WavesWavesCh 12Ch 12

Simple Harmonic Motion (SHM)Simple Harmonic Motion (SHM)

Is Periodic motion that results from a Is Periodic motion that results from a restoring force that is proportional to restoring force that is proportional to displacement, or “back and forth displacement, or “back and forth movement” movement”

SHM is a consistent, repeating motionSHM is a consistent, repeating motion Example 1: a spring with a mass Example 1: a spring with a mass

attached (frictionless)attached (frictionless) Example 2 : a pendulumExample 2 : a pendulum

SHM ForceSHM Force For an object in SHM, the restoring force is For an object in SHM, the restoring force is

trying to restore the object to its trying to restore the object to its equilibrium equilibrium position where there is position where there is zerozero restoring forcerestoring force

As displacement from equilibrium As displacement from equilibrium increases, the force trying to return it increases, the force trying to return it increases tooincreases too

For a For a spring with a massspring with a mass, the , the springspring provides the restoring forceprovides the restoring force

For a simple For a simple pendulumpendulum, , gravitygravity provides provides the restoring forcethe restoring force

SHM summarySHM summary Maximum displacement (Maximum displacement (xx):):

– Max force, accelerationMax force, acceleration– Min velocity (0)Min velocity (0)

Minumum displacement (Minumum displacement (x x =0):=0):– Min force & acceleration (=0)Min force & acceleration (=0)– Max velocityMax velocity– Equilibrium positionEquilibrium position– Human slinshot http://www.youtube.com/watch?v=u2-od4n5Xl0

Simple Harmonic MotionSimple Harmonic Motion

One complete back and forth motion One complete back and forth motion is one is one CYCLECYCLE

The time for 1 cycle of SHM is called The time for 1 cycle of SHM is called the the Period (T)Period (T) measured in seconds measured in seconds

SpringsSprings

Springs can be made from many Springs can be made from many materials in many shapes and sizesmaterials in many shapes and sizes

If friction and material flaws are If friction and material flaws are ignored, the spring is called “Ideal”ignored, the spring is called “Ideal”

Springs are compared by measuring Springs are compared by measuring their force at amount of distance their force at amount of distance stretchedstretched

Ideal springs follow Hooke’s LawIdeal springs follow Hooke’s Law

Hooke’s LawHooke’s Law

Hooke’s Law means the ratio of spring Hooke’s Law means the ratio of spring force force displacement (F/x) is a constant ( displacement (F/x) is a constant (kk))

Formula:Formula:

Where Where FFss = spring force, or elastic force = spring force, or elastic force xx = spring displacement (meter) = spring displacement (meter) kk = spring constant (N/m) = spring constant (N/m)The negative sign indicates the spring pulls The negative sign indicates the spring pulls

in the opposite direction of in the opposite direction of xx

sF k x

ExamplesExamples

A load of 45 N attached to a spring A load of 45 N attached to a spring that is hanging vertically stretches that is hanging vertically stretches the spring 0.14 m. What is the the spring 0.14 m. What is the spring constant?spring constant?

If a mass of 0.55 kg attached to a If a mass of 0.55 kg attached to a vertical spring stretches the spring vertical spring stretches the spring 36 cm from its equilibrium position, 36 cm from its equilibrium position, what is the spring constant?what is the spring constant?

SHM & SpringsSHM & Springs

The The timetime for one back and forth cycle (or for one back and forth cycle (or up and down) is called the up and down) is called the Period (T)Period (T) measured in seconds. It can be calculated measured in seconds. It can be calculated

if if kk is known and the mass on the spring.is known and the mass on the spring.

2m

Tk

What is the period of the spring in the previous example (0.55kg) ?

SHM & PendulumsSHM & Pendulums

SHM can be analyzed using the sine SHM can be analyzed using the sine curvecurve

A pendulum traces out a sine curve A pendulum traces out a sine curve as it swings as it swings

– http://www.walter-fendt.de/ph14e/pendulum.htm

http://www.youtube.com/watch?v=yVkdfJ9PkRQ&feature=youtu.be  

Pendulum Period Changes??Pendulum Period Changes??

