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Chapter 16

Wave Motion

Prof. Raymond Lee,revised 11-21-2010

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• Types of waves

• 2 main types of waves

• Mechanical waves

• A physical medium is disturbed

• Wave is propagation of disturbance throughthe medium

• Examples: sound, water waves

• Electromagnetic waves

• No medium required

• Examples: light, radio waves, x-rays

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• General wave features

• In wave motion, energy is transferred overa distance

• Matter isn’t transferred over a distance

• All waves carry energy, but amount &mechanism of energy transport differ

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• Mechanical wave requirements

• Some source of disturbance

• Medium that can be disturbed

• A physical mechanism via which medium’selements can influence each other

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• Pulse on a rope

• Wave is generated byflicking 1 end of rope,which is under tension

• A single bump (or pulse)forms & travels along rope

(compare Fig. 16-1,p. 414)

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• Pulse on a rope, 2

• Rope is medium via which pulse travels

• Pulse has definite height & propagationspeed within medium

• Continually flicking rope ! periodicdisturbance in form of a wave

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• Transverse wave

• Traveling wave or pulse thatmakes medium’s elements move" propagation direction is a

transverse wave

• Blue arrow shows particle motion

• Red arrow shows propagationdirection

(compare Fig. 16-1,p. 414)

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• Longitudinal wave

• Longitudinal wave makes disturbed medium’s elements

move || propagation direction

• Here, coils’ displacement || to propagation direction

(compare Fig. 16-2, p. 414)

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• Complex waves

• Some waves combine transverse & longitudinalwaves (e.g., surface water waves)

(SJ 2008 Fig. 16.4, p. 451)

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• Example: Earthquake waves

• P (or primary) waves

• Fastest, at 7–8 km/s• Longitudinal

• S (or secondary) waves

• Slower, at 4–5 km/s• Transverse

• Seismograph records waves & helps determineinformation about earthquake’s place of origin

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• Traveling pulse

• Pulse shape at t = 0 isshown

• Represent shape by y(x,0)= f (x), which gives stringelements’ transverse ypositions for all x at t = 0

(compare Fig. 16-4, p. 415)

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• Traveling pulse, 2

• Pulse speed = v

• At time t, pulse hastraveled a distance vt

• Pulse’s shape doesn’tchange, but its position isnow y = f (x–vt)

• So at time t, y(x=0) equalsy(t=0) at a distance vt toleft of x=0 at t=0

(compare Fig. 16-4, p. 415)

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• Traveling pulse, 3

• For pulse traveling to right, y(x, t) = f (x – vt)

• For pulse traveling to left, y(x, t) = f (x + vt)

• Function y also called wave function y(x,t)

• y(x,t) represents y coordinate (or transverseposition) of an element at position x at time t

• For fixed t, y(x,t) is called waveform & itdefines a curve that is pulse’s geometricshape at t

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• Sinusoidal waves

• Wave shown here is asinusoidal wave, the curvegenerated by sin(!) vs. !

• Simplest example of aperiodic continuous wave,which we use to buildmore complex waves

(compare Fig. 16-7, p. 417)

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• Sinusoidal waves, 2

• Wave moves to the right• In last figure, brown wave is initial position

• As wave travels rightward, it eventually gets toblue curve’s position

• Each element moves up & down in SHM

• Distinguish between wave’s motion & motionof medium’s particles

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• Terminology: A & #

• Wave’s crest is where anelement has its maximumdisplacement (amplitudeA) from normal position

• Wavelength " is distancebetween adjacent crests

(compare Figs. 16-4 & 16-5, pp. 415-416)

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• Terminology: # & T

• More generally, " is minimum distancebetween any 2 identical points onadjacent waves

• Period T is time interval required for 2identical points of adjacent waves topass by a point• Wave T is same as T for SHM of 1 element

in the medium

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• Terminology: ƒ

• Frequency ƒ = # of crests (or any point onwave) passing given point in unit time

interval (typical ƒ units = 1/sec = Hz)

• Wave ƒ is same as ƒ for SHM of 1 element inthe medium

• ƒ & T are related by ƒ = 1/T (Eq. 16-9, p. 416)

