Chapter 16Chapter 16

Vibrations and WavesVibrations and Waves

Vibrations/OscillationsVibrations/Oscillations

Object at the end of a springObject at the end of a spring Tuning forkTuning fork PendulumPendulum String of a violinString of a violin Atoms in a crystalAtoms in a crystal Source of a waveSource of a wave

Hooke’s LawHooke’s Law

FFss = - k x = - k x

• FFss is the spring force is the spring force

• k is the spring constantk is the spring constant It is a measure of the stiffness of the springIt is a measure of the stiffness of the spring

• A large k indicates a stiff spring and a small k indicates A large k indicates a stiff spring and a small k indicates a soft springa soft spring

• x is the displacement of the object from its x is the displacement of the object from its equilibrium positionequilibrium position

x = 0 at the equilibrium positionx = 0 at the equilibrium position

• The negative sign indicates that the force is The negative sign indicates that the force is always directed opposite to the displacementalways directed opposite to the displacement

Hooke’s Law ForceHooke’s Law Force

The force always acts toward the The force always acts toward the equilibrium positionequilibrium position• It is called the It is called the restoring forcerestoring force

The direction of the restoring force is The direction of the restoring force is such that the object is being either such that the object is being either pushed or pulled toward the pushed or pulled toward the equilibrium positionequilibrium position

Hooke’s Law Applied to a Hooke’s Law Applied to a Spring – Mass SystemSpring – Mass System

When x is positive When x is positive (to the right), F is (to the right), F is negative (to the negative (to the left)left)

When x = 0 (at When x = 0 (at equilibrium), F is 0equilibrium), F is 0

When x is negative When x is negative (to the left), F is (to the left), F is positive (to the positive (to the right)right)

Motion of the Spring-Mass Motion of the Spring-Mass SystemSystem

Assume the object is initially pulled to a Assume the object is initially pulled to a distance A and released from restdistance A and released from rest

As the object moves toward the As the object moves toward the equilibrium position, F and a decrease, but equilibrium position, F and a decrease, but v increasesv increases

At x = 0, F and a are zero, but v is a At x = 0, F and a are zero, but v is a maximummaximum

The object’s momentum causes it to The object’s momentum causes it to overshoot the equilibrium positionovershoot the equilibrium position

Motion of the Spring-Mass Motion of the Spring-Mass System, contSystem, cont

The force and acceleration start to The force and acceleration start to increase in the opposite direction increase in the opposite direction and velocity decreasesand velocity decreases

The motion momentarily comes to a The motion momentarily comes to a stop at x = - A stop at x = - A

It then accelerates back toward the It then accelerates back toward the equilibrium positionequilibrium position

The motion continues indefinitelyThe motion continues indefinitely

Simple Harmonic MotionSimple Harmonic Motion

Motion that occurs when the net Motion that occurs when the net force along the direction of motion force along the direction of motion obeys Hooke’s Lawobeys Hooke’s Law• The force is proportional to the The force is proportional to the

displacement and always directed displacement and always directed toward the equilibrium positiontoward the equilibrium position

The motion of a spring mass system The motion of a spring mass system is an example of Simple Harmonic is an example of Simple Harmonic MotionMotion

Simple Harmonic Motion, cont.Simple Harmonic Motion, cont.

Not all periodic motion over the same Not all periodic motion over the same path can be considered Simple path can be considered Simple Harmonic motionHarmonic motion

To be Simple Harmonic motion, the To be Simple Harmonic motion, the force needs to obey Hooke’s Lawforce needs to obey Hooke’s Law

AmplitudeAmplitude

Amplitude, AAmplitude, A• The amplitude is the maximum position The amplitude is the maximum position

of the object relative to the equilibrium of the object relative to the equilibrium positionposition

• In the absence of friction, an object in In the absence of friction, an object in simple harmonic motion will oscillate simple harmonic motion will oscillate between the positions x = between the positions x = ±A±A

Period and FrequencyPeriod and Frequency

The period, T, is the time that it takes for The period, T, is the time that it takes for the object to complete one complete cycle the object to complete one complete cycle of motion of motion • From x = A to x = - A and back to x = AFrom x = A to x = - A and back to x = A

The frequency, The frequency, ƒ, is the number of ƒ, is the number of complete cycles or vibrations per unit timecomplete cycles or vibrations per unit time• ƒ = 1 / Tƒ = 1 / T• Frequency is the reciprocal of the periodFrequency is the reciprocal of the period• Unit: Hertz, 1 Hz = 1 cycle/sUnit: Hertz, 1 Hz = 1 cycle/s

Acceleration of an Object in Acceleration of an Object in Simple Harmonic MotionSimple Harmonic Motion

Newton’s second law will relate force and Newton’s second law will relate force and accelerationacceleration

The force is given by Hooke’s LawThe force is given by Hooke’s Law F = - k x = m aF = - k x = m a

