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Chapter 16 Sound and Hearing

16.1 Sound Waves Sounds are longitudinal waves produced by the vibrations of material objects. Your voice results from the vibrations of your vocal cords. The frequency of the sound waves equals the frequency of the vibrating source. The audible range of frequencies (for a loud tone of intensity level 80 dB) by a human of good hearing is from about 20 Hz to about 20,000 Hz. (ii) infrasonic waves: Hzf 20! (earthquakes, thunder) (iii) ultrasonic waves: Hzf 000,20! Acoustics is the branch of physics that deals with the study of sound. We can describe sound waves either as (i) changes in the local pressure in the medium or as (ii) displacements of the air molecules from their

equilibrium positions.

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As the source of sound vibrates, it produces a periodic series of compressions and rarefactions in the medium surrounding it. Compressions are regions of high density and pressure (higher than average), while rarefactions are regions of low density and pressure (lower than average). Ears and microphones detect sound by sensing pressure differences, not displacements, so it is useful to describe sound in terms of pressure fluctuations. Let p(x,t) be an instantaneous pressure fluctuation in a sound wave at any point x at time t. That is, p(x,t) is the amount by which pressure differs from normal atmospheric pressure Po , so p(x,t) is a gauge pressure. One may write the pressure fluctuation p(x,t) in a medium (solid, liquid, or gas) in a sound wave propagating in the positive x-direction as

!

p(x,t) = Bk A sin(kx "#t)

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where B = Bulk modulus of the medium A = displacement amplitude of particle in the medium Note that we can write the maximum pressure fluctuation, also called the pressure amplitude as

!

Pmax = Bk A so that,

!

p(x,t) = Pmax sin(kx "#t) Note that the pressure amplitude is proportional to the displacement amplitude A, and it also depends on wavelength through the wave number k. The greater the wavelength of the sound wave, then the smaller the pressure amplitude for the same displacement amplitude.

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16.2 Speed of Sound Waves As a sound wave travels along a medium, the compressions and rarefactions travel along the medium. A. Speed of sound in a fluid: The speed of sound in a fluid (gas or liquid) is given by

!

v =B"

where B = Bulk modulus of the fluid ρ = density of fluid specifically, the speed of sound in an ideal gas may be written as

!

v =" RTM

where γ = ratio of specific heat capacities. This is a quantity that characterizes the thermal properties of the gas R = universal gas constant = 8.314 J mol-1 K M = molar mass of the gas The speed of propagation v of a sound wave in air depends on wind conditions and air temperature. The

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speed of sound waves in air at temperature Tc in Celsius is given by

!

v " 331+ 0.6 Tc( ) in meters/second

Note that sound travels faster in warm air than in cold air. This can lead to the refraction of sound. Sound refraction refers to the bending in the direction of sound travel when sound travels through a medium of uneven temperature. The speed of propagation of a sound wave in air does NOT depend on loudness or frequency.

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B. Speed of sound in a solid: The speed of sound in a solid rod is given by

!

v =Y"

where Y = Young’s modulus of the solid ρ = density of fluid An echo is a reflected sound wave.

Sound energy dissipates into thermal energy as the sound travels in air. The energy of a high frequency sound wave is transformed more rapidly into thermal energy than the energy of a low frequency sound wave. Thus, sound waves of low frequency travel farther (not faster) through air than sound of high frequencies. This is the reason why the foghorns of ships emit low frequency sounds! Remember that a wave is any disturbance from an equilibrium condition, which travels or propagates with time from one region of space to another.

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16.3 Sound Intensity: Decibels An essential aspect of wave propagation is energy transfer. The intensity I of a traveling wave is defined as the average rate at which energy is transported by the wave, per unit area, across a surface perpendicular to the direction of wave propagation. That is, the intensity is the average power transported per unit area. At the threshold of hearing, the human ear can detect sounds with an intensity of as low as 10-12 W/m2. At the threshold of pain, the intensity of sound is 1 W/m2. Because of the wide range of sound intensities over which the human ear is sensitive, a logarithmic intensity scale rather than a linear intensity scale is convenient. The intensity level β of a sound wave of intensity I is defined as

!

