Waves Interference

Home , Sine wave

Waves Interference

Lana Sheridan

De Anza College

May 22, 2020 Last time

• transverse speed of an element of the medium

• energy transfer by a sine wave Overview

• power of a sine wave

• interference λ 2 2 2 Eλ = µω A cos (kx − ωt) dx Z0 λ = µω2A2 2

Rate of Energy Transfer in Sine Wave

1 dK = µ dx A2ω2 cos2(kx − ωt) 2 1 dU = µA2ω2 cos2(kx − ωt) dx 2

Adding dU + dK gives

dE = µω2A2 cos2(kx − ωt) dx

Integrating over one wavelength gives the energy per wavelength: Rate of Energy Transfer in Sine Wave

1 dK = µ dx A2ω2 cos2(kx − ωt) 2 1 dU = µA2ω2 cos2(kx − ωt) dx 2

Adding dU + dK gives

dE = µω2A2 cos2(kx − ωt) dx

Integrating over one wavelength gives the energy per wavelength:

λ 2 2 2 Eλ = µω A cos (kx − ωt) dx Z0 λ = µω2A2 2 Rate of Energy Transfer in Sine Wave

For one wavelength: 1 E = µω2A2λ λ 2

Power averaged over one wavelength: E 1 λ P = λ = µω2A2 T 2 T

Average power of a wave on a string:

1 P = µω2A2v 2 Question

Quick Quiz 16.51 Which of the following, taken by itself, would be most eﬀective in increasing the rate at which energy is transferred by a wave traveling along a string?

(A) reducing the linear mass density of the string by one half (B) doubling the wavelength of the wave (C) doubling the tension in the string (D) doubling the amplitude of the wave

1Serway & Jewett, page 496. Question

Quick Quiz 16.51 Which of the following, taken by itself, would be most eﬀective in increasing the rate at which energy is transferred by a wave traveling along a string?

(A) reducing the linear mass density of the string by one half (B) doubling the wavelength of the wave (C) doubling the tension in the string (D) doubling the amplitude of the wave ←

1Serway & Jewett, page 496. Interference of Waves (Reminder from Lab)

When two wave disturbances interact with one another they can amplify or cancel out.

Waves of the same frequency that are “in phase” will reinforce, amplitude will increase; waves that are “out of phase” will cancel out. Interference of Waves (Reminder from Lab)

Waves that exist at the same time in the same position in space add together.

superposition principle If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves.

This works because the wave equation we are studying is linear.

This means solutions to the wave equations can be added:

y(x, t) = y1(x, t) + y2(x, t)

y is the resultant wave function. 18.1 Analysis Model: Waves in Interference 535

y a 2 a y 1 18.1 Analysis Model: Waves in Interference 535 y 1 y 2

When the pulses overlap, the When the pulses overlap, the wave function is the sum of wave function is the sum of y the individual wave functions. athe individual wave functions.2 y a 1 18.1 Analysis Model: Waves in Interference y 1 y 2 535

