end and closed on the other.

10.35

Smokers may have vocal folds that are slightly swollen and inflamed, and therefore, perhaps have folds that are more massive than those of nonsmokers. How would this affect the vibration frequency of their vocal folds? Would the voices of smokers be relatively deeper or higher pitched?

10.36

Explain why people speak in a high pitch after taking a breath of helium. The speed of sound in helium is about 970 m/s. (Show this, given $\gamma = c_{p}/c_{v}$ is 5/3 for helium.)

10.37

In the text, the inverse relationship between the vocal-fold frequency and vocal-fold length was explained by considering the resonant frequencies of an oscillating string. Use (4.4) to show that this relationship is also expected if the vocal folds are modeled as a free, spring like object oscillating length-wise.

10.38

Estimate the vibration frequency of the vocal folds, by assuming they are a spring-like object oscillating length-wise that is 1 cm long, 0.3 cm wide, and 0.3 cm thick, with a Young’s modulus of 100 kPa, and a mass density of 1 g/cm$^{3}$ [1]. (See (4.4).)

10.39

We can describe the production of the “m” sound as being voiced bilabial nasal. In what way does the production of the “n” sound differ?

10.40

Compared to men, do women use higher or lower harmonics of their fundamental buzzing frequency to produce the same vowel formant? Why?

10.41

Use the vowel formant plot (Fig. 10.19) to sketch the transmission curve of the vocal tract for three vowels.

10.42

Describe the frequency spectra of the vowels and consonants in Fig. 10.9. Point out their similarities and differences.

10.43

Plot the first- and second-formant frequencies for each vowel in Fig. 10.9 on the same set of axes.

10.44

Plot the first- and second-formant frequencies for the vowel in Fig. 10.20 on the same set of axes.

10.45

(advanced problem) Explain why the two-tube model in Fig. 10.25b, d, f, explains the mode shifts for the “ee” sound as in “see,” which are shifted from the predictions of the one-tube model in Fig. 10.24a.

Hearing Mechanism

10.46

What is the resonant frequency of the 1.3 cm long ear canal of a baby? How does it compare to that of an adult?

10.47

The amplitude of motion of the eardrum is $0.03 \; \upmu$m when measured at 100 dB SPL for 3,000 Hz. Assuming the amplitude is linear with the total force on the eardrum (which is a good assumption for pressures below 130 dB SPL), find the amplitude of eardrum motion at 0 dB SPL in m.

10.48

Model the motion of the eardrum as a flat membrane that is fixed at its ends, with a displacement that increases linearly from the edge to the center of the eardrum. (The motion is really more closely sinusoidal than this triangular mode shape.) If this maximum displacement at the center is 0.03 $\upmu$m, as in Problem 10.47, find the full change of angle the eardrum makes during its motion (in radians and degrees).

10.49

Compare the potential energy of motion of the eardrum at 3,000 Hz to that at 1 Hz for the same amplitude of motion. (Assume the motion can be modeled as a simple harmonic oscillator.)

10.50

Is modeling the eardrum as a freely vibrating object reasonable, given that the stapes touches it? Why?

10.51

Do people have good auditory sensitivity at the fundamental frequencies of their voices? Is this important? Why?

10.52

What is the gain in dB due to the middle ear if only the force enhancement in the middle ear (and not the entire torque enhancement) is considered?

10.53

Sketch the hair cell responses in Fig. 10.39 on a linear–linear plot of acoustic intensity versus frequency.

10.54

How are the three sharp hair cell responses in Fig. 10.39 related to the responses for a oscillator as shown in Figs. 10.7 and D.3?

10.55

Design hair cells composed of keratin with resonant frequencies ranging from 20 Hz to 20 kHz with:

(a) A fixed length of 2 mm

(b) A fixed diameter of $20 \;\upmu$m or

(c) Dimensions of 2 mm length and $20 \;\upmu$m diameter for the hair resonant at 1 kHz, with variations in these two parameters that cause equal changes in frequency for different hair cells.

10.56

Calculate the fundamental frequency for lateral vibrations of a rod of solid bone of length 10 mm and diameter 1 mm ($Y = 1 \times 10^{11}$ Pa and density $\rho = 3,\!000$ kg/m$^{3}$) fixed at one end to an incompliant base.

10.57

(a) Show that 1 mmH$_{2}$O = 0.98 daPa.

(b) Express the tympanogram range of +200 daPa to $-300$ daPa in mmH$_{2}$O.

10.58

Show that the impedance relationship equivalent to the admittance relationship for tympanograms, $Y_{\mathrm {middle \;ear}}=Y_{\mathrm {total}}-Y_{\mathrm {outer \;ear}}$, is $Z_{\mathrm {middle \;ear}}=Z_{\mathrm {outer \;ear}}Z_{\mathrm {total}}/(Z_{\mathrm {outer \;ear}}-Z_{\mathrm {total}})$.

10.59

(advanced problem) For adiabatic conditions (no heat flow) $PV^{\gamma }$ is a constant. Show that (10.59) becomes $\Delta P=-\gamma P(\Delta V)/V$ and that this leads to (10.61).

Hearing Levels and Perception

10.60

A student in a class wants to set the tone of his cell phone ringer so that he could hear it, but his instructor, who is a bit older than he is, cannot. Would a suitable fundamental frequency be 250, 1,000, 17,000, or 30,000 Hz? Why?

