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Chapter 17

Sound Waves

Prof. Raymond Lee,revised 11-26-2013

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• Introduction to sound waves

• Sound waves are longitudinal waves

• Travel through any medium

• v depends on medium’s properties

• Equation for sinusoidal sound wavesvery similar to those for string waves

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• Categories of sound waves

• Categories cover different frequency ranges

• Audible waves are within human ear’ssensitivity, ~ 20 Hz–20 kHz

• Frequencies of infrasonic waves < 20 Hz

• Frequencies of ultrasonic waves > 20 kHz

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• Speed of sound waves

• Use compressible gas incylinder fitted with a piston

• Gas has uniform ! until pistonmoves

• When piston suddenly movesright, adjacent gas iscompressed (darker region)

(compare Fig. 17-3, p. 447)

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• Speed of sound waves, 2

• When piston stops, region ofcompressed gas keepsmoving to right• A longitudinal pulse traveling

down tube at v

• N.B.: Piston speed ! v

(compare Fig. 17-3, p. 447)

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• Speed of sound waves, 3

• Sound’s v in a medium depends on itscompressibility & !

• Write compressibility using material’s elasticmodulus (= stress/strain; see p. 315)

• General form of mechanical wave speeds is v= (elastic property/intertial property)1/2

(Eq. 17-1, p. 446)

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• vsound in materials

• In liquid or gas with material bulkmodulus B & density !, vsound = (B/!)1/2

(Eq. 17-1, p. 446)

• In solid rod with Young’s modulus Y &density !, vsound = (Y/!)1/2 (p. 316)

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• vsound also depends on medium’s temperature T,particularly important for gasses

• For air, vsound(T) is

where 331 m/s = vsound at 0°C, TC is air T in °C(see Table 17-1, p. 447)

• vsound in air

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• vsound in gases

N.B.: speeds in m/s

(compare Table 17-1, p. 447)

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• vsound in liquids

N.B.: speeds in m/s

(compare Table 17-1, p. 447)

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• vsound in solids

N.B.: speeds in m/s; for bulk solids

(compare Table 17-1, p. 447)

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• vsound in Al rod

• Need vsound in a metal rod (for !, see

Table 12-1, p. 317):

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Sound-wave energy

• Piston transmits energyto air element in tube

• This energy propagatesaway from piston in formof a sound wave

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Sound-wave energy, 2

• Air element’s speed v = time derivativeof its displacement s(x,t)

• From air element’s v & maximumdisplacement amplitude smax, find its KE:

volume V = A*!x

(from Eq. 17-12, p. 449)

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Sound-wave energy, 3

• Total KE in 1 wavelength " is K" = (1/4)!A#2smax

2"

• Total PE for 1 " equals KE, so total MEis E" = K" +U" = (1/2)!A#2smax

2"

(compare dK"/dt inEq. 17-33, p. 455)

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• Energy transfer rate = sound wave’s powerP = !E/!t = E" /T = (1/2)!Av #2smax

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(compare Eq. 17-27, p. 452)

• This energy passes by a given point duringone period T

Sound-wave power

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• Wave’s intensity I = power per unit area A,the rate at which sound energytransported by wave passes through A $wave’s direction

• Thus I = P/A (Eq. 17-26, p. 452)

• For a wave in air, I = (1/2)!v#2smax2

• In terms of max pressure amplitude !Pmax,I = (!Pmax)

2/(2!v) since !Pmax = !v # smax

Sound-wave intensity

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Point-source I

• A point source emits sound wavesisotropically, resulting in a sphericalwave front (Fig. 17-9, p. 453)

• Identify an imaginary sphere of radius rcentered on source

• Power is distributed equally acrosssphere’s surface

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Point-source I, 2

• With result I = Pavg/(4%r 2) (Eq. 17-28, p. 453)

• This inverse-square relationship exists sincea fixed amount of radiated power passesthrough ever-larger imaginary sphericalshells (see Fig. 17-9, p. 453)