• a = -kx / ma = -kx / m The acceleration is a function of positionThe acceleration is a function of position

• Acceleration is Acceleration is notnot constant and therefore the constant and therefore the uniformly accelerated motion equation cannot uniformly accelerated motion equation cannot be appliedbe applied

Elastic Potential EnergyElastic Potential Energy

A compressed spring has potential A compressed spring has potential energyenergy• The compressed spring, when allowed to The compressed spring, when allowed to

expand, can apply a force to an objectexpand, can apply a force to an object• The potential energy of the spring can The potential energy of the spring can

be transformed into kinetic energy of be transformed into kinetic energy of the objectthe object

Elastic Potential Energy, contElastic Potential Energy, cont

The energy stored in a stretched or The energy stored in a stretched or compressed spring or other elastic compressed spring or other elastic material is called material is called elastic potential energyelastic potential energy• PEPEss = = ½kx½kx22

The energy is stored only when the spring The energy is stored only when the spring is stretched or compressedis stretched or compressed

Elastic potential energy can be added to Elastic potential energy can be added to the statements of Conservation of Energy the statements of Conservation of Energy and Work-Energyand Work-Energy

Energy in a Spring Mass Energy in a Spring Mass SystemSystem

A block sliding on a A block sliding on a frictionless system frictionless system collides with a light collides with a light springspring

The block attaches The block attaches to the springto the spring

The system oscillates The system oscillates in Simple Harmonic in Simple Harmonic MotionMotion

Energy TransformationsEnergy Transformations

The block is moving on a frictionless surfaceThe block is moving on a frictionless surface The total mechanical energy of the system is the The total mechanical energy of the system is the

kinetic energy of the blockkinetic energy of the block

Energy Transformations, 2Energy Transformations, 2

The spring is partially compressedThe spring is partially compressed The energy is shared between kinetic energy and The energy is shared between kinetic energy and

elastic potential energyelastic potential energy The total mechanical energy is the sum of the The total mechanical energy is the sum of the

kinetic energy and the elastic potential energykinetic energy and the elastic potential energy

Energy Transformations, 3Energy Transformations, 3

The spring is now fully compressedThe spring is now fully compressed The block momentarily stopsThe block momentarily stops The total mechanical energy is stored as The total mechanical energy is stored as

elastic potential energy of the springelastic potential energy of the spring

Energy Transformations, 4Energy Transformations, 4

When the block leaves the spring, the total When the block leaves the spring, the total mechanical energy is in the kinetic energy of the mechanical energy is in the kinetic energy of the blockblock

The spring force is conservative and the total The spring force is conservative and the total energy of the system remains constantenergy of the system remains constant

Graphical Representation of Graphical Representation of MotionMotion

When x is a maximum When x is a maximum or minimum, velocity or minimum, velocity is zerois zero

When x is zero, the When x is zero, the velocity is a maximumvelocity is a maximum

When x is a maximum When x is a maximum in the positive in the positive direction, a is a direction, a is a maximum in the maximum in the negative directionnegative direction

Period of a SHMPeriod of a SHM

It can be shown thatIt can be shown that

k

mTPeriod 2

Simple PendulumSimple Pendulum The simple The simple

pendulum is pendulum is another example another example of simple of simple harmonic motionharmonic motion

The force is the The force is the component of the component of the weight tangent to weight tangent to the path of the path of motionmotion• FFtt = - m g sin = - m g sin θθ

Simple Pendulum, contSimple Pendulum, cont

In general, the motion of a pendulum In general, the motion of a pendulum is not simple harmonicis not simple harmonic

However, for small angles, it However, for small angles, it becomes simple harmonicbecomes simple harmonic• In general, angles < 15° are small In general, angles < 15° are small

enoughenough

g

lTPeriod 2

Period of Simple PendulumPeriod of Simple Pendulum

This shows that the period is This shows that the period is independent of the amplitudeindependent of the amplitude

The period depends on the length of The period depends on the length of the pendulum and the acceleration of the pendulum and the acceleration of gravity at the location of the gravity at the location of the pendulumpendulum

Measure gMeasure g

g

lTPeriod 2

ExampleExample

A body with a mass of 5.0 kg is A body with a mass of 5.0 kg is suspended by a spring, which suspended by a spring, which stretches 10 cm when the body is stretches 10 cm when the body is attached. The body is then pulled attached. The body is then pulled downward an additional 5 cm and downward an additional 5 cm and released. Find k, T, f, E, vreleased. Find k, T, f, E, vmax max and aand amax.max.