When the pulses overlap, the bWhen the pulses overlap, the b y ϩ y wave function isy the1ϩ y sum2 of wave function1 is2 the sum of y the individual wave functions. athe individual wave functions.2 When the crests of the two y 18.1 Analysis Model: Waves in Interference 535 a When the1 crests of the two pulses align,y 1 the amplitudey 2 is Interferencethe of Waves:sum of the Constructive individual Interference pulses align, the amplitude is Whenamplitudes. the pulses overlap, the bWhenthe difference the pulses between overlap, the the b y ϩ y y 1ϩ y 2 waveindividual function amplitudes.1 is2 the sum of wave function is the sum of y 2 the individual wave functions. athe individual wave functions. a y 1 When they crests of they two pulses align,1 the amplitude2 is cWhen the crests of the two c pulses align, they ϩ amplitudey is the sum of they individualϩ y 1 2 When the pulses1 overlap,2 the Whenthe difference the pulses between overlap, the the bamplitudes. b wave function isy theϩ y sum of wave functiony ϩ y is the sum of 1 2 Whenindividual the pulsesamplitudes.1 2 no longer theWhen individual the pulses wave no functions. longer the individual wave functions. overlap, they have not been Whenoverlap, the they crests have of notthe beentwo permanently affected by the pulsespermanently align, theaffected amplitude by the is cWhen the crests of the two interference. cinterference. pulses align, they ϩ amplitudey is the sum of they individualϩ y 1 2 b 1 2 bthe difference between the amplitudes. y ϩ y y ϩ y 1 2 individual amplitudes.1 2 When the pulses no longer When the pulses no longer y 2 overlap,When the they crests have of notthe beentwo doverlap, they have not been d When the crests of they two pulses align,y the amplitudey is permanently affected 1by the permanently 2affected by 1the cpulses align, the amplitude is cthe sum of the individual interference.y ϩ y interference. y ϩ y the difference1 between2 the Figureamplitudes. 18.1 Constructive1 2 interfer- Figure 18.2 Destructive interfer- individual amplitudes. ence. Two positive pulses travel on ence. Two pulses, one positive and When the pulses no longer When the pulses no longer a stretched string in opposite direc- one negative, travely on a stretched overlap, they have not been doverlap, they2 have not been tionsd and overlap. stringcpermanently in opposite affected directionsy by1 the and permanentlyy 2affected byy 1the c overlap.interference. y 1ϩ y 2 interference. y 1ϩ y 2 Figure 18.1 Constructive interfer- Figure 18.2 Destructive interference.When Two positivethe pulses pulses no longer travel on ence.When Two thepulses, pulses one no positive longer and overlap, theyy 2 have not been Q uick Quiza stretchedoverlap, 18.1 string they Two have inpulses opposite not been move direc- in oppositeone directionsd negative, travelon a onstring a stretched and are iden- tionsdpermanently and overlap. affected by the stringpermanently in opposite affected directionsy 1 by the and y 2 y 1 tical in shapeinterference. except that one has positive displacementsoverlap.interference. of the elements of the stringFigure and the 18.1 other Constructive has negative interfer- displacements.Figure At 18.2 the Destructivemoment the interfer- two pulses completelyence. Two overlap positive on pulses the travelstring, on what happens?ence. Two(a) pulses,The energy one positive associated and with y the pulsesa stretched has stringdisappeared. in opposite (b) direc- The string isone notd negative, moving. travel2 (c) on The a stretched string forms a Q uick Quiztionsd and18.1 overlap. Two pulses move in oppositestring directions in opposite on directions a stringy 1 andand are iden- straight line. (d) They 2 pulsesy 1have vanished and will not reappear. tical in shape except that one has positive displacementsoverlap. of the elements of the stringFigure and the 18.1 other Constructive has negative interfer- displacements.Figure At 18.2 the Destructivemoment the interfer- two pulses completelyence. Two overlap positive on pulses the travelstring, on what happens?ence. Two(a) Thepulses, energy one positive associated and with Superpositionthe pulsesa stretched has stringdisappeared. of Sinusoidalin opposite (b) direc- The Waves string isone not negative, moving. travel (c) on The a stretched string forms a Q uick Quiztions and18.1 overlap. Two pulses move in oppositestring directions in opposite on adirections string and and are iden- Letstraighttical us now in shape line. apply (d) except the The principle thatpulses one haveof has superposition vanishedpositive displacements andoverlap. to willtwo notsinusoidal reappear. of the waveselements traveling of the in thestring same and direction the other in ahas linear negative medium. displacements. If the two Atwaves the aremoment traveling the twoto the pulses right andcompletely have the overlapsame frequency, on the string, wavelength, what happens? and amplitude (a) The energybut differ associated in phase, with we can express their individual wave functions as SuperpositionQ uickthe pulses Quiz 18.1has disappeared. Twoof Sinusoidal pulses move (b) The inWaves oppositestring is notdirections moving. on (c) a Thestring string and formsare iden- a Letticalstraight us now in shape line. apply (d) except they The1 5principle Athatpulses sin one (kx haveof 2has superposition v vanished tpositive) y2 5 displacementsAand sin to will (twokx not2 sinusoidal v reappear.t 1of fthe) waveselements traveling of the in thewhere,string same as and usual,direction the k other 5 in2p a /has llinear, v negative 5 2medium.pf, and displacements. f If is the the two phase Atwaves constantthe aremoment traveling as discussed the twoto the pulses in rightSec- andtioncompletely have16.2. theHence, overlapsame the frequency, resultanton the string, wavewavelength, whatfunction happens? and y is amplitude (a) The energybut differ associated in phase, with we canSuperpositionthe express pulses their has disappeared. individualof Sinusoidal wave (b) functionsThe Waves string as is not moving. (c) The string forms a y 5 y1 1 y2 5 A [sin (kx 2 vt) 1 sin (kx 2 vt 1 f)] Letstraight us now line. apply (d) the The principle pulses haveof superposition vanished and to willtwo notsinusoidal reappear. waves traveling in y1 5 A sin (kx 2 vt) y2 5 A sin (kx 2 vt 1 f) theTo simplify same direction this expression, in a linear we medium.use the trigonometric If the two waves identity are traveling to the right where, as usual, k 5 2p/l, v 5 2pf, and f is the phase constant as discussed in Sec- and have the same frequency, wavelength, and amplitude but differ in phase, we tion 16.2. Hence, the resultant wave functiona 2 y isb a 1 b Superpositioncan express their individualofsin Sinusoidal a 1 wavesin b 5functions Waves2 cos as sin y 5 y 1 y 5 A [sin (kx 2 vt)2 1 sin (kx 22 vt 1 f)] Let us now apply they 51principle A sin2 (kx of 2 superposition vt) y 5 A sin to (twokx 2 sinusoidal vt 1 f) waves traveling in 1 2a b a b Tothewhere, simplify same as usual,direction this expression,k 5 in2p a/ llinear, v we5 2medium.usepf, theand trigonometric f If is the the twophase waves identityconstant are traveling as discussed to the in rightSec- andtion have16.2. Hence,the same the frequency, resultant wavewavelength, function and y is amplitude but differ in phase, we can express their individual wave functionsa as2 b a 1 b y 5 ysin 1 a y1 5sin A [sinb 5 (2kx cos 2 vt) 1 sin sin (kx 2 vt 1 f)] 1 2 2 2 y1 5 A sin (kx 2 vt) y2 5 A sin (kx 2 vt 1 f) To simplify this expression, we use the trigonometrica b identitya b where, as usual, k 5 2p/l, v 5 2pf, and f is the phase constant as discussed in Sec- tion 16.2. Hence, the resultant wave functiona 2 y isb a 1 b sin a 1 sin b 5 2 cos sin y 5 y1 1 y2 5 A [sin (kx 2 vt)2 1 sin (kx 22 vt 1 f)] To simplify this expression, we use the trigonometrica b identitya b

