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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)