Class 38 - Waves IClass 38 - Waves IChapter 16 - Wednesday November 24thChapter 16 - Wednesday November 24th

Reading: pages 413 to 423 (chapter 16) in HRWReading: pages 413 to 423 (chapter 16) in HRWRead and understand the sample problemsRead and understand the sample problemsAssigned problems from chapter 16 (due Dec. Assigned problems from chapter 16 (due Dec. 2nd):2nd):

6, 20, 22, 24, 30, 34, 42, 44, 66, 70, 78, 826, 20, 22, 24, 30, 34, 42, 44, 66, 70, 78, 82

•Traveling waves•Waves on a string•The wave equation•Sample problems

• Exam 3: Friday December 3rd, 8:20pm to 10:20pmExam 3: Friday December 3rd, 8:20pm to 10:20pm• You must go to the following locations based on the 1st letter of your You must go to the following locations based on the 1st letter of your

last name:last name:

• Review sessions: Tues. Nov. 30 and Thurs. Dec. 2, 6:15 to 8:10pmReview sessions: Tues. Nov. 30 and Thurs. Dec. 2, 6:15 to 8:10pm

Waves I - types of wavesWaves I - types of waves1.1. Mechanical waves:Mechanical waves: water waves, sound waves, seismic water waves, sound waves, seismic

waves.waves.2.2. Electromagnetic waves:Electromagnetic waves: radio waves, visible light, ultraviolet radio waves, visible light, ultraviolet

light, x-rays, gamma rays.light, x-rays, gamma rays.3.3. Matter waves:Matter waves: electrons, protons, neutrons, anti-protons, electrons, protons, neutrons, anti-protons,

etc..etc..1.1. These are the most familiar. We encounter them every day. These are the most familiar. We encounter them every day. The common feature of all mechanical waves is that they The common feature of all mechanical waves is that they are governed entirely by Newton's laws, and can exist only are governed entirely by Newton's laws, and can exist only within a material medium.within a material medium.

2.2. All electromagnetic waves travel through vacuum at the All electromagnetic waves travel through vacuum at the same speed same speed cc, the speed of light, where , the speed of light, where cc = 299 792 458 = 299 792 458 m/s. Electromagnetic waves are governed by Maxwell's m/s. Electromagnetic waves are governed by Maxwell's equations (PHY 2049).equations (PHY 2049).

3.3. Although one thinks of matter as being made up from Although one thinks of matter as being made up from particles, it is in fact made up from fundamental matter particles, it is in fact made up from fundamental matter waves that travel in vacuum. Matter waves are governed by waves that travel in vacuum. Matter waves are governed by the laws of quantum mechanics, or the Schrödinger and the laws of quantum mechanics, or the Schrödinger and Dirac equations.Dirac equations.

Wave interferenceWave interference

Matter-wave interference

Waves I - types of wavesWaves I - types of wavesTransverse waves Transverse waves (2 polarizations)(2 polarizations)

Longitudinal wavesLongitudinal waves

Waves I - wavelength and Waves I - wavelength and frequencyfrequency

Wavelength (consider wave at Wavelength (consider wave at tt = 0 = 0):):

( ,0) sin sin

sin

m m

m

y x y kx y k x

y kx k

You can always add You can always add 22 to the phase to the phase of a wave without changing its of a wave without changing its displacement, displacement, i.e.i.e. 22 ork k

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wav

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sinu

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( , ) sin( )my x t y kx t

phaseshift

Waves I - wavelength and Waves I - wavelength and frequencyfrequency

Wavelength (consider wave at Wavelength (consider wave at tt = 0 = 0):):22 ork k

We call We call kk the the angular wavenumberangular wavenumber..The SI unit is The SI unit is radian per meterradian per meter, or , or metermeter-1-1..This This kk is NOT the same as spring is NOT the same as spring constant.constant.

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( , ) sin( )my x t y kx t

phaseshift

Waves I - wavelength and Waves I - wavelength and frequencyfrequency

Period and frequency (consider wave Period and frequency (consider wave at at x x = 0 = 0):):

(0, ) sin sin

sin

m m

m

y t y t y t

y t T

Again, we can add Again, we can add 22 to the phase, to the phase,22 orT T

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sinu

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( , ) sin( )my x t y kx t

phaseshift

Waves I - wavelength and Waves I - wavelength and frequencyfrequency

Period and frequency (consider wave Period and frequency (consider wave at at x x = 0 = 0):):

22 orT T

We call We call the the angular frequencyangular frequency..The SI unit is The SI unit is radian per secondradian per second..The The frequency frequency ff is defined as is defined as 1/1/TT..

12

fT

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sinu

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( , ) sin( )my x t y kx t

phaseshift

The speed of a traveling waveThe speed of a traveling wave•A fixed point on a wave has a A fixed point on a wave has a constant value of the phase, constant value of the phase, i.e.i.e.

constantkx t

0 ordx dxk vdt dt k

OrOrv f

k T

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The speed of a traveling waveThe speed of a traveling wave•For a wave traveling in the opposite For a wave traveling in the opposite direction, we simply set time to run direction, we simply set time to run backwards, backwards, i.e.i.e. replace replace tt with with tt..

constantkx t

0 ordx dxk vdt dt k

( , ) sinmy x t y kx t

•So, general sinusoidal solution is:So, general sinusoidal solution is: ( , ) sinmy x t y kx t

•In fact, any function of the formIn fact, any function of the form ( , ) my x t y f kx t

is a solution.is a solution.

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Traveling waves on a stretched Traveling waves on a stretched stringstring

m l

is the string's linear density, is the string's linear density, or force per unit length.or force per unit length.

Dimensional analysisDimensional analysis

•Tension Tension provides the restoring force (kg.m.s provides the restoring force (kg.m.s-2-2) in the string. ) in the string. Without tension, the wave could not propagate.Without tension, the wave could not propagate.•The mass per unit length The mass per unit length (kg.m (kg.m-1-1) determines the response ) determines the response of the string to the restoring force (tension), through Newtorn's of the string to the restoring force (tension), through Newtorn's 2nd law.2nd law.•Look for combinations of Look for combinations of and and that give dimensions of that give dimensions of speed (m.sspeed (m.s-1-1).). v C

dm

F net

Traveling waves on a stringTraveling waves on a string

•The tension in the string is The tension in the string is ..•The mass of the element The mass of the element dmdm is is dldl, where , where is the is the mass per unit length of the mass per unit length of the string.string.

cos cos

sin sin

x

y

F d

F d

x

y

x+ dx

y+ d yd l

+ d

x

y

Traveling waves on a stringTraveling waves on a string

cos cos

sin sin

x

y

F d

F d

•In the small In the small limit... limit... cos 1 sin sin tan yd d

x

2

20x yyF F d d ma dx

t

x

y

x+ dx

y+ d yd l

+ d

x

y

2 2

2 2

y yx x t

The wave equationThe wave equation2 2

2 2

y yx t

•General solution:General solution:

( , ) sin or ( , )m my x t y kx t y x t y f kx t

2 2

2 22 2, ,y yk y x t y x t

x t

22 2 2

2ork vk

. .i e v