VIBRATIONS AND WAVES

Hooke's LawOne of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law.

                                                           

The idea behind Hooke's The idea behind Hooke's Law:Law:

Any object that is initially Any object that is initially displaced slightly from a displaced slightly from a stable equilibrium point stable equilibrium point will oscillate about its will oscillate about its equilibrium position. It equilibrium position. It will, in general, experience will, in general, experience a restoring force that a restoring force that depends on the depends on the displacement x from displacement x from equilibrium. equilibrium.

Harmonic Motion Oscillator

http://www.kettering.edu/~drussell/Demos/SHO/mass.html

Energy Conversion

http://www.kettering.edu/~drussell/Demos/SHO/mass.html

Harmonic Motion

Simple Harmonic Motion (SHM)

• Motion that occurs when the restoring force acting on an object is proportional to the object’s displacement from its resting position

• Harmonic part if SHM means the motion repeats itself

• Objects at the end of springs move in SHM when they are displaced from their rest position and bounce up and down on the spring, or oscillate.

Sine Curve• To and fro vibration motion of swinging

pendulum in small arc is called SHM

Sine Curve

Sine Curve

• Pictorial representation of a wave

Simple Tire Swing

Harmonic motion in a Bottle…making a bottle “sing”

http://arts.ucsc.edu/ems/music/tech_background/TE-12/teces_12.htmlhttp://arts.ucsc.edu/ems/music/tech_background/TE-12/teces_12.html

Flow of air across bottle top, pulls air out of bottle reducing mass of Flow of air across bottle top, pulls air out of bottle reducing mass of air in bottle which reduces pressure within bottle. The low pressure air in bottle which reduces pressure within bottle. The low pressure pulls air back into bottle causing oscillations. pulls air back into bottle causing oscillations.

VIBRATION OF A PENDULUM

• What does the period (T) depend upon?

• Length of the pendulum (l).• Acceleration due to gravity (g).• Period does not depend upon

the bob mass.

Pendulum

When oscillations are

small, the motion is called

simple harmonic motion (shm)

and can be described by a

simple sine curve.

T l g2

Vibrations and Waves

• Waves transmit energy and information.

–Sound and Light are examples of waves.

There are two ways to transmit information/energy

in our universe:

Particle Motion

and

Wave Motion Wave Simulation: http://phet.colorado.edu/simulations/sims.php?sim=Wave_Interference

Mechanical Waves

• Requires a medium• Ex. Water waves, sound• Two different material objects cannot be in the

same place at the same time…however mechanical waves displace matter to transfer energy and thus can be in the same place at the same time.

Electromagnetic Waves

•Do not require a medium (can move through empty space, a vacuum)

•Ex. Radio waves, light waves, microwaves

WAVES

•Transfer energy not matter from one place to another

•Disturbance that moves through space or through a medium (material)

Wave Pulse

• Single Disturbance

http://www.colorado.edu/physics/phys4830/phys4830_fa01/lab/n0911.htm

Wave Train (Continuous Wave)

• Series of pulses at regular intervals

Sine Curve

AmplitudeA

Wavelength

Picture of a Transverse Wavel

WavelengthWavelength

ll

A - AmplitudeA - Amplitude

AA

CrestCrest

TroughTrough

2. WAVE DESCRIPTION

TRANSVERSE WAVESParticles vibrate perpendicular to

the wave motionTransverse waves can be polarized

string musical instruments

ripples on water

electromagnetic waves e.g. Light waves, x-rays, radio waves

Picture of a Transverse Wave

CrestCrest

TroughTrough

WavelengthWavelength

AA

A - AmplitudeA - Amplitude

Transverse Wave

http://dev.physicslab.org/Document.aspx?http://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_IntroductionWaves.xmldoctype=3&filename=WavesSound_IntroductionWaves.xml

Transverse Wave

• Particles vibrate perpendicular to the direction the wave travels

• Ex. Vibrating string of musical instrument, radio waves

Longitudinal Wave

• Particles vibrate parallel to the direction of wave travel

• Ex. Sound

Longitudinal Wave

• Compression- (similar to crests) – particles close together (high density)

• Rarefactions (similar to troughs)- particles spread out or rarefied (low density)

LONGITUDINAL WAVESParticles vibrate parallel to the motion

of the waves

Ex: Sound Waves

http://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_IntroductionWaves.xmlhttp://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_IntroductionWaves.xml

Rarefactions are regions of low density.

Compressions (condensations) are regions of high density.

is the length of one rarefaction plus one compression

Animated comparison of transverse & longitudinal waves: http://members.aol.com/nicholashl/waves/movingwaves.html

Time required to make one vibration.

• Time required to generate one wave

• Time required for the wave to travel one

wavelength.

Period (T)

Frequency (f)

The number of vibrations per unit of

time made by the vibrating source.