10.61

The acoustic power incident on the eardrum at threshold (0 dB SPL) is equivalent to how many optical photons [52]? (Hint: Use the known threshold intensity and the dimensions of the eardrum.)

10.62

Compare the intensity of the crack of a bat hitting a baseball as heard by the catcher and by a fan in the bleachers.

10.63

A noisy elevated train (in the open air) in Brooklyn in New York City causes acoustic discomfort to those 5 ft away from it. How many city blocks away can it be heard? (Treat the train as a point source, even though this is clearly an approximation. There are 20 canonical city blocks in a mile. Also, remember that background noises in the city could mask the sound of the train, so the threshold intensity for hearing is much above the threshold of hearing pure tones in a quiet room.)

10.64

The explosion of 23 kg of TNT creates a sound level of 200 dB SPL a distance 3 m away from the detonation. Assume the total acoustic energy produced in such explosions is proportional to the mass of the TNT:

(a) How far away must you be to avoid the threshold of pain from a blast from 100 kg of TNT?

(b) What fraction of the energy released from the detonation is in the form of acoustic energy over the hemispherical release region? Assume the blast is 50 ms long and a ton of TNT releases $4.18 \times 10^{9}$ J.

(c) It has been reported that 1 ton of TNT produces 120 dB SPL at 15 km. Is this consistent with the data given in this problem?

10.65

A static pressure of $8 \times 10^{3}$ Pa across the eardrum can cause it to rupture [11]. How does this compare to the sound pressure from a 160-dB SPL sound that can also cause the eardrum to rupture?

10.66

According to Fig. 10.30, what range of frequencies is needed to hear speech and over what overall volume range in dB SPL is this required?

10.67

According to Fig. 10.30, what range of frequencies is needed to hear orchestral music and over what overall volume range in dB SPL is this required?

10.68

Resketch the normal hearing range in Fig. 10.30 along with the music range for rock and roll music (which did not exist in 1934 when this curve was made).

10.69

Show that (10.65) is referenced to a loudness corresponding to 40 phons.

10.70

Show that the loudness in phons, $L_{\mathrm {p}}$, and sones, $L_{\mathrm {s}}$, are related by: $L_{\mathrm {p}} = 33.3 \log L_{\mathrm {s}} + 40$, by using (10.64) and (10.65).

10.71

Which is loudest and which is the most quiet for these three sounds: one at 7,000 Hz with loudness of 60 phons, one at 4,000 Hz with a loudness of 8 sones, or one at 1,000 Hz with intensity 50 dB SPL? (As part of this problem, express each of the three sets of data into phon and sone loudness units.)

10.72

Which 1,000 Hz sound is louder 50 ft from its isotropic source: one with an intensity 80 dB SPL a distance 5 ft from the source or one that is 65 dB SPL a distance 25 ft from the source? (In both cases, calculate the intensity in dB SPL at 50 ft.)

10.73

Approximately 80 dB SPL is needed to achieve a loudness of 80 phons at 100 and 1,000 Hz. To achieve 40 phons, 40 dB SPL is needed at 1,000 Hz. How much more acoustic intensity is needed to attain this loudness at 100 Hz? Express your answer in dB and by the factor increase in intensity needed at 100 Hz relative to 1,000 Hz.

10.74

You have a radio with inexpensive speakers that produce sound only in the 250–5,000 Hz frequency range. Will you be able to hear notes with 100 Hz fundamental frequencies, and if so, why?

10.75

Does the relative loudness scale given by sones agree with the power law dependence described in Table 1.14?

10.76

You want to design an organ with a 55 Hz tone:

(a) Show that you need to use a 3-m long length of pipe (open at both ends) to produce this resonant frequency.

(b) Say this 3-m long pipe is too long to use in the organ, but you still want to perceive a 55-Hz note tone. Explain why you can play notes from pipes that are 1.0 and 1.5 m long at the same time and hear a 55-Hz tone.

10.77

The musical scale of “Just Intonation” consists of tones that sound pleasing when sounded together or immediately after one another. Such pleasing combinations occur when the notes are harmonics of each other or have frequencies that are related by fractions with relative small integral numerators and denominators [54]. The frequencies of the notes in one octave in the Tonic C in the major scale of Just Intonation are f for a C tone, 9f / 8 for D, 5f / 4 for E, 4f / 3 for F, 3f / 2 for G, 5f / 3 for A, 15f / 8 for B, and 2f for C:

(a) Find the frequencies in this scale if the A tone has a frequency of 440 Hz.

(b) Calculate the lengths of pipes with fundamental frequencies at the frequencies of each of these notes (with the tubes open on both ends).

10.78

In the text, the difference limen for a 1,000 Hz tone was cited to be 5 dB SPL for a 5 dB SPL sound and a much smaller fraction, 6 dB SPL, for the much louder 100 dB SPL sound. Determine and then compare the absolute increase in acoustic intensity (in W/m$^{2}$) in both cases.

10.79

A person, 2 m from your left ear, speaks to you, and his voice reaches your left ear with an intensity 60 dB SPL:

(a) If your right ear receives his voice delayed by 0.05 ms, where is this speaker located relative to you? (Say a person directly in front of you is at $0^{\circ }$ and immediately to your right is at $90^{\circ }$. Assume your ears are separated by 20 cm.)

(b) If the intensity of the voice decreases inversely as the square of the propagation distance, how much lower (in dB) is the sound arriving in the right ear?