Chapter 17Chapter 17

WavesWaves

Wave MotionWave Motion

Fundamental to physics (as important Fundamental to physics (as important as particles)as particles)

A wave is the motion of a disturbanceA wave is the motion of a disturbance All waves carry energy and momentumAll waves carry energy and momentum Mechanical waves requireMechanical waves require

• Some source of disturbanceSome source of disturbance• A medium that can be disturbedA medium that can be disturbed• Some physical connection between or Some physical connection between or

mechanism though which adjacent portions mechanism though which adjacent portions of the medium influence each otherof the medium influence each other

Types of Waves – Traveling Types of Waves – Traveling WavesWaves

Flip one end of a Flip one end of a long rope that is long rope that is under tension and under tension and fixed at one endfixed at one end

The pulse travels The pulse travels to the right with a to the right with a definite speeddefinite speed

A disturbance of A disturbance of this type is called a this type is called a traveling wavetraveling wave

Types of Waves – TransverseTypes of Waves – Transverse

In a transverse wave, each element that is In a transverse wave, each element that is disturbed moves in a direction disturbed moves in a direction perpendicular to the wave motionperpendicular to the wave motion

Types of Waves – LongitudinalTypes of Waves – Longitudinal

In a longitudinal wave, the elements of the In a longitudinal wave, the elements of the medium undergo displacements parallel to medium undergo displacements parallel to the motion of the wavethe motion of the wave

A longitudinal wave is also called a A longitudinal wave is also called a compression wavecompression wave

Other Types of WavesOther Types of Waves

Waves may be a combination of Waves may be a combination of transverse and longitudinaltransverse and longitudinal

Mainly consider periodic sinusoidal Mainly consider periodic sinusoidal waveswaves

Waveform – A Picture of a Waveform – A Picture of a WaveWave

The brown curve is a The brown curve is a “snapshot” of the “snapshot” of the wave at some wave at some instant in timeinstant in time

The blue curve is The blue curve is later in timelater in time

The high points are The high points are crestscrests of the wave of the wave

The low points are The low points are troughstroughs of the wave of the wave

Longitudinal Wave Represented as Longitudinal Wave Represented as a Sine Curvea Sine Curve

A longitudinal wave can also be represented as a A longitudinal wave can also be represented as a sine curvesine curve

Compressions correspond to crests and stretches Compressions correspond to crests and stretches correspond to troughscorrespond to troughs

Also called density waves or pressure wavesAlso called density waves or pressure waves

Amplitude and WavelengthAmplitude and Wavelength

Amplitude is the Amplitude is the maximum maximum displacement of string displacement of string above the equilibrium above the equilibrium positionposition

Wavelength, Wavelength, λ, is the λ, is the distance between two distance between two successive points that successive points that behave identicallybehave identically

Speed of a WaveSpeed of a Wave

v = ƒ v = ƒ λλ• Is derived from the basic speed equation Is derived from the basic speed equation

of distance/timeof distance/time This is a general equation that can This is a general equation that can

be applied to many types of wavesbe applied to many types of waves

Speed of a Wave on a StringSpeed of a Wave on a String

The speed of wave on a stretched The speed of wave on a stretched rope under some tension, Frope under some tension, F

• is called the linear densityis called the linear density The speed depends only upon the The speed depends only upon the

properties of the medium through properties of the medium through which the disturbance travelswhich the disturbance travels

F mv where

L

ExampleExample

String vibrates at 10 hz and a String vibrates at 10 hz and a snapshot. Determine wavelength, snapshot. Determine wavelength, period, amplitude, speed.period, amplitude, speed.

ExampleExample

Mass and length of Mass and length of the string are 0.9 the string are 0.9 kg and 8 m. What kg and 8 m. What is the speed of is the speed of wave on the wave on the string?string?

Wave fronts & raysWave fronts & rays

Wave frontsWave fronts – locate – locate crests of waves crests of waves • Ripples from a pebble Ripples from a pebble

dropping in a pond dropping in a pond • concentric arcsconcentric arcs• The distance between The distance between

successive wave fronts is successive wave fronts is the wavelengththe wavelength

RaysRays are the radial lines are the radial lines pointing out from the pointing out from the source and perpendicular source and perpendicular to the wave frontsto the wave fronts

Plane WavePlane Wave

Far away from the Far away from the source, the wave source, the wave fronts are nearly fronts are nearly parallel planesparallel planes

The rays are nearly The rays are nearly parallel linesparallel lines

A small segment of A small segment of the wave front is the wave front is approximately a approximately a plane waveplane wave

Reflection of WavesReflection of Waves

Waves reflect when they hit Waves reflect when they hit boundariesboundaries• Fixed end: wave inverts upon reflectionFixed end: wave inverts upon reflection• Free end: no inversionFree end: no inversion

Superposition PrincipleSuperposition Principle

Two traveling waves can meet and pass Two traveling waves can meet and pass through each other without being through each other without being destroyed or even altereddestroyed or even altered