ExampleExample

Period of a simple pendulum is 2s on Period of a simple pendulum is 2s on Earth. What would be the period of Earth. What would be the period of the same pendulum on the moon? the same pendulum on the moon? (g(gmoonmoon=1.67 m/s=1.67 m/s22))

Forced Vibrations and Forced Vibrations and ResonanceResonance

Shaking (vibration) as specific Shaking (vibration) as specific frequencyfrequency

Pushing child on swingPushing child on swing glassglass Tacoma Narrows BridgeTacoma Narrows Bridge ! Every object has its natural ! Every object has its natural

frequenciesfrequencies

Wave MotionWave Motion

A wave is the motion of a disturbanceA wave is the motion of a disturbance 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

All waves carry energy and momentumAll waves carry energy and momentum

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

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?

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

Chapter 17Chapter 17

SoundSound

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

Intensity of Sound WavesIntensity of Sound Waves

The average The average intensityintensity of a wave is the rate of a wave is the rate at which the energy flows through a unit at which the energy flows through a unit area, A, oriented perpendicular to the area, A, oriented perpendicular to the direction of travel of the wavedirection of travel of the wave

The rate of energy transfer is the powerThe rate of energy transfer is the power Units are W/mUnits are W/m22

Various Intensities of SoundVarious Intensities of Sound

Threshold of hearingThreshold of hearing• Faintest sound most humans can hearFaintest sound most humans can hear• About 1 x 10About 1 x 10-12-12 W/m W/m22

Threshold of painThreshold of pain• Loudest sound most humans can tolerateLoudest sound most humans can tolerate• About 1 W/mAbout 1 W/m22

The ear is a very sensitive detector of The ear is a very sensitive detector of sound wavessound waves• It can detect pressure fluctuations as small as It can detect pressure fluctuations as small as

about 3 parts in 10about 3 parts in 101010

Intensity Level of Sound WavesIntensity Level of Sound Waves

The sensation of loudness is The sensation of loudness is logarithmic in the human hearlogarithmic in the human hear

β is the intensity level or the decibel β is the intensity level or the decibel level of the soundlevel of the sound

IIoo = 1 x 10 = 1 x 10-12-12 W/m W/m22 is the threshold of is the threshold of hearinghearing

10 logo

II

Various Intensity LevelsVarious Intensity Levels

Threshold of hearing is 0 dBThreshold of hearing is 0 dB Threshold of pain is 120 dBThreshold of pain is 120 dB Jet airplanes are about 150 dBJet airplanes are about 150 dB Table of next slide lists intensity Table of next slide lists intensity

levels of various soundslevels of various sounds• Multiplying a given intensity by 10 adds Multiplying a given intensity by 10 adds

10 dB to the intensity level10 dB to the intensity level

ExampleExample

Quite automobile: 10Quite automobile: 10-7-7W/mW/m22

1000 automobiles:?1000 automobiles:?

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

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

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?

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?

BeatsBeats BeatsBeats are alternations in loudness, due to interference are alternations in loudness, due to interference Waves have slightly different frequencies and the time Waves have slightly different frequencies and the time

between constructive and destructive interference between constructive and destructive interference alternatesalternates

The The beat frequencybeat frequency equals the difference in equals the difference in frequency between the two sources: frequency between the two sources:

2 1ƒ ƒ ƒb

Doppler EffectDoppler Effect

A Doppler effect is experienced A Doppler effect is experienced whenever there is relative motion whenever there is relative motion between a source of waves and an between a source of waves and an observer.observer.• When the source and the observer are When the source and the observer are

moving toward each other, the observer moving toward each other, the observer hears a higher frequencyhears a higher frequency

• When the source and the observer are When the source and the observer are moving away from each other, the moving away from each other, the observer hears a lower frequencyobserver hears a lower frequency

Doppler Effect, General CaseDoppler Effect, General Case

Both the source and the observer could be Both the source and the observer could be movingmoving

Use positive values of vUse positive values of voo if observer if observer moving toward sourcemoving toward source• Frequency appears higherFrequency appears higher

Use positive values of vUse positive values of vss if source moving if source moving away from observeraway from observer• Frequency appears lowerFrequency appears lower

)(s

oso vv

vvff

ExampleExample

FFs s = 300 Hz and v=300m/s= 300 Hz and v=300m/s

a.a. Observer moves 30m/s away from Observer moves 30m/s away from source.source.

b.b. Source moves 30 m/s toward Source moves 30 m/s toward observer.observer.

c.c. Both movingBoth moving

Police radarPolice radar““Redshift” of distant galaxies Redshift” of distant galaxies

Shock WavesShock Waves

A shock wave A shock wave results when the results when the source velocity source velocity exceeds the exceeds the speed of the speed of the wave itselfwave itself

The circles The circles represent the represent the wave fronts wave fronts emitted by the emitted by the sourcesource

Shock Waves, finalShock Waves, final

Shock waves carry Shock waves carry energy energy concentrated on concentrated on the surface of the the surface of the cone, with cone, with correspondingly correspondingly great pressure great pressure variationsvariations

A jet produces a A jet produces a shock wave seen shock wave seen as a fogas a fog