Mass of pendulum?Mass of pendulum? Amount of swing?Amount of swing? Length of Pendulum?Length of Pendulum? Why were pendulums Why were pendulums

used in clocks?used in clocks?

Length

mass

Amount of swing

Mass & Spring and Pendulum Mass & Spring and Pendulum Period formulaePeriod formulae

Spring & mass Simple Pendulum

T 2m

k

T 2L

g

ExamplesExamples

You are designing a pendulum clock You are designing a pendulum clock to have a period of 1.0 s. How long to have a period of 1.0 s. How long should the pendulum be?should the pendulum be?

A mass of 0.30 kg is attached to a A mass of 0.30 kg is attached to a spring and is set into vibration with spring and is set into vibration with a period of 0.24 s. What is the a period of 0.24 s. What is the spring constant of the spring?spring constant of the spring?

Human slinshot http://www.youtube.com/watch?v=u2-od4n5Xl0 http://www.youtube.com/watch?v=yVkdfJ9PkRQ&feature=youtu.be    

ExamplesExamples A 0.75 kg mass attached to a vertical spring A 0.75 kg mass attached to a vertical spring

stretches the spring 0.30 m.stretches the spring 0.30 m.– What is the spring constant?What is the spring constant?– The mass-spring system is now placed on a horizontal The mass-spring system is now placed on a horizontal

surface and set vibrating. What is the period of the surface and set vibrating. What is the period of the vibration?vibration?

Calculate the period of a 3.500 m long pendulum Calculate the period of a 3.500 m long pendulum at the following locations at the following locations – The North Pole, where g = 9.832 m/sThe North Pole, where g = 9.832 m/s22

– Chicago, where g = 9.803 m/sChicago, where g = 9.803 m/s22

– Jakarta, Indonesia, where g = 9.782 m/sJakarta, Indonesia, where g = 9.782 m/s22

DefinitionDefinition WaveWave

– A moving disturbance that transports A moving disturbance that transports ENERGY, but NO massENERGY, but NO mass

– Examples:Examples:Water (ocean) Water (ocean) http://www.youtube.com/watch?v=XWZAz9Qbzos SoundSoundLightLightMicrowavesMicrowaves

A sine curve is the model for a waveA sine curve is the model for a wave

Waves Information Waves Information

Waves can be:Waves can be:– A single vibration, a PulseA single vibration, a Pulse– A repeating, or periodic, stream of vibrations A repeating, or periodic, stream of vibrations

Two groups of waves will be studied:Two groups of waves will be studied:– Mechanical Waves (sound, springs, pendulum, Mechanical Waves (sound, springs, pendulum,

ocean)ocean)– Electromagnetic (light, microwave, radio)Electromagnetic (light, microwave, radio)

We’ll study mechanical waves We’ll study mechanical waves

Mechanical WavesMechanical Waves

Examples: Waves in string or springs, Examples: Waves in string or springs, ocean, sound waves, seismic waves ocean, sound waves, seismic waves (earthquakes)(earthquakes)

These waves REQUIRE some material, These waves REQUIRE some material, called a Medium, to transport the wavecalled a Medium, to transport the waveWavesWaves MediumMediumRopeRope roperopeSoundSound airairSeismicSeismic earth (ground) or waterearth (ground) or water

The Medium determines how the wave The Medium determines how the wave moves (ex: how fast) moves (ex: how fast)

Wave TypesWave Types

There are two forms of waves There are two forms of waves which we will study:which we will study:– Transverse (next slide)Transverse (next slide)– Longitudinal (in 2 slides)Longitudinal (in 2 slides)

Transverse WavesTransverse Waves Side to side vibration in a Side to side vibration in a

direction perpendicular to the direction perpendicular to the wave's direction of travel wave's direction of travel

Examples: Examples: – waves on a ropewaves on a rope– string musical instruments musical instruments– One type of slinky waveOne type of slinky wave