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Sound level

• Range of intensities detectable by humanear actually is quite large

• Convenient to use a logarithmic scale todetermine intensity level &, where& = 10dB*log10(I/I0) (Eq. 17-29, p. 454)

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Sound level, 2

• I0 is a reference intensity, taken as thethreshold of hearing (I0 = 1.00 x 10-12 W/m2)& I is intensity of sound of interest

• & is in decibels (dB)

• Pain threshold: I = 1.0 W/m2 or & = 120 dB

• Hearing threshold: I0 = 1.0 x 10-12 W/m2 or& = 0 dB

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Sound level, 3

• What sound level corresponds to I = 2.0x 10-7 W/m2 ?

• & = 10*log10(2.0 x 10-7 W/m2/1.0 x 10-12

W/m2) = 10dB*log10(2.0 x 105) ~ 53 dB

• Perceptually, a doubling in loudnesscorresponds to any increase of 10 dB

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Sound level, 4

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Perceived loudness & &

• Sound levels in decibels are a physical measureof a sound’s strength

• We can also make a psychological measure ofthis strength

• Auditory system assesses a sound’s perceivedloudness by comparison to a reference sound

• Latter is the threshold of hearing, nominally atƒ= 1000 Hz

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Perceived loudness & &, 2

• Complicatedrelationship existsbetween loudness & ƒ

• White area’s lowerbound is the thresholdof hearing

• Its upper bound isthreshold of pain

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• Periodic sound waves

• Compression pulse moves through amaterial, continuously compressingmaterial just in front of pulse

• Yields alternating areas of compressed &rarefied P & !

• Compressions & rarefactions move with v

= medium’s vsound

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• Periodic sound waves, 2

• Longitudinal wave propagatesthrough gas-filled tube

• Wave’s source is oscillatingpiston

• Distance between 2 successivecompressions (or rarefactions) =wavelength "

(compare Fig. 17-4, p. 448)

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• Periodic sound waves, 3

• As wave traverses tube, each element of mediummoves with SHM || wave’s propagation direction

• Harmonic position function is:s (x,t) = smax cos(kx–#t) (Eq. 17-12, p. 449)

• smax is maximum displacement from equilibriumposition (AKA wave’s displacement amplitude)

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• Periodic sound waves, 4

• Gas pressure variation 'P is also periodic:'P = 'Pmax sin(kx–#t), (Eq. 17-13, p. 449)

where 'Pmax = pressure amplitude or'Pmax = !v#smax (Eq. 17-14, p. 449)

{check units: kg*m/(s2*m2) = (kg/m3)*(m/s)*(1/s)*m}

• In both equations, k = wave number &# = angular frequency

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• Periodic sound waves, 5

• Model sound wave as aP or s wave

• Phases of P & s wavesdiffer by 90° (e.g., P =max when s = 0)

(compare Fig. 17-6, p. 449)

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• Waves on a string, harmonic series

• Fundamental ƒ corresponds to n = 1 & islowest frequency, ƒ1

• ƒ of remaining natural modes are integermultiples of ƒ1 (i.e., ƒn = nƒ1)

• Such normal-mode ƒ form a harmonic series& its normal modes are called harmonics

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• Musical note of a string

• Fundamental ƒ defines amusical note

• Change string’s ƒ eitherby changing its L or FT

(SJ 2004 Ex. 18.4, p. 556)

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• Harmonics example

• Piano middle “C” has ƒ1 = 262 Hz.What are string’s next 2 harmonics?