Waves obey the Waves obey the Superposition PrincipleSuperposition Principle• If two or more traveling waves are moving If two or more traveling waves are moving

through a medium, the resulting wave is found through a medium, the resulting wave is found by adding together the displacements of the by adding together the displacements of the individual waves point by pointindividual waves point by point

• Actually only true for waves with small Actually only true for waves with small amplitudesamplitudes

Constructive InterferenceConstructive Interference

Two waves, a and Two waves, a and b, have the same b, have the same frequency and frequency and amplitudeamplitude• Are Are in phasein phase

The combined The combined wave, c, has the wave, c, has the same frequency same frequency and a greater and a greater amplitudeamplitude

Constructive Interference in a Constructive Interference in a StringString

Two pulses are traveling in opposite directionsTwo pulses are traveling in opposite directions The net displacement when they overlap is the The net displacement when they overlap is the

sum of the displacements of the pulsessum of the displacements of the pulses Note that the pulses are unchanged after the Note that the pulses are unchanged after the

interferenceinterference

Destructive InterferenceDestructive Interference

Two waves, a and b, Two waves, a and b, have the same have the same amplitude and amplitude and frequencyfrequency

They are 180° out of They are 180° out of phasephase

When they combine, When they combine, the waveforms the waveforms cancelcancel

Destructive Interference in a Destructive Interference in a StringString

Two pulses are traveling in opposite directionsTwo pulses are traveling in opposite directions The net displacement when they overlap is The net displacement when they overlap is

decreased since the displacements of the pulses decreased since the displacements of the pulses subtractsubtract

Note that the pulses are unchanged after the Note that the pulses are unchanged after the interferenceinterference

Standing WavesStanding Waves

When a traveling wave reflects back When a traveling wave reflects back on itself, it creates traveling waves in on itself, it creates traveling waves in both directionsboth directions

The wave and its reflection interfere The wave and its reflection interfere according to the superposition according to the superposition principleprinciple

With exactly the right frequency, the With exactly the right frequency, the wave will appear to stand stillwave will appear to stand still• This is called a This is called a standing wavestanding wave

Standing Waves, contStanding Waves, cont

A A nodenode occurs where the two traveling occurs where the two traveling waves have the same magnitude of waves have the same magnitude of displacement, but the displacements are displacement, but the displacements are in opposite directionsin opposite directions• Net displacement is zero at that pointNet displacement is zero at that point• The distance between two nodes is The distance between two nodes is ½λ½λ

An An antinodeantinode occurs where the standing occurs where the standing wave vibrates at maximum amplitudewave vibrates at maximum amplitude• The distance between two antinodes is The distance between two antinodes is ½λ½λ

Distance between node and antinode Distance between node and antinode λ/4λ/4

Standing Waves on a StringStanding Waves on a String

Nodes must occur at the ends of the string Nodes must occur at the ends of the string because these points are fixedbecause these points are fixed

Standing Waves, cont.Standing Waves, cont.

The pink arrows The pink arrows indicate the direction indicate the direction of motion of the parts of motion of the parts of the stringof the string

All points on the string All points on the string oscillate together oscillate together vertically with the vertically with the same frequency, but same frequency, but different points have different points have different amplitudes of different amplitudes of motionmotion

ResonanceResonance

Can have resonance in strings (these Can have resonance in strings (these are actually standing waves)are actually standing waves)

Amplitude increasesAmplitude increases How to determine resonance How to determine resonance

frequencies?frequencies?

Standing Waves on a String, Standing Waves on a String, finalfinal

The lowest The lowest frequency of frequency of vibration (b) is vibration (b) is called the called the fundamental fundamental frequencyfrequency

n

LnL n

n 2

2

12nf

L

nvvf

nn

Standing Waves on a String – Standing Waves on a String – FrequenciesFrequencies

ƒƒ11, ƒ, ƒ22, ƒ, ƒ33 form a harmonic series form a harmonic series

• ƒƒ1 1 is the fundamental and also the first is the fundamental and also the first harmonicharmonic

• ƒƒ22 is the second harmonic (1 is the second harmonic (1stst overtone) overtone)

Waves in the string that are not in the Waves in the string that are not in the harmonic series are quickly damped harmonic series are quickly damped outout• In effect, when the string is disturbed, it In effect, when the string is disturbed, it

“selects” the standing wave frequencies“selects” the standing wave frequencies

ExampleExample

A guitar has 0.6 m long string. Wave A guitar has 0.6 m long string. Wave speed on the string is 420 m/s. What speed on the string is 420 m/s. What are the frequencies of the first few are the frequencies of the first few harmonics?harmonics?

ExampleExample

String 80 cm long is driven with String 80 cm long is driven with frequency of 120 Hz when both ends frequency of 120 Hz when both ends fixed. There are 4 nodes in the fixed. There are 4 nodes in the middle of the string. Find speed of middle of the string. Find speed of wave on string?wave on string?