Units -cycles/sec or hertz (Hz)

Examples of Frequency

• What is the frequency of the second hand of a clock?

Frequency = 1cycle/60 secFrequency = 1cycle/60 sec Period = 60 secPeriod = 60 sec

What is the frequency of US Presidential What is the frequency of US Presidential elections?elections?

Frequency = 1 election/4 yrsFrequency = 1 election/4 yrs Period = 4 yrsPeriod = 4 yrs

FrequencyPeriod

1

WAVE MOTION• Energy is transported by particles or waves. A wave

is a disturbance transmitted through a medium.

• Exception: light does not require a medium.

• A disturbance moves through the medium. Causing

elements of the medium vibrate.

– Example: ripples on water

WAVE MOTION

• Medium is disturbed, energy is imparted to it

WAVE SPEED

The average speed is defined as

time

distancev

For a wave, if the distance traveled is a

wavelength (), then the time to travel

this distance is the period (T). Thus

Tv

fv or

is true for all waves.

Note:

v is dictated by the medium

f is dictated by the source.

fv

Surface Water Waves

http://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_IntroductionWaves.xmlhttp://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_IntroductionWaves.xml

SuperpositionTwo or more waves overlapping in

some way

The overlapping causes interference

Animation courtesy of Dr. Dan Russell, Kettering University

Wave Interactions• Because waves are not matter but rather

displacement of matter, two waves can occupy the same space at the same time

• Combination of two overlapping waves is called superposition (causes interference)

Superposition Principle

• Displacement of a medium (material) caused by 2 or more waves is the sum of the displacements of the individual waves at each point

• Holds true for all types of waves

Interference

• Interference is a characteristic of all waves.

• Result of superposition of 2 or more waves

• Constructive- (crest meets crest or trough meets trough) amplitudes added

• Destructive- (crest meets trough) amplitudes subtract

Interference Pattern

The pattern formed by superposition of different sets of waves that produce mutual reinforcement in some places and cancellation in others.

Constructive Interference

Reinforcement

†

Maximum

In phase displacement

ConstructiveConstructive

Constructive Interference

• When the crest of one wave overlaps the crest of another

Destructive- Crests and Troughs overlap

CANCELLATIONCANCELLATION

Zero Zero DisplacementDisplacement

Destructive: crests & troughs overlap

Interference –Destructive

• When the crest of one wave overlaps the trough of another

Destructive Interference

Standing Waves• When two sets of waves of equal amplitude

and wavelength pass through each other in opposite directions, it is possible to create an interference pattern that looks like a wave that is “standing still.”

Standing Wave

V

V

V

V

Standing Wave

Incident Wave

Reflected Wave

Standing Waves

Standing Waves

• Result of interference and reflection

• When 2 sets of waves of equal amplitude and λ pass through each other in opposite directions, the waves are steadily in and out of phase with each other.

• Consists of nodes (0 amplitude) and antinodes (max. amplitude)

• Wave looks as if it is standing still

Standing Waves

• Nodes- point in standing wave that undergoes complete deconstructive interference and is therefore stationary (no vibration)

• Antinode- a point in a standing wave (1/2way between 2 nodes) at which the largest amplitude occurs (maximum vibration)

Waves on a String: http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_String

Standing Waves

Standing Waves

• Since the two identical waves travel in opposite directions, the net energy flow is zero; the energy is “standing” in the loops

Standing Wave Harmonics

There is no vibration at a node.• There is maximum vibration at an

antinode.

ll

Standing Waves on a StringA string that's held very A string that's held very tightly at both ends can only tightly at both ends can only vibrate at very particular vibrate at very particular

wavelengths.wavelengths.

It can vibrate in halves, It can vibrate in halves, with a node at the middle of with a node at the middle of the string as well as each the string as well as each end, or in thirds, fourths, end, or in thirds, fourths, and so on. and so on.

Any wavelength that doesn't Any wavelength that doesn't have a node at each end of have a node at each end of the string, can't make a the string, can't make a standing wave on the string.standing wave on the string.

Sound

• Travels as a longitudinal wave

• sounds audible to humans range from 20 - 20,000 Hz

• as f increases, pitch rises

• Pitch refers to how f is observed

• speed of sound depends on medium and temperature

• speed of sound in air @ 0 ºC is 331 m/s…it increases by 0.59 m/s for every 1 ºC increase

Sound Simulation: http://phet.colorado.edu/simulations/sims.php?sim=Sound

ORIGIN OF SOUND

• Sound Waves are produced by a vibrating object

• Are longitudinal, mechanical waves

A noise is a jumble of sound waves.

A tone is a very regular set of waves, all the same size and same distance apart distance

apart.