" =10 log IIo

#

$ %

&

' (

where Io is an arbitrary reference intensity, takes as Io = 10-12 W/m2, corresponding to the intensity of the faintest sound which can be heard. Intensity level is measured in units of decibels (abbreviated dB).

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If the intensity level of a sound wave is Io , or 10-12 W/m2, its intensity level is 0 dB. The maximum intensity which the ear can tolerate is 1 W/m2 , which corresponds to an intensity level of 120 dB. We can express the intensity I of the sound wave in terms of the pressure amplitude Pmax of the sound wave by

!

I =Pmax( )2

2" v

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16.4 Standing Sound Waves and Normal Modes A. String Instruments As discussed in the previous chapter, the frequencies of the normal modes of vibrations may be calculated using

!"

#$%

&=

Lvnfn 2

( ),...4,3,2,1=n

Because µ

=Tensionv , one can express the natural

frequencies of vibration of a stretched string as

!

fn =n2L

Tensionµ

( ),...4,3,2,1=n

The lowest allowed natural frequency of vibration is called the fundamental frequency. Any integer multiple of the fundamental frequency is called a harmonic. Thus,

!1f fundamental frequency µ

=!tension

Lf

21

1

!= 12 2 ff second harmonic

!= 13 3 ff third harmonic, and so on.

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All the even and odd harmonics are present. B. Open Pipe (pipe open at both ends)

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! same results as stretched string clamped at both ends:

!"

#$%

&=

Lvnfn 2

( ),...4,3,2,1=n

where now v is the speed of sound in the air column. C. Stopped Pipe (column of air in a pipe open at one end, closed at the other end)

In general,

!

L = " n #4

where ( " n =1,3,5,7,...) so that

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!

" =4 L# n

and the normal mode frequencies

!

f =v"

are thus

!

fn = " n v4 L#

$ %

&

' (

!

" n =1, 3, 5,...( )

!1f fundamental frequency Lvf41 =!

!= 13 3 ff third harmonic

!= 15 5 ff fifth harmonic, and so on.

Only the odd harmonics are present. 16.7 Beats Beats are variations in loudness. They occur whenever two waves of slightly different frequencies interfere.

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The number of beats heard per second = 12 ff ! , that is, the difference between the frequencies of the two interfering waves. 16.8 The Doppler Effect for Sound Christian Doppler (1803 – 1853) The positive reference direction is always taken from the listener to the source! Let, v = speed of sound in air = 343 m/s at room temperature (always positive). vs ! speed of the source vL ! speed of the listener (observer)

!

fs " frequency of the sound emitted by the source

!

fL " frequency of the sound heard by the listener. The master equation for the Doppler effect for sound is

!

fLv ± vL

=fs

v ± vs

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When you solve a problem having to do with the Doppler effect, there would be only four possibilities. These are: (a) source and listener traveling toward each other in

opposite directions

!

fLv + vL

=fs

v " vs

(b) source and listener traveling away from each other in opposite directions

!

fLv " vL

=fs

v + vs

(c) source and listener traveling in the same direction with the listener following the source

!

fLv + vL

=fs

v + vs

(d) source and listener traveling in the same direction with the source following the listener

!

fLv " vL

=fs

v " vs

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Doppler Effect for Electromagnetic Waves In the frame of reference in which the receiver (listener or observer) is at rest, the source of EM waves (light) is moving relative to the receiver with speed v. fS = frequency of the EM waves emitted by the source fR = frequency of the EM waves measured or received by the receiver c = speed of the EM waves (light) v = speed of the source relative to the receiver There are two possibilities: (a) Source approaching the receiver: here fR > fS

(Blue Shift)

!

fR = fSc+ vc" v

(b) Source receding away from the receiver: here fR < fS (Red Shift)

!

fR = fSc" vc+ v

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We show in problem 16.78 that when v << c, one can write the Doppler effect formula as

!

fR " fS 1±vc

# $ %

& ' (

where we use the positive sign when we have a blue shift (approaching), and the negative sign when we have a red shift (moving away).