topic 4.4 - wave behavior

7/21/2019 Topic 4.4 - Wave Behavior

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Essential idea: Waves interact with media and each

other in a number of ways that can be unexpected

and useful.

Nature of science:Competing theories: The conflicting

work of Huygens and Newton on their theories oflight and the related debate between resnel!

"rago and #oisson are demonstrations of two

theories that were valid yet flawed and incomplete.

This is an historical example of the progress ofscience that led to the acceptance of the duality of

the nature of light.

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Understandings:

& 'eflection and refraction

& (nell)s law! critical angle and total internal reflection

& *iffraction through a single+slit and around ob,ects

& -nterference patterns& *ouble+slit interference

& #ath difference

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Applications and skills:

& (ketching and interpreting incident! reflected and

transmitted waves at boundaries between media

& (olving problems involving reflection at a plane

interface& (olving problems involving (nell)s law! critical angle

and total internal reflection

& *etermining refractive index experimentally

& ualitatively describing the diffraction pattern formedwhen plane waves are incident normally on a

single+slit

& uantitatively describing double+slit interference

intensity patterns

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Guidance:

& uantitative descriptions of refractive index are limited

to light rays passing between two or more

transparent media. -f more than two media! only

parallel interfaces will be considered& (tudents will not be expected to derive the double+slit

e/uation

& (tudents should have the opportunity to observe

diffraction and interference patterns arising frommore than one type of wave

Topic 4: Waves

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Data booklet reference:

& n0/ n12 sin 1/sin 0= v1/v0

& s2 D / d

& Constructive interference: path difference 2 n

& *estructive interference: path difference 2 3n 4 56International-indedness:

& Characteristic wave behavior has been used in many

cultures throughout human history! often tying

closely to myths and legends that formed the basisfor early scientific studies

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T!eor" of kno#ledge:

& Huygens and Newton proposed two competing

theories of the behavior of light. How does the

scientific community decide between competing

theories7Utili$ation:

& " satellite footprint on 8arth is governed by the

diffraction at the dish on the satellite

& "pplications of the refraction and reflection of lightrange from the simple plane mirror through the

medical endoscope and beyond. 9any of these

applications have enabled us to improve and

extend our sense of vision.

Topic 4: Waves

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Utili$ation:

& The simple idea of the cancellation of two coherent

light rays reflecting from two surfaces leads to data

storage in compact discs and their successors

& The physical explanation of the rainbow involvesrefraction and total internal reflection. The bright

and dark bands inside the rainbow!

supernumeraries! can be explained only by the

wave nature of light and diffraction.

Topic 4: Waves

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Ais:

& Ai %: the historical aspects of this topic are still

relevant science and provide valuable insight into

the work of earlier scientists

& Ai &: experiments could include: determination ofrefractive index and application of (nell)s law

determining conditions under which total internal

reflection may occur examination of diffraction

patterns through apertures and around obstaclesinvestigation of the double+slit experiment

& Ai ': the increasing use of digital data and its

storage density has implications on individual

privacy through the permanence of a digital foot+

print

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Reflection and refraction

Wave reflectionoccurs when a wave meets a

boundary! such as a solid ob,ect! or a change in the

medium! and is at least partially diverted backwards.

The angles of the rays are always measured with

respect to the noralwhich is

perpendicular to the surface.

The relationship between

the angle of incidenceincidentand the angle of

reflectionreflect is simple:

Topic 4: Waves

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reflective

surfa

ce

incidentray

reflectedra

y

incident

reflect

incident2 reflect reflected waves

normal


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We can also look at the wavefronts: ;bserveN *"'?

IN(IDENTWA)E

*E+*A(TEDWA)E

normal

incidence

angle of

incidence

refraction angle of

refraction


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-t may help to imagine the ranks of a marching band.

;bviously! the

cadence does

not change.

Thus the

period and the

fre/uency do

not change.

@ut the speed

and the

wavelength do

change.

C;NC'8T8

*88# 9>*

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*uring refraction the fre/uency and the period do not

c!ange.