Longitudinal WavesLongitudinal Waves Back and forth vibration in a direction parallel Back and forth vibration in a direction parallel

to the wave's motion to the wave's motion Compressed region Compressed region followed by expandedfollowed by expanded region, often called region, often called

pressure wavespressure waves

Examples: Examples: – Sound waves Sound waves – One type of slinky waveOne type of slinky wave

http://einstein.byu.edu/~masong/HTMstuff/WaveTrans.html http://www.wimp.com/slinkyanswer/

Longitudinal WavesLongitudinal Waves These are difficult to draw, especially for These are difficult to draw, especially for

soundsound Can be also represented by Sine curveCan be also represented by Sine curve

– Dense or compressed areas (peaks of curve) Dense or compressed areas (peaks of curve) are called “Compressions”are called “Compressions”

– Lows are “Rarefactions”Lows are “Rarefactions”

Measuring WavesMeasuring Waves Waves are looked at in one “cycle”, Waves are looked at in one “cycle”,

where the pattern starts againwhere the pattern starts again Highs & Lows are called crests and Highs & Lows are called crests and

troughstroughs

cycleCrest or Peak

Troughcycle

Measuring WavesMeasuring Waves Typically look at waves as they Typically look at waves as they

move, so time is needed move, so time is needed Wave Wave Period(T)Period(T) - the amount of - the amount of

time for a wave to repeat itself at a time for a wave to repeat itself at a specific point in space (1 cycle time)specific point in space (1 cycle time)– It is measured in secondsIt is measured in seconds

time

Period,Tvelocity

Frequency(Frequency(ff)) - is the number of wave - is the number of wave crests (cycles) passing a given point crests (cycles) passing a given point per unit time (per second)per unit time (per second)

– It is measured in Hertz (cycles per It is measured in Hertz (cycles per second), or Hzsecond), or Hz

– Frequency = reciprocal of PeriodFrequency = reciprocal of Periodff = 1/T or T = 1/ = 1/T or T = 1/ff

– ff does not depend on the mediumdoes not depend on the medium

Measuring WavesMeasuring Waves

High Frequency Lower Frequency

Measuring WavesMeasuring WavesHow Big is a Wave?

Amplitude is the deviation from equilibrium, and the amount of Energy in the wave This This corresponds to intensity or loudness for a to intensity or loudness for a

sound wave… and intensity for light wavessound wave… and intensity for light waves Wavelength is the length of 1 cycle (meters)

+ Amplitude

- Amplitude

Wavelength

Wavelength

Wavelength

Wave MeasurementsWave Measurements Wavelength(Wavelength()) - the distance between two - the distance between two

successive corresponding points in a wave, successive corresponding points in a wave, or one cycle.or one cycle.– Can use meters or prefix with meters

Wave Velocity(Wave Velocity(vv)) - is the speed the waves - is the speed the waves moves at, or the speed a wave crest moves at, or the speed a wave crest passes by a particular point in spacepasses by a particular point in space– It is typically measured in meters/secondIt is typically measured in meters/second– Ex: Speed of sound, speed of lightEx: Speed of sound, speed of light

Wave simulationsWave simulations

Ripple Tank (water waves) http://www.falstad.com/ripple/

http://www.walter-fendt.de/ph14e/pendulum.htm

THE wave equation:THE wave equation:

Wave Speed = Wavelength Wave Speed = Wavelength Frequency Frequency

vv = = ff

or or vv = = /T/T

Wave SpeedWave Speed

Example ProblemsExample Problems

Suppose the water waves crash onto a Suppose the water waves crash onto a shore every five seconds. The crest to shore every five seconds. The crest to crest measurement is 10 meters.crest measurement is 10 meters.

What is the wave frequency?What is the wave frequency? Frequency = 1/5 Hertz Frequency = 1/5 Hertz

What is the wavelength?What is the wavelength? Wavelength = 10 mWavelength = 10 m

What is the wave speed?What is the wave speed? V = V = f = 1/5 Hertz f = 1/5 Hertz 10 meters = 2 m/s 10 meters = 2 m/s

Examples Examples

A piano emits frequencies that range A piano emits frequencies that range from a low of about 28 Hz to a high of from a low of about 28 Hz to a high of about 4200 Hz. Find the range of about 4200 Hz. Find the range of wavelengths in air attained by this wavelengths in air attained by this instrument when the speed of sound in instrument when the speed of sound in air is 340 m/s.air is 340 m/s.