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• Wave speed

• Waves travel with speed determined byproperties of medium being disturbed

• Wave function is given by:

for a rightward-moving wave. For a leftward-moving wave, replace x–vt with x+vt

(Eq. 16-2, p. 415)

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• Alternative wave function

• Since speed is distance/time, v = "/T

• Then rewrite wave function as:

• This form shows y’s periodic nature

(see Eq. 16-2,p. 415)

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• Wave equations (on pp. 415-417)

• Define angular wave number k $ 2%/# (Eq. 16-5)

• Rewrite angular frequency & $ 2%/T (Eq. 16-8)

• Then write wave function as y = A sin(kx–#t) {Eq.

16-2} & wave speed v = "ƒ {Eq. 16-13}

• If y ! 0 at t = x = 0, generalized wave function isy = A sin (kx – #t + $) {Eq. 16-10}, with phaseconstant $

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• Sinusoidal wave on string

• To create a series ofpulses, attach string tooscillating blade

• Wave consists of seriesof identical waveforms

• All relationships betweenv, T, ƒ, & & hold

(SJ 2008 Fig. 16.10a,p. 457)

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• Sinusoidal wave on string, 2

(SJ 2008 Fig. 16.10b,p. 457)

• Each string element (e.g., pointP) oscillates vertically in SHM

• Treat these as SHO vibratingwith ƒ = ƒ(blade oscillation)

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• Sinusoidal wave on string, 3

• Element’s transversespeed vy is:

or vy = -#A cos(kx – #t)

• vy differs from wavespeed v itself (SJ 2008 Fig. 16.10c,

p. 457)

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• Sinusoidal wave on string, 4

• Element’s transverseacceleration ay is:

or ay = -#2A sin(kx – #t)

(SJ 2008 Fig. 16.10d,p. 457)

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• Sinusoidal wave on string, 5

• vy & ay maxima are:• vy, max = #A (SJ 2008 Eq. 16.16, p. 457)

• ay, max = #2A (SJ 2008 Eq. 16.17, p. 457)

• vy & ay maxima don’t occur simultaneously:• v maximum at y = 0

• a maximum at y = ±A

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• v depends on (1) string’s physicalcharacteristics & (2) string tension T

for µ $ linear mass density (= mass/length)

• Eq. 16-26 assumes:(1) pulse doesn’t affect T(2) no particular pulse shape

• Wave speed on a string

(Eq. 16-26, p. 421)

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• Energy in string waves

• Waves transport energy in propagatingthrough a medium

• Model each string element as a SHOwith oscillation in y-direction

• Each element has same total energy

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• Energy in string waves, 2

• Each element has mass = 'm

• Each 'm’s KE is 'K = 1/2('m)vy2

• Now 'm = µ'x for µ $ linear mass density

• As string element 'x ! 0, above equationbecomes dK = 1/2(µdx)vy

2

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• Energy in string waves, 3

• Integrating over all elements, total KE in 1wavelength is K" = (1/4)µ#2A2" (compare Eq. 16-29, p. 422)

• Total PE in 1 wavelength is U" = (1/4)µ#2A2"

• So total ME of E" = K" + U" = (1/2)µ#2A2"(units: (kg/m)*(m/s)2*m = kg*m2/s2 = J)

• Next consider wave power "E/"t as opposed towave energy ...

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• Wave power

• Power is rate at which energy is transferred:

• Thus sinusoidal wave transfers power (:&2, A2, & wave speed v

(Eq. 16-33,p. 423)

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Waves vs. particles

Multiple waves cancombine at a point insame medium – canexist at same location

Multiple particles canexist only at differentlocations

Waves have acharacteristic size –their #

Particles have 0 size

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• Superposition principle

• If # 2 traveling waves move through amedium, wave function’s resultant at a point= algebraic sum of individual wave functions

• Waves obeying this superposition principle arelinear waves (for mechanical waves, linear

waves’ amplitudes « their #s)

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• Superposition & interference

• 2 traveling waves can pass through each otherw.o. being destroyed or altered, a result ofsuperposition principle

• Interference: Combining separate waves insame region of space to ! resultant wave