• ƒ1 = 262 Hz

• ƒ2 = 2ƒ1 = 524 Hz

• ƒ3 = 3ƒ1 = 786 Hz

(SJ 2008 Ex. 18.3, p. 510)

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• Standing waves in air columns

• Can set up standing waves in aircolumns via interference betweenlongitudinal sound waves traveling inopposite directions

• Phase relationship between incident &reflected waves depends on whetherpipe’s end is open or closed

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• Standing waves in air columns, 2

• Pipe’s closed end is a displacementnode in standing wave• End wall doesn’t let air move longitudinally

• Reflected wave is 180° out of phase w.r.t.incident wave

• Closed end is a pressure antinode (i.e.,a site of maximum pressure variations)

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• Standing waves in air columns, 3

• Pipe’s open end is a displacement antinodein standing wave• As wave’s compression region exits pipe’s open

end, pipe constraint ends & compressed airexpands freely into atmosphere

• Open end is a pressure node (i.e., a site ofno pressure variation)

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• Standing waves in open tube

(Fig. 17-14,p. 457)

• Both ends are displacement antinodes

• Fundamental frequency ƒ1 = v/2L (1st diagram below)

(Eq. 17-39, p. 458()

• Higher harmonics ƒn = nƒ1 = n(v/2L) for n = 1, 2, 3, …

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• Standing waves in tube with1 closed end

• Closed end is displacement node

• Open end is displacement antinode, so " forfundamental ƒ1 is "1 = 4L (( Eq. 17-41, p. 458)

• Higher harmonics ƒn = nƒ = n(v/4L) for n = 1, 3, 5,… {note only odd n

values above}

(Fig. 17-14,p. 457)

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• Standing waves in tubes, summary

• In pipe open at both ends, oscillation ƒn) harmonic series that includes allinteger multiples n of ƒ1

• In pipe closed at 1 end, oscillation ƒn )harmonic series that includes only oddinteger multiples n of ƒ1

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• Musical-instrument notes

• As Tair *:

• Sounds from air columns become “sharp”; i.e.,both ƒ & v * due to Tair * (v = 331m/s*(1+T(°C)/273°C)1/2)

• Sounds from strings become “flat”; i.e., ƒ ( sinceTair * makes strings expand & + string FT (

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• Musical-instrument notes, 2

• In general, resonance excites air-columninstruments

• Air column responds to sound wavescontaining many ƒ

• Sound sources:• Vibrating reed in woodwinds

• Vibrating lips for brass players

• Blowing over edge of flute mouthpiece

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• Air-column resonance(SJ 2008 Ex. 18.6, p. 515)

• Place tuning fork near tube’s top

• If L corresponds to 1 of pipe’sƒn, then sound intensity *

• Water acts as tube’s closed end

• Calculate " from those L whereresonance occurs

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• Standing waves in rods

• Clamp rod’s middle, then slidehand along rod’s length )audible-ƒ oscillation in rod

• Clamp ) displacement node

• Rod ends can freely vibrate & sobecome displacement antinodes

(SJ 2008 Fig. 18.15a, p. 516)

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• Standing waves in rods, 2

• Clamping rod away frommidpoint ) other normal-mode oscillations

• Here, clamp rod at L/4from one end ) 2ndnormal mode (i.e., 2ndharmonic)

(SJ 2008 Fig. 18.15b,p. 516)

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• Standing waves in membranes

(SJ 2008Fig. 18.16,p. 517)

• Set up 2-dimensional oscillations in flexible membranestretched over a circular hoop (e.g., drumhead)

• Resulting sound isn’t harmonic since ƒs of standingwaves are not integer multiples of ƒ1

• Here ƒ1 defines a nodal curve on surface rather than anodal point on a string

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• Spatial & temporal interference

• Spatial interference: medium’s oscillation Avaries with element’s spatial position (type ofinterference considered so far)

• Temporal interference: medium’s waves areperiodically in & out of phase; i.e., a temporalalternation between constructive & destructiveinterference

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• Beat frequency

• Temporal interference occurs forinterfering waves with slightly different ƒ

• Beating is periodic A-variation at fixed xdue to superposition of 2 waves withslightly different ƒ