The red light emitted by a He-Ne laser The red light emitted by a He-Ne laser has a wavelength of 633 nm in air and has a wavelength of 633 nm in air and travels at 3.00 x 10travels at 3.00 x 1088 m/s. Find the m/s. Find the frequency of the laser light.frequency of the laser light.

Mechanical Waves InformationMechanical Waves Information Waves transport Waves transport EnergyEnergy, and Energy in a , and Energy in a

wave is proportional to wave is proportional to AmplitudeAmplitude SquaredSquared

For most waves, as they travel the waves For most waves, as they travel the waves “spread out” more, which also spreads out “spread out” more, which also spreads out the the EnergyEnergy

EnergyEnergy will be lost due to friction in the will be lost due to friction in the medium, so as the wave travels farther it medium, so as the wave travels farther it loses more Energy (sounds get quieter) loses more Energy (sounds get quieter)

MediumMedium affects wave speed, length and affects wave speed, length and amplitude, amplitude, but but not frequencynot frequency

Frequency doesn’t depend on the medium, Frequency doesn’t depend on the medium, only on the source of vibrationsonly on the source of vibrations

1.1. Waves can move from one medium Waves can move from one medium to another (media is plural) at a to another (media is plural) at a boundary—transmission (refraction) boundary—transmission (refraction) and reflectionand reflection

2.2. Two or more waves can be in the Two or more waves can be in the same space at the same time and same space at the same time and interact—they get addedinteract—they get added—”superposition”—”superposition”

3.3. Waves can spread out and bend—Waves can spread out and bend—diffraction!diffraction!

1. Changing Medium1. Changing Medium

What happens when a wave is What happens when a wave is moving and “runs into” a different moving and “runs into” a different medium? medium?

Waves can:Waves can:– be transmitted (continues on in new be transmitted (continues on in new

medium) also known as Refractionmedium) also known as Refraction– change direction, or reflect (Reflection)change direction, or reflect (Reflection)– or some of bothor some of both

1. Boundary Behavior1. Boundary Behavior

A Boundary is where the A Boundary is where the Medium changes from Medium changes from one to anotherone to another

If the wave is transmitted If the wave is transmitted (refraction), the “new” (refraction), the “new” medium will result in a medium will result in a different wave speed and different wave speed and wavelength (wavelength (ff doesn’t doesn’t change)change)

Refraction

1.1. Examples at BoundaryExamples at Boundary

ReflectionIf the wave is If the wave is reflected, it can be reflected, it can be identical or inverted identical or inverted depending on the depending on the density change density change between mediumsbetween mediums..\old\Reflection of Waves from Boundaries.htm

..\downloads\string-wave.swf

http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html

Ripple Tank (water waves) http://www.falstad.com/ripple/Inverted

Free EndFixed End

1. More Examples: Change in 1. More Examples: Change in MediumMedium

Reflection and Refraction/Transmission

Inverted !

Light density medium to more dense

More dense medium to less

2. Behavior of waves together2. Behavior of waves together

Waves in the same place at the same Waves in the same place at the same time add together to make a new time add together to make a new “total” wave, but the original waves “total” wave, but the original waves still exist (waves move through each still exist (waves move through each other)other)

Adding waves is called Adding waves is called SuperpositionSuperposition– At anyAt any position, add the amplitudes of the position, add the amplitudes of the

waves to get total, new amplitudewaves to get total, new amplitude http://www.udel.edu/idsardi/sinewave/sinewave.htmlhttp://

www.acs.psu.edu/drussell/Demos/superposition/superposition.html

2. Superposition2. Superposition example 1example 1

A

B

+

=

total

Amplitude =1

Amplitude =2

Amplitude =3

A

2. Superposition2. Superposition example 2example 2

B

+

=

total

Amplitude =1

Amplitude =1

Amplitude =0

2. Superposition of waves2. Superposition of waves

When waves come together and are When waves come together and are added by superposition, the result is added by superposition, the result is interferenceinterference– DestructiveDestructive interference is when interference is when

opposite amplitudes (+, -) add for a opposite amplitudes (+, -) add for a reducedreduced amplitude. amplitude.