Producing a Sound WaveProducing a Sound Wave

Sound waves are longitudinal waves Sound waves are longitudinal waves traveling through a mediumtraveling through a medium

A tuning fork can be used as an example A tuning fork can be used as an example of producing a sound waveof producing a sound wave

Using a Tuning Fork to Produce Using a Tuning Fork to Produce a Sound Wavea Sound Wave

A tuning fork will produce a A tuning fork will produce a pure musical notepure musical note

As the tines vibrate, they As the tines vibrate, they disturb the air near themdisturb the air near them

As the tine swings to the As the tine swings to the right, it forces the air right, it forces the air molecules near it closer molecules near it closer togethertogether

This produces a high density This produces a high density area in the airarea in the air• This is an area of compressionThis is an area of compression

Using a Tuning Fork, cont.Using a Tuning Fork, cont.

As the tine moves As the tine moves toward the left, the air toward the left, the air molecules to the right molecules to the right of the tine spread outof the tine spread out

This produces an area This produces an area of low densityof low density• This area is called a This area is called a

rarefactionrarefaction

Using a Tuning Fork, finalUsing a Tuning Fork, final

As the tuning fork continues to vibrate, a succession As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the of compressions and rarefactions spread out from the forkfork

A sinusoidal curve can be used to represent the A sinusoidal curve can be used to represent the longitudinal wavelongitudinal wave• Crests correspond to compressions and troughs to Crests correspond to compressions and troughs to

rarefactionsrarefactions

Categories of Sound WavesCategories of Sound Waves

Audible wavesAudible waves• Lay within the normal range of hearing of the Lay within the normal range of hearing of the

human earhuman ear• Normally between 20 Hz to 20,000 HzNormally between 20 Hz to 20,000 Hz

Infrasonic wavesInfrasonic waves• Frequencies are below the audible rangeFrequencies are below the audible range• Earthquakes are an exampleEarthquakes are an example

Ultrasonic wavesUltrasonic waves• Frequencies are above the audible rangeFrequencies are above the audible range• Dog whistles are an exampleDog whistles are an example

Applications of UltrasoundApplications of Ultrasound

Can be used to produce images of Can be used to produce images of small objectssmall objects

Widely used as a diagnostic and Widely used as a diagnostic and treatment tool in medicinetreatment tool in medicine• Ultrasounds to observe babies in the wombUltrasounds to observe babies in the womb• Cavitron Ultrasonic Surgical Aspirator (CUSA) used to Cavitron Ultrasonic Surgical Aspirator (CUSA) used to

surgically remove brain tumorssurgically remove brain tumors

Ultrasonic ranging unit for camerasUltrasonic ranging unit for cameras

Speed of Sound, GeneralSpeed of Sound, General

The speed of sound is higher in solids The speed of sound is higher in solids than in gasesthan in gases

The speed is slower in liquids than in The speed is slower in liquids than in solidssolids

Speed of Sound in AirSpeed of Sound in Air

331 m/s is the speed of sound at 0°C 331 m/s is the speed of sound at 0°C and 1 atmand 1 atm

Changes with temperatureChanges with temperature

T in °C T in °C At 20 °C, 343 m/sAt 20 °C, 343 m/s In other substancesIn other substances

m/s)(in 6.0331 TvT

in He: 1000 m/sin He: 1000 m/s

in Water: 1500 m/sin Water: 1500 m/s

in Al: 5000 m/sin Al: 5000 m/s

Standing Waves in Air ColumnsStanding Waves in Air Columns

If one end of the air column is closed, If one end of the air column is closed, a node must exist at this end since a node must exist at this end since the movement of the air is restrictedthe movement of the air is restricted

If the end is open, the elements of If the end is open, the elements of the air have complete freedom of the air have complete freedom of movement and an antinode existsmovement and an antinode exists

Tube Open at Both EndsTube Open at Both Ends

Resonance in Air Column Open Resonance in Air Column Open at Both Endsat Both Ends

In a pipe open at both ends, the In a pipe open at both ends, the natural frequency of vibration forms natural frequency of vibration forms a series whose harmonics are equal a series whose harmonics are equal to integral multiples of the to integral multiples of the fundamental frequencyfundamental frequency

1ƒ ƒ 1, 2, 3,2n

vn n n

L

Tube Closed at One EndTube Closed at One EndClosed pipeClosed pipe

Resonance in an Air Column Resonance in an Air Column Closed at One EndClosed at One End

The closed end must be a nodeThe closed end must be a node The open end is an antinodeThe open end is an antinode

There are no even multiples of the There are no even multiples of the fundamental harmonicfundamental harmonic

1ƒ 1, 3, 5,4n

vf n n n

L

ExampleExample

An open organ pipe has a fundamental An open organ pipe has a fundamental frequency of 660 Hz at 0 C and 1 frequency of 660 Hz at 0 C and 1 atm. atm.

a.a. Frequency of 2Frequency of 2ndnd overtone? overtone?

b.b. Fundamental at 20 C?Fundamental at 20 C?

c.c. Replacing air with He?Replacing air with He?