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Physics 213

Waves, Fluids and Thermal Physics

Summer 2007Lecturer: Mike Kagan

Today’s Discussion:Today’s Discussion:

Elasticity Elasticity –– how to stress and strain over physicshow to stress and strain over physics

Waves Waves –– Longitudinal vs. Transverse vs. Longitudinal vs. Transverse vs.

Standing waves Standing waves

Superposition of wavesSuperposition of waves

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Elasticity. Elasticity. Getting into Hook’s law.Getting into Hook’s law.

Rubber BandsRubber Bands

relaxedrelaxed

stretchedstretchedPair of forces at each point!Pair of forces at each point!

HookHook’’s Laws Law

We simply say: We simply say: band is under tensionband is under tension

spring constantspring constant(depends on geometry/shape and material)(depends on geometry/shape and material)

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Elasticity. Elasticity. Getting into Hook’s law.Getting into Hook’s law.

Rubber BandsRubber Bands

B. Series ConnectionB. Series Connection

samesame

EACH band stretches by !EACH band stretches by !

A. Parallel ConnectionA. Parallel Connection

samesame

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Elasticity. Elasticity. Getting into Hook’s law.Getting into Hook’s law.

Dependence on GeometryDependence on Geometry

andand

depends on materialdepends on material

Compare with HookCompare with Hook’’s Laws Law

oror

EE =Young modulus, =Young modulus, [[EE]=]=GPaGPa

strainstrainstressstress

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The Moduli of Elasticity

Tensile stress

A = area of surface,

∆∆∆∆V = |∆∆∆∆V|

Shear stressHydraulic

stress

In general, one would need more numbers: In general, one would need more numbers:

33 possible force directions possible force directions x 3x 3 possible plane orientationspossible plane orientations

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Example Problem:

Quantitative

4.8cm

5.3cm

1200kg

Aluminum

cylinder,

E=30GPa,

attached to

wall.

Find shear

stress on rod

and its vertical

deflection.

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Oscillations. Oscillations. Reminder.Reminder.

Oscillation = any periodic motionOscillation = any periodic motion

1) Pendulum1) Pendulum 2) Bouncing ball2) Bouncing ball

3) Uniform circular motion3) Uniform circular motion 4) LC 4) LC -- circuitcircuit

CC

LL

Period Period TT determined by parameters of the systemdetermined by parameters of the system

Oscillations occur around Oscillations occur around stablestable equilibria equilibria –– returning forcereturning force FF

Conversion of one type of energy into anotherConversion of one type of energy into another

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Oscillations. Oscillations. Mathematical Mathematical DecriptionDecription..

Harmonic process Harmonic process (for some physical quantity y)

Harmonic Motion = building block for oscillationsHarmonic Motion = building block for oscillations

‘‘any periodic process = sum of sinusoidal (harmonic) processesany periodic process = sum of sinusoidal (harmonic) processes

with multiple frequencieswith multiple frequencies’’

Fourier: Fourier:

amplitudeamplitude initial phaseinitial phase

frequencyfrequency

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Waves (Review)Waves (Review)

• Transverse and longitudinal waves

• Wavelength and frequency

• Wave speed

• Superposition and interference of waves

• Standing waves and resonance

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Waves. Waves. Definitions.Definitions.

1) water waves1) water waves

Examples: Examples:

2) sound2) sound 3) light3) light 4) 4) ““green wavegreen wave””

Wave = propagation of Wave = propagation of ““somethingsomething”” through through spacespace with with timetime

(always carries energy/information)(always carries energy/information)

Generally: Generally: ““somethingsomething”” = physical state = physical state described by physical quantitydescribed by physical quantity

phase phase

We will call this quantity We will call this quantity

1) surface elevation1) surface elevation 2) air pressure/density2) air pressure/density

3) electric field3) electric field 4) color of a traffic light4) color of a traffic light

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Waves. Waves. Classification.Classification.

3) Polarization.3) Polarization. Determined by direction of Determined by direction of y(x,ty(x,t))

in the transversal planein the transversal plane

changes in xchanges in x--directiondirection

2) Longitudinal & Transversal2) Longitudinal & Transversal ..

changes in an orthogonal directionchanges in an orthogonal direction

1)1) Shape.Shape. Determined by surfaces of constant Determined by surfaces of constant

wavefrontswavefronts

spherical, cylindrical, spherical, cylindrical, planeplane etc.etc.

propagates in one direction, say, along propagates in one direction, say, along xx--axisaxis

plane harmonic wave:plane harmonic wave:(has fixed temporal and spatial period = wavelength)(has fixed temporal and spatial period = wavelength)building block for other waves!building block for other waves!