– ConstructiveConstructive interference is when interference is when amplitudes of the same sign add for an amplitudes of the same sign add for an increasedincreased amplitude amplitude

2. Interference2. Interference Special terms for interferenceSpecial terms for interference

– When a peak is aligned to a trough and the When a peak is aligned to a trough and the waves add to zero, its waves add to zero, its Total Destructive Total Destructive InterferenceInterference

– When the peaks match and add to a maximum, When the peaks match and add to a maximum, its its Total Constructive InterferenceTotal Constructive Interference

– Examples:Examples:Sound canceling headphones http://thespoon7.tripod.com/wave.htm

http://www.amazon.com/Sennheiser-Active-Noise-Canceling-Headphones/http://www.amazon.com/Sennheiser-Active-Noise-Canceling-Headphones/

dp/tech-data/B000089GN2/ref=de_a_smtddp/tech-data/B000089GN2/ref=de_a_smtd Ripple Tank (water waves) http://www.falstad.com/ripple/

..\downloads\string-wave.swf

http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.htmlhttp://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html

2. Standing Still 2. Standing Still

Interfering waves with the Interfering waves with the same same frequencyfrequency and wavelength but travel in and wavelength but travel in the opposite direction, the result appears the opposite direction, the result appears to stand still, which is called a to stand still, which is called a Standing Standing WaveWave

This often happens with a reflectionThis often happens with a reflection http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.htmlhttp://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html

Violin Violin http://id.mind.net/~zona/mstm/physics/waves/standingWaves/stanhttp://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.htmldingWaves1/StandingWaves1.html

2. Standing Waves2. Standing Waves Frequency and Length are the keys to standing Frequency and Length are the keys to standing

waves—they must be the same for both waveswaves—they must be the same for both waves Vocabulary for standing waves:Vocabulary for standing waves:

– Node is point(s) of zero amplitude, looks like not movingNode is point(s) of zero amplitude, looks like not moving– Anti-node is the crest, peak, or maximum amplitude Anti-node is the crest, peak, or maximum amplitude

point(s)point(s)

Node 2Node 1 Node 3

Anti-Node 1 Anti-Node 2

Anti-Node Anti-Node

We’ll see this more in the next chapter on Sound

Example: Standing Wave with Length = 1

3. Bending waves3. Bending waves

Waves can “bend” around obstacles, Waves can “bend” around obstacles, or spread out, which is called or spread out, which is called DiffractionDiffraction– The longer the wavelength of the wave The longer the wavelength of the wave

the larger the amount of diffractionthe larger the amount of diffraction

Diffraction angle

3. Diffraction 3. Diffraction

Diffraction spreads out EnergyDiffraction spreads out Energy Math can be used to determine the Math can be used to determine the

amount of diffractionamount of diffraction Diffraction is used with ocean waves Diffraction is used with ocean waves

and the study of interference and and the study of interference and lightlight

3. Diffraction through Gaps3. Diffraction through GapsWavelength similar to the size of one gap:

Large amount of DIFFRACTION

Wavelength NOT similar to the size of one gap:

Small amount of DIFFRACTION

Two gaps or slits make a pattern of interferenceRipple Tank (water waves) http://www.falstad.com/ripple/

Behavior SummaryBehavior Summary

1.1. Medium/Boundary changeMedium/Boundary change– Refraction: changes v and Refraction: changes v and , not f, not f– Reflection: Reflection:

From free end or less dense medium, no change in waveFrom free end or less dense medium, no change in wave From fixed end or more dense medium, invertedFrom fixed end or more dense medium, inverted

2.2. Superposition: adding wavesSuperposition: adding waves– Destructive interference: reduced ADestructive interference: reduced A

Total destructive = zero ATotal destructive = zero A– Constructive interference: increased AConstructive interference: increased A

Total Constructive = maximum A (crests align)Total Constructive = maximum A (crests align)

3.3. Diffraction: bending wavesDiffraction: bending waves– Depends on Depends on compared to object size compared to object size

Examples Examples A wave of amplitude 0.30 m interferes with a A wave of amplitude 0.30 m interferes with a

second wave of amplitude 0.20 m. What is the second wave of amplitude 0.20 m. What is the largest resultant displacement that may occur?largest resultant displacement that may occur?