a 2 b a 1 b sin a 1 sin b 5 2 cos sin 2 2 a b a b 18.1 Analysis Model: Waves in Interference 535

y a 2 a y 1 18.1 Analysis Model: Waves in Interference y 1 y 2 535

When the pulses overlap, the When the pulses overlap, the wave function is the sum of wave function is the sum of y the individual wave functions. athe individual wave functions.2 a y 1 18.1 Analysis Model: Waves in Interference 535 y 1 y 2 bWhen the pulses overlap, the bWhen the pulses overlap, the y ϩ y y ϩ y wave function is 1the 2sum of wave function1 is2 the sum of y the individual wave functions. athe individual wave functions.2 When the crests of the two y a When the1 crests18.1 of theAnalysis two Model: Waves in Interference 535 pulsesy 1 align, the amplitudey 2 is the sum of the individual Interferencepulses of Waves: align, the Destructive amplitude is Interference amplitudes. bthe difference between the Whenb the pulses overlap, the When the pulsesϩ overlap, the y 1ϩ y 2 individual yamplitudes.1 y 2 wave function is the sum of wave function is the sumy of a 2 the individualWhen the crestswave offunctions. the two the individualy wave functions. a When the1 crests of the two pulses align,y 1 the amplitudey 2 is c pulses align, the amplitude is cthe sum of the individual y 1ϩ y 2 y 1ϩ y 2 Whenamplitudes. the pulses overlap, the Whenthe difference the pulses between overlap, the the b bwaveindividual function amplitudes. is the sum of Whenwave function the pulsesy isϩ ytheno longersum of When the pulsesy ϩ y no longer 1 2 the individual1 wave2 functions. overlap,the individual they have wave not functions. been overlap, they have not been Whenpermanently the crests affectedof the two by the cpermanently affected by the interference. interference.c When the crestsy 1ϩ yof2 the two pulses align, the amplitudey 1ϩ y 2 is the sumb of the individual bpulses align, the amplitude is y ϩ y y 1ϩ y 2 amplitudes. 1 2 theWhen difference the pulses between no longer the When the pulses no longer y overlap, they have not been dindividualoverlap, they amplitudes.2 have not been When the crests of the two y d Whenpermanently the crests affected of the 1by two the pulsespermanently align,y 2theaffected amplitude byy 1 the is interference. theinterference. sum of the individual pulses align, the amplitude is c Figureamplitudes. 18.1 Constructive interfer- Figurethe difference18.2 Destructive between interfer-the c individual amplitudes.y 1ϩ y 2 ence. Two positivey 1ϩ ypulses2 travel on ence. Two pulses, one positive and y a stretched string in opposite direc- oned negative, travel2 on a stretched tionsd and overlap. stringWhen in opposite the pulses directions noy 1longer and When the pulsesy 2 no longery 1 c overlap.overlap, they have not been overlap,c they have not been y ϩ y y ϩ y 1 2 permanentlyFigure 18.1 affectedConstructive1 2 by the interfer- Figurepermanently 18.2 Destructive affected interfer-by the interference.ence. Two positive pulses travel on ence.interference. Two pulses, one positive and When the pulses no longer When the pulses no longer a stretched string in opposite direc- one overlap,negative, they travel have on not a stretched been Q uick Quiztionsoverlap, and18.1 overlap. they Two have pulses not beenmove in oppositestring directions in opposite on directionsa string andand are iden- permanently affected by the permanently affected by the tical in shape except that one has positive displacementsoverlap. y of the elements of the interference. dinterference.2 stringd and the other has negative displacements. At the moment ythe1 two pulses completely overlapy 2 on they 1string, what happens? (a) The energy associated with y 2 Q theuickFigure pulses Quiz 18.1 has18.1 Constructivedisappeared. Two pulses interfer- move (b) The in opposite string isFigure directionsnotd moving. 