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Waves and Wave Waves and Wave Motion Motion in elastic media in elastic media

# Waves and Wave Motion in elastic media Simple Harmonic Motion Any object moving under the influence of Hooke’s Law type forces exhibits a particular

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Waves and Wave Waves and Wave MotionMotion

in elastic mediain elastic media

Simple Harmonic MotionSimple Harmonic Motion

Any object moving under the Any object moving under the influence of Hooke’s Law type forces influence of Hooke’s Law type forces exhibits a particular type of motion exhibits a particular type of motion about its equilibrium position.about its equilibrium position.

This motion is called “sinusoidal” This motion is called “sinusoidal” and has certain specific and has certain specific characteristics.characteristics.

NomenclatureNomenclature

A - Amplitude - the maximum A - Amplitude - the maximum displacement from equilibriumdisplacement from equilibrium

- Wavelength - Distance between - Wavelength - Distance between any two successive identical points.any two successive identical points.

f - Frequency - Number of cycles or f - Frequency - Number of cycles or oscillations per second. ( = 1/T)oscillations per second. ( = 1/T)

T - Period - Time to complete one full T - Period - Time to complete one full oscillation or cycle. (=1/f)oscillation or cycle. (=1/f)

Plot of Amplitude vs. Plot of Amplitude vs. PositionPosition

Mechanical WavesMechanical Waves

Waves in an elastic (or deformable) Waves in an elastic (or deformable) mediamedia

Originate as a displacement of some Originate as a displacement of some portion of the media from its portion of the media from its equilibrium conditionequilibrium condition

This portion then begins to oscillate This portion then begins to oscillate about its equilibrium position.about its equilibrium position.

The elastic properties of the media The elastic properties of the media transmit this disturbance from one transmit this disturbance from one layer to the next.layer to the next.

Types of mechanical Types of mechanical waveswaves

Transverse - Transverse - oscillation oscillation to to direction of direction of propagationpropagation

Longitudinal - Longitudinal - oscillation oscillation to to direction of direction of propagationpropagation

One, two, or three One, two, or three dimensionaldimensional

PulsePulse PeriodicPeriodic HarmonicHarmonic

Any combination of Any combination of these!these!

One, Two, & Three One, Two, & Three DimensionsDimensions

Two DimensionTwo Dimension Ripples on waterRipples on water

Three DimensionThree Dimension Sound / Earth QuakeSound / Earth Quake

Pulse wavePulse wave

A single oscillation sent through a A single oscillation sent through a media due to a non-repeating event.media due to a non-repeating event.

e.g. Sound from an explosion or crash. e.g. Sound from an explosion or crash. Splash from a stone dropped in Splash from a stone dropped in water.water.

Periodic WavePeriodic Wave

Wave produced by an oscillation of the Wave produced by an oscillation of the media which repeats with a regular media which repeats with a regular period.period.

e.g. Sound from a steady drum beate.g. Sound from a steady drum beat

Harmonic WaveHarmonic Wave

Created by a simple harmonic Created by a simple harmonic oscillations of the media.oscillations of the media.

e.g. Musice.g. Music

v = f

Different types of wavesDifferent types of waves

Reflection of WavesReflection of Waves

When a wave hits a “hard” surface it When a wave hits a “hard” surface it reflects back “inverted”. ( Newton’s reflects back “inverted”. ( Newton’s 3rd law)3rd law)

When a wave hits a “soft” surface it When a wave hits a “soft” surface it reflects back in a “non-inverted” reflects back in a “non-inverted” manner.manner.

““Hard” surfaces have lower velocity for Hard” surfaces have lower velocity for that type of wave.that type of wave.

Wave Wave reflectionreflection

Reflection of a wave Reflection of a wave from a free endfrom a free end

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Reflection of a wave Reflection of a wave from a fixed endfrom a fixed end

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

TransmissionTransmission

When a wave hits a surface or When a wave hits a surface or interface not all of the energy is interface not all of the energy is reflected. Some is transmitted into reflected. Some is transmitted into the next medium. the next medium.

The transmitted wave is “non-The transmitted wave is “non-inverted” and the “harder” the inverted” and the “harder” the surface the less is transmitted.surface the less is transmitted.