A string is rigidly attached to a post at one end. A string is rigidly attached to a post at one end. Several pulses of amplitude 0.15 m sent down Several pulses of amplitude 0.15 m sent down the string are reflected at the post and travel the string are reflected at the post and travel back down the string without a loss of back down the string without a loss of amplitude. What is the amplitude at a point on amplitude. What is the amplitude at a point on the string where the initial maximum the string where the initial maximum displacement points of the two pulses cross? displacement points of the two pulses cross? What type of interference is this?What type of interference is this?

How would your answer to the above problem How would your answer to the above problem change if the same pulses were sent down a change if the same pulses were sent down a string whose end is free? What type of string whose end is free? What type of interference is this?interference is this?

End Ch 12End Ch 12

Doppler Effect, Bow & Shock Waves

Doppler• Christian Doppler figured it out in 1840’s• Trains had become popular, and the whistle

sounded different as they moved• Effect: frequency a listener hears is different due

to motion at a high enough speed• Doppler Effect works for all waves• Doppler effect involves motion of either the wave

source or the wave observer, or both

Doppler Effect and Sound

• When the “cause of the sound” moves or the listener moves (or both), the sound “seems” different to the listener

• How does it sound when a car is coming toward you with the horn blowing?

• Going away?

Car horn ExamplesCar moving, you aren’t Both moving together

Both moving opposite

What’s happening?• When there is relative motion between the

cause and listener, the frequency and wavelength are different to the listener

Stationary “cause” or source

Stationary listener

Wavelength and frequency

Moving “cause”

Stationary listener

Bigger smaller (to listener)

Smaller f bigger f

Sound waves

Frequency and Wavelength Changes

• When approaching, the “apparent” frequency increases and the wavelength decreases (to the listener)

• When separating, the “apparent” frequency decreases and the wavelength increases

• See it! • Doppler found a formula to calculate the change in

frequency• Uses the speed of sound, vs in calculation (about

340 m/s)

http://www.lon-capa.org/~mmp/applist/doppler/d.htm

More uses of Doppler Effect

• Radar: airplanes, speed traps, weather, …

• Bats hunting

• Dolphins, etc hunting

• Universe—expanding or contracting (light has blue shift—approach, or red shift—depart)

• Medical testing like Ultrasound

• Sound barrier ??

Sound “barrier”

• Doppler equation predicts a problem if the speed of “cause” (vc) = the speed of sound in air (vs )

As vc gets close to vs, the sound pressure waves begin to add up—the “barrier”

http://www.lon-capa.org/~mmp/applist/doppler/d.htm

soundlistener cause

sound cause

vf f v v

Breaking the Sound Barrier• When vc > vs, “break” the sound barrier• The sound waves add up to make one large

amplitude wave—a high pressure wave• Hear a sonic “boom” when a jet breaks the sound

barrier (goes faster than vs)• Hear the “crack” of a whip when the tip breaks the

sound barrier• Resulting wave is a shock wave, or bow wave• Ratio of vc/vs is called Mach number• Mach number >1 is “supersonic” or faster than

sound

What’s happening?

• In England during WWII, explosions started occurring before any bombs were heard. What was going on?

• Germans had developed rockets which traveled faster than sound. Sound arrives after the bomb!

Why the Jet is gone

listener

Movie

\\f14.mpeg

The wave is shaped like a sideways “V”, which is called a BOW wave

The faster the jet goes, the more pointy (smaller angle)

F-14 Tomcat Sonic Boom.gvi

The shock

• The “barrier”, or start of the bow wave when going close to or greater than the speed of sound is called the shock wave

• It is made of a high pressure wave followed by a low pressure wave, which sometimes causes the condensation