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Fishbone & Slinky DemoFishbone & Slinky Demo

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The Nittany SinusoidThe Nittany Sinusoid

We will use the form:

y(x,t) = ymsin(kx-ωωωωt)

displacement amplitude phase

oscillating term

angularwave # position angular

frequency

time

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Freezing a Sinusoid in TimeFreezing a Sinusoid in Time• Set t=constant (t=0 works) to better understand spatial properties: k & λλλλ� y(x,0) = ymsin(kx)� After one “wavelength” require that we have

• ymsin(kx1) = ymsin(k(x1+λλλλ)) = ymsin(kx1+kλλλλ)

� Hence, kλλλλ = 2ππππ, or k = 2ππππ/λ λ λ λ (k=angular wave number)

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Speed of a WaveSpeed of a Wave• For Our Sinusoid, with y(x,t) = ymsin(kx-

ωωωωt), keeping the same y-value means� (kx-ωωωωt) = constant

• Note: as t increases, so must x—as expected for a wave moving in the +x direction

� d/dt(kx-ωωωωt) = kdx/dt – ωωωω = 0� dx/dt = v = ωωωω/k

• Using ω ω ω ω = 2ππππ/T and k = 2ππππ/λλλλ, have� v = ωωωω/k = λλλλ/T = λλλλf

• For a wave traveling in the –x direction, use (kx+ωωωωt) instead

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Derivatives of the WaveDerivatives of the Wave

Traveling Displacement:

y(x,t) = ymcos(wt+φφφφ)Traveling Velocity:

v(t) = dy/dt = -ωωωωymsin (ωωωωt+φφφφ)

Traveling Acceleration:a(t) = d2222y/dt2222 = -ωωωω2222 ymcos (ωωωωt+φφφφ)

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ExampleExample

• A traveling harmonic wave is described by:� y(x,t) = (1 m) sin (10x - 5πt)

• What is the direction of propagation?• What is the amplitude?• What is the wavelength?• What is the wave number?• What is the angular frequency?• What is the frequency?• What is the period?• What is the speed?• What is the traveling speed?

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F.Y.I.: We’re skipping 16:7-8F.Y.I.: We’re skipping 16:7-8

• 16:7 covers energy and power of a traveling string wave

• 16:8 covers the wave equation

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Superposition of WavesSuperposition of Waves

• Principle of superposition basically says that things add linearly

• For waves, this means that if two waves y1(x,t) and y2(x,t) overlap, the resultant wave is a linear sum of the two:� y(x,t) = y1(x,t) + y2(x,t)

• Not, say, y(x,t) = [y1(x,t) + y2(x,t)]½

• This also means that overlapping waves do not in any way alter the travel of one another

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Interference of Waves: QualitativeInterference of Waves: Qualitative• Imagine two waves traveling on a string at different speeds. What does the resultant wave look like if they are in phase, ππππ rads out of phase, or 2ππππ/3 rads out of phase?

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Interference of Waves: QuantitativeInterference of Waves: Quantitative• Two waves, same amplitude, wavelength and speed, traveling in same direction:� y1(x,t) = ymsin(kx-ωωωωt)� y2(x,t) = ymsin(kx-ωωωωt+φφφφ))))

• Result:

• Values of φφφφ:• φφφφ=0: Amplitude = 2ym (fully constructive)• φφφφ=ππππ: Amplitude = 0 (fully destructive)• φφφφ=15.1: Amplitude = 0.6ym (intermediate)

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F.Y.I.: We’re skipping 16:11F.Y.I.: We’re skipping 16:11

• Covers “phasors,” a technique used to combine waves even if their amplitudes differ

• Unrelated to Star Trek

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Standing Waves: QualitativeStanding Waves: Qualitativesum wave 2 w

ave 1

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Standing Waves: QuantitativeStanding Waves: Quantitative

• Add the following two waves together algebraically:� y1(x,t) = ymsin(kx-ωωωωt)� y2(x,t) = ymsin(kx+ωωωωt))))

• This gives� y’(x,t) = [2ymsin(kx)]cos(ωωωωt)

• This is NOT a traveling wave!

http://webphysics.davidson.edu/Applets/Applets.html

Nodes at kx=nππππ;Antinodes at kx=(n+½)ππππ.n=0,1,2,…

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Creating a Standing WaveCreating a

Standing Wave

• Use reflectionsat a Boundary� This is how, say, a guitar string works

• Two kinds of reflections:� “hard” or ½λλλλ phase shift

� “soft” or 0 phase shift

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Standing Waves on a StringStanding Waves on a String

Clearly, must havenλλλλ/2 = L orλ= 2L/n to have astanding wave.

Frequencies: f=nv/2L

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Transverse Standing Wave DemoTransverse Standing Wave Demo

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What have we learnedWhat have we learned

• How to stress/strain� F/A = E ∆∆∆∆L/L

• What a wave isy(x,t) = ymsin(kx-ωωωωt)

• How waves interfereadd ‘em up

• Standing wavesf=nv/2L

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Next TimeNext Time

• Fluids� Fluids at rest, pressure and Pascal’s principle

� Fluids in motion, Archimedes principle

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Creative Problems(for Thursday recitation)Creative Problems

(for Thursday recitation)

1) Using equipment at hand (e.g. ruler, object of known mass etc.) find Young’s modulus of SLINKY. (The less sofisticated your equipment the better.)

2) Propose how, in principle, one can measure the operating frequency of a microwave using marshmallows and a ruler.