Approximate Speeds of Sound in Various Mediums

• 0º C air = 331 m/s

• 25º C Saltwater = 1533 m/s

• 25º C Freshwater = 1493 m/s

• Steel = 5790 m/s

• Iron = 5130 m/s

• Rubber = 1600 m/s

• Diamond = 12,000 m/s

SOUND FREQUENCY• Determines pitch

– frequency and pitch are directly related

• 20-20,000 Hz are audible to avg. person

• < 20 Hz are infrasonic

• > 20,000 Hz are ultrasonic

ULTRASONIC WAVES

• A) sonar showing Hamilton schooner, sunk in Lake Ontario during War of 1812 lies under 91 m of water

• B) Bats (as well as dolphins) use echolocation

• C) ultrasonic waves travel through tissue get reflected by tumors or amniotic fluid

Applications of Sound

SONAR:  How deep is the water; where are the fish?

Ultrasound is sound at a frequency whichis outside of the range of human hearing

What is the Mosquito Ring tone?

• High pitched ring tone that most adults cannot hear

• “teen buzz”

• originally designed as a repellant

• Mosquito ringtone is 17 kHz (that’s 17,000 Hz for those of you who forgot what a kilo is)

• The highest note on a piano is only 4 kHz

Hearing Range by Age

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

18-21 yrs 30-39 yrs 40-49yrs 50-59 yrs

Hearing

Range

kHz

2-16 kHz 2-14

kHz 2-12 kHz 2-11

kHz

Frequencies

• http://www.opti.dyndns.org/~dng/SOUND/ReferenceSound/refsound.html

• http://www.freemosquitoringtone.org/

Ringtones

• Beats• When two sound waves of different frequency approach your ear,

the alternating constructive and destructive interference causes the sound to be alternatively soft and loud - a phenomenon which is called "beating" or producing beats. The beat frequency is equal to the absolute value of the difference in frequency of the two waves. Arising from simple interference, the applications of beats are extremely far ranging.

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/beat.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/sound/beat.html

Pulsating Loudness due to Difference in Frequencies

• beat frequency or the different between the two frequencies beat frequency = |f2 - f1|

• beat pitch at which those beats are heard is the average of the two frequencies beat pitch = ½(f2+f1)

http://library.thinkquest.org/C005705/English/sound/sound3.htm

Examples of Beat Frequencies: http://www.school-for-champions.com/science/sound_beat.htm

Practice

• Suppose you sound two tuning forks simultaneously: one fork has a frequency of 256 hz and the other has a frequency of 260 hz.

a) How many beats would be heard each second?

b) What is the pitch of these beats?

Try to determine the unknown frequencies: http://www.hazelwood.k12.mo.us/~grichert/explore/dswmedia/tonebeat.htm

DOPPLER EFFECT• Refers to the change in frequency when

there is relative motion between an observer of waves and the source of the waves

Source moving with vsource < vsound ( Mach 0.7 )

Source moving with vsource = vsound ( Mach 1 - breaking the sound barrier )

DOG IN WATER

Dog treads water (no velocity)

Dog swims to the right

Dog swims at same speed as water waves he is creating

Doppler Effect

• A change in frequency due to a change in motion of either the vibration source or the observer

Red & Blue Shifts

• Light from moving objects will appear to have different wavelengths depending on the relative motion of the source and the observer.

Explain the following Bumper Sticker viewed on the back of a

passing car:

If this sticker appears blue you are driving too

fast!

http://www.artunderthesun.com/page3.htm

• Objects in motion compress the light waves in front of them making them appear more blue (blue shift), the light waves behind are stretched out and appear more red (red shift).

http://cse.ssl.berkeley.edu/bmendez/ay10/2002/notes/lec8.html

Reflection

• Turning back of a wave at the boundary of a new medium

• ex. Light off a mirror; sound echo

Law of Reflection

• States that the angle of incidence equals the angle of reflection

• Өi = Өr

Refraction• Bending of a wave path as it enters a new

medium at an angle

• Caused by difference in speed of wave in new medium

• Fast to slow- bends towards the normal

• Slow to fast- bends away from normal

More on Refraction

Drawing the Refracted Wave

Refraction • Bending of waves due to change in speed as it passes through a new medium

• During World War I drivers returning to the rear from the front encountered alternating regions of intense artillery sound/ no sound caused by refraction,then reflection, then refraction, etc.

• On some evenings the whistle of trains miles away can be heard; however the train whistle is not heard during the day.

Sound speed is greater in warm air.

Diffraction

• Spreading of waves around edges or through an opening of a boundary

• Is greatest when size of opening is smaller than wavelength of wave passing through

Diffraction

A wave spreads out (noticeable diffraction) when the size of the slit is comparable to or smaller than the wavelength

A wave goes straight when the size of the slit is much larger than the wavelength

Interference & Diffraction

Resonance

• Condition that exists when the frequency of an applied force is the same as the natural frequency of vibration of an object or system

What is the Natural Frequency?