Travel between media of Travel between media of different densitydifferent density

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Longitudinal WaveLongitudinal Wave

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

SoundSound

Pressure wave - longitudinalPressure wave - longitudinal

Frequency = pitchFrequency = pitch

v = 334 m/s in air at room temperaturev = 334 m/s in air at room temperature

Velocity is dependent upon the Velocity is dependent upon the materialmaterial

Sound is a pressure Sound is a pressure wavewave

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Tuning fork Tuning fork creating a sound wavecreating a sound wave

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Guitar String Guitar String creating a sound wavecreating a sound wave

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Sound wave and Sound wave and EardrumEardrum

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

EarEar

Reverberation vs EchoReverberation vs Echo

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Standing WavesStanding Waves Created by the interference of two waves of the Created by the interference of two waves of the

same frequency traveling in opposite directions.same frequency traveling in opposite directions. In a string or tube of limited length the In a string or tube of limited length the

reflections off the ends will create two traveling reflections off the ends will create two traveling waves moving in opposite directions.waves moving in opposite directions.

Only certain wavelengths (Only certain wavelengths () will create ) will create standing waves in a string or tube of limited standing waves in a string or tube of limited length .length .

These wavelengths (These wavelengths () correspond to frequencies ) correspond to frequencies (f = v/ (f = v/ ) called natural or resonant frequencies) called natural or resonant frequencies

Traveling WaveTraveling Wave

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Standing WaveStanding Wave

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Fundamental Fundamental FrequencyFrequency

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Second HarmonicSecond Harmonic

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Third HarmonicThird Harmonic

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Fourth HarmonicFourth Harmonic

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Fifth HarmonicFifth Harmonic

Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

HarmonicsHarmonics

Speeds of SoundSpeeds of Soundat T = 20 Cat T = 20 C

Air Air HeliumHelium HydrogenHydrogen WaterWater Sea waterSea water Iron/SteelIron/Steel GlassGlass AluminumAluminum

343 m/s343 m/s 1005 m/s1005 m/s 1300 m/s1300 m/s 1440 m/s1440 m/s 1560 m/s1560 m/s ≈≈5000 m/s5000 m/s ≈ ≈ 4500 m/s4500 m/s ≈ ≈ 5100 m/s5100 m/s

Intensity of SoundIntensity of Sound

Unit is the “Bel”. Named after Unit is the “Bel”. Named after Alexander Graham Bell Alexander Graham Bell

More commonly used is the decibel More commonly used is the decibel (dB)(dB)

= 0.1 Bel= 0.1 Bel (in dB) = 10 log(I/I(in dB) = 10 log(I/I00)) I is the intensity ( I is the intensity ( AA22)) II0 0 = 1.0 x 10 = 1.0 x 10 -10-10 W/m W/m22 the “threshold of the “threshold of

hearing”hearing”

Some Intensities (in dB)Some Intensities (in dB)

Jet plane at 30 mJet plane at 30 m Threshold of painThreshold of pain Indoor rock concertIndoor rock concert Auto interior Auto interior

(50mph)(50mph) Street trafficStreet traffic Conversation (50cm)Conversation (50cm) WhisperWhisper Rustle of leavesRustle of leaves

140140 120120 120120 7575 7070 6565 1x101x10-10-10

1x101x10-11-11

InterferenceInterference

When two (or more) waves pass When two (or more) waves pass through each other they “interfere”. through each other they “interfere”.

Simultaneous displacements of the Simultaneous displacements of the medium will add together. -Principle medium will add together. -Principle of Superposition-of Superposition-

Constructive and/or Destructive Constructive and/or Destructive interference.interference.

Interference - two wavesInterference - two waves 11 = 2 m, = 2 m, 22 = 1.5 m = 1.5 m

Interference - two Interference - two waveswaves

11 = 2 m, = 2 m, 22 = 1.5 m = 1.5 m

Interference - two Interference - two waveswaves

11 = 2 m, = 2 m, 22 = 1.5 m = 1.5 m

Beats and Beat Beats and Beat FrequencyFrequency

Caused by two traveling wave of different Caused by two traveling wave of different frequencies, ffrequencies, f11 and f and f22..

Carrier frequency = Average (fCarrier frequency = Average (f11 and f and f22))

Beat frequency = | fBeat frequency = | f11 - f - f22 | |

DemoDemo

Doppler EffectDoppler Effect Apparent change in frequency (pitch) of Apparent change in frequency (pitch) of

a sound from a moving source.a sound from a moving source. Source moving toward observer:Source moving toward observer:

f’ = f / (1-vf’ = f / (1-vss/v)/v)

Source moving away from observer:Source moving away from observer: f’ = f / (1+vf’ = f / (1+vss/v)/v)

f = frequency of source, f = frequency of source,

f’=frequency heard by observer, f’=frequency heard by observer,

vvss = velocity of source, v = velocity of = velocity of source, v = velocity of soundsound