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• Superposition

• 2 pulses travel in opposite directions

• y1 is wave function of right-moving pulse; y2 is thatof left-moving pulse

• Pulses have same v but different shapes; eachelement’s displacement > 0

(compare Fig. 16-11, p. 425)

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• Superposition, 2

• When waves start tooverlap (fig. b), resultantwave function = y1 + y2

• When crests meet (fig. c),resultant wave has A > Aof either original wave

(compare Fig. 16-11, p. 425)

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• Superposition, 3

• 2 pulses separate & keep moving intheir original directions

• Pulse shapes are unchanged

(compare Fig. 16-11, p. 425)

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• Superposition in stretched spring

• 2 equal, symmetric pulsesthat travel in oppositedirections along stretchedspring obey superpositionprinciple

(SJ 2004 Fig. 18.1e)

time)

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• Types of interference

• Constructive interference: 2 pulses’displacements are in same direction• Resultant pulse’s A > A of either individual pulse

• Destructive interference: 2 pulses’displacements are in opposite directions• Resultant pulse’s A < A of either individual pulse

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• Destructive interference

• 2 pulses travel in oppositedirections

• Displacements are invertedw.r.t. each other

• When pulses overlap (fig.c), displacements partiallycancel each other

(SJ 2008 Fig. 18.2, p. 502)

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• Superposition of sinusoidal waves

• Assume 2 waves travel in samedirection with the same &, #, & A

• But, waves differ in phase

• y1 = A sin(kx - #t) (Eq. 16-47, p. 425)

• y2 = A sin(kx - #t + $) (Eq. 16-48, p. 425)

• y = y1+y2 (Eq. 16-51, p. 426)

= 2A cos($/2) sin(kx - #t + $/2)

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• Superposition of sinusoidal waves, 2

• Resultant wave function y is alsosinusoidal

• y has same & & # as original waves

• Resultant wave’s amplitude =2Acos($/2) & its phase = $/2

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• Sinusoidal waves with constructive

interference

• If $ = 0 or an even multiple of %,then cos($/2) = 1

• Resultant wave amplitude = 2A• Crests of 1 wave coincide

with crests of other wave

• Waves are in phaseeverywhere & interfereconstructively

(compare Fig. 16-13, p. 426)

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• Sinusoidal waves with destructiveinterference

• When $ = any odd multiple of %,then cos($/2) = 0

• Resultant wave’s amplitude = 0• Crests of 1 wave coincide with

troughs of other wave

• Waves interfere destructively

(compare Fig. 16-13, p. 426)

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• Sinusoidal waves, general interference

• When $ is other than 0or integer multiple of %,

get 0 < Aresultant < 2A

• Wave functions still add

(compare Fig. 16-13, p. 426)

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• Summary: Sinusoidal wave interference

• Constructive interference occurs when phasedifference $ = 0 or any even multiple of %• Resultant’s amplitude Aresultant = 2Acos($/2) = 2A

• Destructive interference occurs when$ = any odd multiple of %• Aresultant = 2Acos($/2) = 0

• General interference occurs when0 < $ < n% for integer n• Aresultant is 0 < Aresultant < 2A

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• Standing waves

• Assume 2 waves with same &, #, & Atravel in opposite directions in a medium

• y1 = A sin(kx – #t) & y2 = A sin(kx + #t)

• Waves interfere according to superpositionprinciple

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• Standing waves, 2

• Resultant is y = 2A sin(kx)cos(#t)(Eq. 16-60, p. 431)

• ! Wave function of standing wave (nokx–#t term since not a traveling wave)

• While y = f (t) everywhere sin(kx) ! 0,at any time t, ymax = 2A cos(#t) is nolonger a f (x) {i.e., a standing wave}

• Standing wave ! no apparent motion inoriginal waves’ propagation directions

(SJ 2008 Fig. 18.6, p. 505)

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• Note on amplitudes

• Use 3 A types to describe standingwaves• Individual waves’ A

• SHM A for medium’s elements, 2A sin(kx)

• Standing-wave amplitude, 2A• standing-wave elements vibrate within

constraints of envelope function 2A sin(kx),where x is an element’s position in medium

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• Standing waves, particle motion