• The frequency at which an elastic object naturally vibrates when hit, struck, or somehow disturbed.

• At this frequency, a forced vibration occurs with minimal energy.

Natural Frequency...

• The natural frequency of an object depends on its ...– elasticity– size– shape

Resonance

• If a singer produces a note at the same frequency as the resonant frequency of say a wine glass, they can set up vibrations in the glass itself. If they sing loud enough, they will produce larger vibrations and can shatter the glass!

http://www.scienceyear.com/outthere/index.html?page=/outthere/sound_check/http://www.scienceyear.com/outthere/index.html?page=/outthere/sound_check/sound_music.htmlsound_music.html

Natural Frequency

• Examples– Mass on a Spring– Ringing Small and Large Bells– Xylophone – Rubbing the rim of a Glass

What is Forced Vibrations?

• The tendency of one object to force another adjoining object into vibrational motion is referred to as a forced vibration

• During forced vibration sound is intensified because a larger surface area is available to vibrate air molecules.

• http://mmem.spschools.org/grade3science/3.sound/musicvsnoise.html

http://upload.wikimedia.org/wikibooks/en/e/e3/TuningFork.gif

Speed of Sound in Various Mediums

Medium Velocity of sound at 0C(m/s)

Air 331

Oxygen 460

Alcohol 1213

Water 1435

Copper 3560

Iron 5130

Speed of Sound at Various Temperatures

Temperature (C) Velocity of sound (m/s) 0 331

20 344

100 386

500 553

1000 700

Forced Vibrations...

Few examples:– Holding a tuning fork against a table– Vibration of factory floor from heavy

machinery

What is Resonance?

• This occurs when the frequency of forced vibrations on an object matches the objects natural frequency.

• Results in an increase in amplitude that can destroy the object that is vibrating

Examples of Resonance

• swinging your legs in a swing

• breaking a glass using sound

• a tuning fork exciting a guitar string

• a truck driving on a rough road

Video of glass shattering: http://www.wellesley.edu/Physics/Rberg/glassdemo.html

SINGING IN THE SHOWER

• An experiment you can do is to stand in the shower (they reflect sound well) and start singing while changing the pitch slowly. At certain pitches the sound will suddenly amplify, because the sound waves fit an even number of times between the walls.

Tacoma Narrows Bridge

• Why did the Tacoma Narrows Bridge fall?

Video of bridge: Video of bridge: http://www.archive.org/stream/SF121/SF121_256kb.mp4

Tacoma Narrows Bridge• Collapsed Nov 7, 1940• The failure of the

bridge occurred when a never-before-seen twisting mode occurred. This is called a torsional mode whereby when the left side of the roadway went down, the right side would rise, and vice-versa, with the centerline of the road remaining still.

• The original bridge was a suspended plate girder type that caught the wind, rather than allowing it to pass through.

• The wind produced a force in resonance with the natural frequency of the bridge.

• Increasing the amplitude until the bridge collapsed.

• What one student did after he learned of the Tocoma Bridge colapse: http://video.popularmechanics.com/services/player/bcpid1214137061?bctid=1233395616

• http://www.answers.com/topic/tacoma-narrows-bridge

Current Tocoma Narrows Bridge

• The MRI machine applies an RF (radio frequency) pulse that is specific only to hydrogen.

• The pulse causes the protons in that area to absorb the energy required to make them spin.

• The RF pulse forces them to spin at a particular frequency of resonance, in a particular direction.

Amplitude vs. Frequency

Resonance is all around us…

• …from a simple mass on a spring to Nuclear Magnetic Resonance…

• …from a “singing” glass to the collapse of the Tacoma Narrows Bridge.

•

Shock WaveThe cone-shaped wave made by an object moving at supersonic speed through a fluid.

The source is moving faster than the wave speed, which is the speed of sound!

(Super-sonic speed)

SHOCK WAVES

• There are two booms, one from the front of the flying object and one from the back.

If the plane breaks the sound barrier and flies faster If the plane breaks the sound barrier and flies faster than the speed of sound, it produces a sonic boom than the speed of sound, it produces a sonic boom when it flies past. The boom is the "wake" of the when it flies past. The boom is the "wake" of the plane's sound waves.plane's sound waves.

http://hyperphysics.phy-astr.gsu.edu/Hbase/sound/soubar.html#c1

Source moving with vsource > vsound (Mach 1.4 - supersonic)

Shock Wave

The temperature in the low pressure regions must drop, leading to condensation of the water vapor present: Ideal Gas Law

Shock wave and Sonic Booms

•

Sonic Boom

The loud sound resulting from the incidence of a shock wave.

This is the result of the pile up of many wave fronts which produces a sonic boom

- slower than the speed of sound

Subsonic

• Supersonic - faster than the speed of sound

• Mach Number =speed of sound

speed of object