18.2 onDestructive (c) a stringThe string andinterfer- are forms iden- a d y 1 straightticalence. in Two shapeline. positive (d) except The pulsesy 2 thatpulses travel oney 1have on has vanished positive displacementsandence. will Two not pulses, reappear. of one the positive elements and of the stringa stretched and the string other in opposite has negative direc- displacements.one negative, At the travelmoment on a thestretched two pulses Figure 18.1 Constructive interfer- Figure 18.2 Destructive interfer- tions and overlap. string in opposite directions and completelyence. Two overlap positive on pulses the travelstring, on what happens?ence. Two(a) Thepulses, energy one positive associated and with overlap. Superpositionthe pulsesa stretched has stringdisappeared. of Sinusoidal in opposite (b) direc- The Waves string isone not negative, moving. travel (c) on The a stretched string forms a Letstraight us nowtions line.apply and overlap.(d) the The principle pulses ofhave superposition vanished andstring to will two in oppositenot sinusoidal reappear. directions waves and traveling in overlap. the same direction in a linear medium. If the two waves are traveling to the right Q uickand Quizhave 18.1the same Two frequency,pulses move wavelength, in opposite and directions amplitude on but a differstring inand phase, are iden-we ticalcanSuperposition inexpress shape their except individual of that Sinusoidal one wave has functions positiveWaves asdisplacements of the elements of the Q uick Quiz 18.1 Two pulses move in opposite directions on a string and are iden- stringLet us and now the apply other the has 5principle negative of2 superposition displacements.v 5 to two At 2thesinusoidalv moment1 f waves the traveling two pulses in tical in shape excepty1 Athat sin one (kx has tpositive) y2 displacementsA sin (kx t of the) elements of the completelythe same directionoverlap on in thea linear string, medium. what happens?If the two (a)waves The are energy traveling associated to the right with where,string as andusual, the k other5 2p /hasl, v negative 5 2pf, and displacements. f is the phase At theconstant moment as discussed the two pulses in Sec- and have the same frequency, wavelength, and amplitude but differ in phase, we thetion pulsescompletely 16.2. hasHence, overlapdisappeared. the resultanton the (b)string, wave The whatfunction string happens? is y isnot (a)moving. The energy (c) The associated string forms with a can express their individual wave functions as straightthe pulsesline. (d) has The disappeared. pulses have (b) Thevanished string and is not will moving. not reappear. (c) The string forms a y 5 y1 1 y2 5 A [sin (kx 2 vt) 1 sin (kx 2 vt 1 f)] straight line. (d) yThe1 5 Apulses sin (kx have 2 v vanishedt) y2 5 Aand sin will (kx not2 v reappear.t 1 f) Towhere, simplify as usual, this expression,k 5 2p/l, v we 5 use2pf ,the and trigonometric f is the phase identity constant as discussed in Sec- tion 16.2. Hence, the resultant wave function y is Superposition of Sinusoidal Wavesa 2 b a 1 b Superposition 5ofsin Sinusoidal1 a 15sin b 5 Waves2 cos2 v 1 sin 2 v 1 f Let us now apply they principle y1 y2 of A superposition[sin (kx t)2 to sin two (kx sinusoidal 2 t )] waves traveling in Let us now apply the principle of superposition to two sinusoidal waves traveling in the sameTo simplify direction this expression, in a linear we medium. use the trigonometric Ifa the twob waves identitya areb traveling to the right and thehave same the directionsame frequency, in a linear wavelength, medium. If andthe two amplitude waves are but traveling differ toin thephase, right we can expressand have their the sameindividual frequency, wave wavelength, functions aas and2 b amplitudea 1 butb differ in phase, we can express their individualsin a 1 sinwave b 5functions2 cos as sin 2 2 y1 5 A sin (kx 2 vt) y2 5 A sin (kx 2 vt 1 f) y1 5 A sin (kx 2 vt) y2a 5 A sinb (kx a2 vt 1b f) where,where, as usual, as usual, k 5 k 25p /2lp/, lv, 5v 52p 2fp, fand, and f f is is the the phasephase constantconstant as as discussed discussed in inSec- Sec- tion tion16.2. 16.2. Hence, Hence, the the resultant resultant wave wave function function y y isis 5 1 5 2 v 1 2 v 1 f y y y51 y1 1y2 y2 5A A[sin [sin ( kx(kx 2 vt)t) 1 sinsin (kx 2 vt t1 f )])] To simplifyTo simplify this this expression, expression, we we use use the the trigonometric trigonometric identity