• Each element in medium oscillates inSHM with same frequency #

• However, A of SHM depends onelement’s location within medium

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• Standing-wave definitions

• Nodes occur wherever A = 0 (i.e., xpositions s.t. x = n#/2 for n = 0, 1, 2, 3, ...)(Eq 16-62, p. 432)

• Antinodes occur at maximum displacements2A (i.e., at x = n#/4 for n = 1, 3, 5, ...)(Eq. 16-64, p. 432)

• Distance between adjacent: antinodes = "/2,

nodes = "/2, node & adjacent antinode = "/4

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• Nodes & antinodes

• Above: standing-wave patterns produced by 2waves of equal A traveling in opposite directions

• In standing wave, medium’s elements alternatebetween extremes shown in (a) & (c)

(compareFig. 16-16,

p. 431)

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• Wave reflection: Fixed end

• When pulse reaches fixedsupport, it moves back alongstring in opposite direction

• This is pulse’s reflection,which inverts the pulse

(compare Fig. 16-18, p. 432)

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• Wave reflection: Free end

• With a free end,string is free tomove vertically

• Pulse is reflectedbut not inverted

(compare Fig. 16-18, p. 432)

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• Wave transmission

• If boundary is between fixed & free, part of incidentpulse’s energy is reflected & part is transmitted(i.e., some energy passes through boundary)

Consider 2 cases:• Light string attached toheavier string: pulse’s

reflected portion is invertedbut with smaller A thanoriginal (SJ 2008 Fig. 16.15,p. 462)

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• Wave transmission, 2

• Heavier string attachedto light string: pulse’s

reflected portion is not inverted(SJ 2008 Fig. 16.16, p. 462)

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• Wave transmission, 3

• Energy conservation governs pulse:If boundary breaks up a pulse intoreflected & transmitted parts, sum oftheir energies = original pulse’s energy

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• Standing waves in a string

• Consider string of length Lfixed at both ends

• Set up standing waves bycontinuous superposition ofwaves incident on &reflected from ends

• Boundary condition existson these waves (compare Fig. 16-20, p. 433)

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• Standing waves in a string, 2

• Strings ends are necessarily nodes, since they’refixed & * have 0 displacement

• This boundary condition ! string having a set ofnormal modes of vibration:• Each mode has characteristic ƒ

• String’s normal oscillation modes follow from imposingrequirements that (1) string ends are nodes &(2) nodes & antinodes are separated by "/4

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• Standing waves in a string, 3

• First normal modeconsistent with boundaryconditions

• Nodes at both ends & 1antinode in middle

• Longest wavelength mode,

so 1/2" = L or " = 2L

(1st harmonic)(compare Fig. 16-20, p. 433)

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• Standing waves in a string, 4

• Consecutive normal modesadd 1 antinode for eachstep: n = # antinodes

• 2nd mode corresponds to" = L (2nd harmonic)

• 3rd mode corresponds to" = 2L/3 (3rd harmonic)

(compare Fig. 16-20, p. 433)

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• Standing waves in a string, 5

• Normal-mode " for string of length L fixed

at both ends: "n = 2L/n for n = 1, 2, 3, …

• n is nth normal mode of oscillation

• Above defines all possible modes for string

• Corresponding natural ƒ are:

(Eqs. 16-66, p. 434& 16-26, p. 421)

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• Resonance

• System can oscillate in # 1 normalmodes

• If we apply a periodic F to thissystem, then resulting motion’samplitude is greatest whenapplied F’s & = &0 (a systemnatural &)

• Since oscillating system has largeA when driven at any of its &0,call such ƒ the system’s resonancefrequencies ƒ0

• System friction limits Amax

(SJ 2004 Fig. 18.14,p. 558)

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• Resonance, example 1

• If pendulum A swings,other pendulums start tooscillate due to wavestransmitted through beam

• Pendulum C has amplitude> B or D amplitude

• C’s length is closest to A’s& so C’s ƒ0 is closest to A’sdriving frequency

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• Resonance, example 2

• Set up standing waves in astring if 1 end is connectedto a vibrating blade

• When blade vibrates at astring’s ƒ0, large-amplitudestanding waves result

(SJ 2008 Fig. 18.12, p. 512)