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• Beat frequency, 2

• (# of amplitude maxima)/sec = beat frequency

• ƒbeat = difference between 2 different source ƒs

• Humans can detect ƒbeat " 20 beats/sec

(compare Fig.17-17, p. 459)

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• Beat frequency, 3

• Resulting wave’s A(t) varies as:

& so intensity also varies as f(t)

• ) beat frequency ƒbeat = |ƒ1 – ƒ2|

(SJ 2008Eq. 18.11,p. 518)

(Eq. 17-46,p. 460)

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• Doppler effect

• Doppler effect is apparent change in ƒ or "due to motion of sound-wave source(moving at vS) or observer (moving at vO)• If vsound + vO = vrelative > vsound, ƒ appears to *

(requires vector vO to point opposite vsound ;i.e., observer & source are nearing)

• If vrelative < vsound, ƒ appears to ( (requires vO

to point in same direction as vsound; i.e.,observer & source are diverging)

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• Moving-observer Doppler effect

• Observer moves at speed vO

• Assume a point source ofsound stationary relative to air

• Can represent waves as seriesof circles concentric aboutsource (call each circle a wavefront)

(SJ 2008 Fig. 17.8, p. 483)

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• Moving-observer Doppler effect, 2

• Distance between adjacent wave fronts= wavelength ", sound speed = v,frequency = ƒ

• When observer moves toward source, vof waves relative to observer v ´ is:v ´ = v + vO (" is unchanged)

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• Observer hears frequency ƒ´, which * asobserver nears source:

• Conversely, ƒ´ will (as observer moves awayfrom source:

• Moving-observer Doppler effect, 3

(Eq. 17-50, p. 462)

(Eq. 17-51, p. 462)

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• Moving-source Doppler effect

• Consider a moving source andan observer at rest

• As source nears observer A, " (

• As source moves away fromobserver B, " *

(SJ 2008 Fig. 17.9a, p. 484)

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• When source nears observer, apparent ƒ *

• When source moves away from observer,apparent ƒ (

• Moving-source Doppler effect, 2

(Eq. 17-53, p. 463)

(Eq. 17-54, p. 463)

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• Combined Doppler effect

• Combining effects of observer & source motion:

• vO & vS signs depend on direction of relative v• use + for observer or source moving toward each other

• use – for observer or source moving away from each other

• N.B.: +/– signs in Eq. 17-47 don’t change, but signs of vO& vS do change

(Eq. 17-47, p. 461)

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• Combined Doppler effect, 2

• Convenient rule for signs:• “toward” means observed ƒ *• “away from” means observed ƒ (

• Doppler effect is common to all waves &doesn’t depend on source-observer distance

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• Doppler effect, water example

• Point sourcemoves to right

• Wave fronts arecloser on right &farther apart onleft

(SJ 2008 Fig. 17.9, p. 484)

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• Doppler effect, submarine example

• Sub A (source) has v = 8.00 m/s &emits at ƒ = 1400 Hz; vsound = 1533 m/s

• Sub B (observer) has v = 9.00 m/s• What is apparent ƒ´ heard by B as subs

approach? Then as they recede fromeach other?

(SJ 2008 Ex. 17.6, pp. 486-487)

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• Doppler effect, submarine example 2

• Approaching each other:

• Receding from each other:

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• Shock wave

• Source speed vs can exceedsound wave speed v

• Envelope of resulting wavefronts is cone with apex half-angle given by sin($) = v/vS ,the Mach angle

(compare Fig. 17-22, p. 465)

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• Mach number

• Ratio vs /v is called Mach number (notevs = source speed)

• Relationship between Mach angle & theMach number is sin(,) = vt/(vSt) = v/vS

(Eq. 17-57, p. 465)

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• Shock wave, 2

• Conical wave front producedwhen vs > v is a supersonicshock wave

• Shock wave carries muchenergy concentrated oncone’s surface, & great Pvariations exist along surface

(SJ 2008 Fig. 17.10b, p. 487)