aa22bb aa 11bb sinsin a 1a 1sinsin b 5b 52 2cos cos sinsin 22 22 aa bb aa bb Superposition of Sine Waves

Consider two sine waves with the same wavelength and amplitude, but diﬀerent phases, that interfere.

y1(x, t) = A sin(kx − ωt) y2(x, t) = A sin(kx − ωt + φ)

Add them together to ﬁnd the resultant wave function, using the identity:

θ − ψ θ + ψ sin θ + sin ψ = 2 cos sin 2 2

Then φ φ y(x, t) = 2A cos sin(kx − ωt + ) 2 2 New amplitude Sine oscillation φ = 0 or φ = 2π, −2π, 4π, etc.

The waves are “in phase” and constructively interfere. 536 Chapter 18 Superposition and Standing Waves

Figure 18.3 The superposition y The individual waves are in phase of two identical waves y and y y 1 2 and therefore indistinguishable. (blue and green, respectively) to yield a resultant wave (red-brown). a x Constructive interference: the amplitudes add. f ϭ 0°

y1 y The individual waves are 180° out y 2 y of phase. b x Destructive interference: the waves cancel. f ϭ 180° y y This intermediate result is neither y1 y2 constructive nor destructive. c x

f ϭ 60°

Letting a 5 kx 2 vt and b 5 kx 2 vt 1 f, we find that the resultant wave function y reduces to

f f Resultant of two traveling y 5 2A cos sin kx 2vt 1 sinusoidal waves 2 2 a b a b This result has several important features. The resultant wave function y also is sinusoidal and has the same frequency and wavelength as the individual waves because the sine function incorporates the same values of k and v that appear in the original wave functions. The amplitude of the resultant wave is 2A cos (f/2), and its phase constant is f/2. If the phase constant f of the original wave equals 0, then cos (f/2) 5 cos 0 5 1 and the amplitude of the resultant wave is 2A, twice the amplitude of either individual wave. In this case, the crests of the two waves are at the same locations in space and the waves are said to be everywhere in phase and A sound wave from the speaker therefore interfere constructively. The individual waves y1 and y2 combine to form (S) propagates into the tube and the red-brown curve y of amplitude 2A shown in Figure 18.3a. Because the indi- splits into two parts at point P. vidual waves are in phase, they are indistinguishable in Figure 18.3a, where they

Path length r 2 appear as a single blue curve. In general, constructive interference occurs when cos (f/2) 5 61. That is true, for example, when f 5 0, 2p, 4p, . . . rad, that is, when f is an even multiple of p. When f is equal to p rad or to any odd multiple of p, then cos (f/2) 5 cos (p/2) 5 S 0 and the crests of one wave occur at the same positions as the troughs of the sec- ond wave (Fig. 18.3b). Therefore, as a consequence of destructive interference, the P resultant wave has zero amplitude everywhere as shown by the straight red-brown R line in Figure 18.3b. Finally, when the phase constant has an arbitrary value other than 0 or an integer multiple of p rad (Fig. 18.3c), the resultant wave has an amplitude whose value is somewhere between 0 and 2A. In the more general case in which the waves have the same wavelength but dif- Path length r 1 ferent amplitudes, the results are similar with the following exceptions. In the in- The two waves, which combine phase case, the amplitude of the resultant wave is not twice that of a single wave, at the opposite side, are but rather is the sum of the amplitudes of the two waves. When the waves are p rad detected at the receiver (R). out of phase, they do not completely cancel as in Figure 18.3b. The result is a wave whose amplitude is the difference in the amplitudes of the individual waves. Figure 18.4 An acoustical system for demonstrating interfer- Interference of Sound Waves ence of sound waves. The upper

path length r2 can be varied by One simple device for demonstrating interference of sound waves is illustrated in sliding the upper section. Figure 18.4. Sound from a loudspeaker S is sent into a tube at point P, where there is

Dependence on Phase Diﬀerence

0 φ The amplitude of the resultant wave is A = 2A cos 2 , where φ is the phase diﬀerence.

For what value of φ is A0 maximized? Dependence on Phase Diﬀerence

0 φ The amplitude of the resultant wave is A = 2A cos 2 , where φ is the phase diﬀerence.

For what value of φ is A0 maximized? φ = 0 or φ = 2π, −2π, 4π, etc.

The waves are “in phase” and constructively interfere. 536 Chapter 18 Superposition and Standing Waves

y1 y The individual waves are 180° out y 2 y of phase. b x Destructive interference: the waves cancel. f ϭ 180° y y This intermediate result is neither y1 y2 constructive nor destructive. c x

f ϭ 60°

Letting a 5 kx 2 vt and b 5 kx 2 vt 1 f, we find that the resultant wave function y reduces to

Figure 18.3 The superposition y The individual waves are in phase of two identical waves y and y y 1 2 and therefore indistinguishable. (blue and green, respectively) to yield a resultant wave (red-brown). 0 a If φ = π, −π, 3π, −3π, etc. A = 0.x DestructiveConstructive interference. interference: the amplitudes add. f ϭ 0°

y1 y The individual waves are 180° out y 2 y of phase. b x Destructive interference: the waves cancel. f ϭ 180° y y This intermediate result is neither y1 y2 constructive nor destructive. c x

f ϭ 60°

Letting a 5 kx 2 vt and b 5 kx 2 vt 1 f, we find that the resultant wave function y reduces to

y1 y The individual waves are 180° out y 2 y of phase. Interference of Two Sine Waves (equal wavelength) b x Destructive interference: the φ φ y(x, t) = 2A cos sin(kx − ωt + ) waves cancel. 2 2 f ϭ 180° y y This intermediate result is neither y1 y2 constructive nor destructive. c x

f ϭ 60°

Letting a 5 kx 2 vt and b 5 kx 2 vt 1 f, we find that the resultant wave function y reduces to

We can count phase diﬀerences in terms of wavelengths also.

If two waves have a phase diﬀerence of 1 wavelength then φ = 2π. Constructive interference.

If two waves have a phase diﬀerence of half a wavelength then φ = π. Destructive interference. Summary

• energy transfer by a sine wave • interference

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