-t should be clear that the incident wave is faster than

the refracted wave in this example.

Topic 4: Waves

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C=8"' W"T8' 9>**? W"T8'

@;>N *"'?

normal

incidence

angle ofincidence

refraction angle of

refraction


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8A"9#=8: " wave traveling at 1B. ms+0hits a medium

boundary at an angle of 1D relative to the normal to theboundary! and is refracted at E1D. What is the wave)s

speed in the new medium7

rom (nell)s law we have v1/v02 sin1/sin0so

v1/1B 2 sinE1D/sin1D2 +0

Snells law

(nell)s law relates the

wave)s velocities in two

mediums to their angles:

Topic 4: Waves

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@;>N *"'?

normal

1

2

v1/ v02 sin1/ sin0 (nell)s law for refracted waves


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Snells law

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#'"CT-C8: " wave

traveling at FB. ms+0strikes

a boundary as shown. #art

of the wave energy isreflected and part

is refracted.

3a6 What is the angle of reflection7

(;=>T-;N:

(ince all angles are measured wrt the normal! the

incident angle is GBD% D 2 11D.

This is also the reflected angle since incident2 reflect.

@;>N *"'?

normal 68

75

2222


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Snells law

Topic 4: Waves

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#'"CT-C8: " wave

traveling at FB. ms+0strikes

a boundary as shown. #art

of the wave energy isreflected and part

is refracted.

3b6 What is the speed of the transmitted wave7

(;=>T-;N:

The angle of refraction is GBD % IFD 2 0FD.

v1/v0 2 sin1/sin0

v1

/FB 2 sin

0FD/sin

11D

v 2 EF m s+0.

@;>N *"'?

normal 68

75

2222 15


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8A"9#=8: (how that the relationship between the

indices of refraction and the angles is given by n0sin02

n1sin1for two transparent mediums.

(;=>T-;N:

n02 c / v0so that v02 c / n0. (imilarly! v12 c / n1.

rom (nell)s law we have v1/v02 sin1/sin0so that

sin1/sin02 v1/v02 3c / n16/ 3c / n06 2 3n0/ n16.

Topic 4: Waves

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Snells law% the refractive index

" refractive inde,nmis often used when dealing with

light waves. -t is defined to be the ratio of the speed of

light in vacuum c to the speed of light in the medium vm.

nm2 c/vm index of refraction

0

1n0 n1

n0sin02 n1sin1


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Topic 4: Waves

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" refractive inde,nmis often used when dealing with

light waves. -t is defined to be the ratio of the speed of

light in vacuum c to the speed of light in the medium vm.

nm2 c/vm index of refraction

8A"9#=8: Jarious indexes

of refraction are shown here:

(ince n2 c / v!

then v2 c / n.

(ince all of the n)s are

greater than 0! all of the v)s

are less than c.

This means that light slows down in transparent media.

0

1n0 n1

n0sin02 n1sin1


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#'"CT-C8: =ight enters a glass sample and is

refracted as in the figure.

3a6 What is the index of refraction of the glass sample7

n0sin0 2 n1sin1

0 sin$BD 2 n1sinED

n12 0.0 3unitless6.

3b6 What is the speed of light in the glass sample7 Thespeed of light in air is c 2

E.B0Bms+0.

rom the definition of the index of refraction n12 c/v1.

v1 =c/ n1

Topic 4: Waves

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8A"9#=8: The index of refraction for a particular glass

is 0.F. What is the speed of light in it7

(;=>T-;N: >se n0/n12 v1/v0. Then

0/0.F 2 v1/E.B0Bso that v12 0.0B

ms+0.


'ecall (nell)s law:

rom n0sin

02 n

1sin

1we get

n0/n12sin1/sin0.

(nell)s law can be expanded:

Topic 4: Waves

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v1/v02 sin1/sin0 (nell)s law for refracted waves

n0/n12 v1/v02 sin1/sin0 (nell)s law for

refracted waveswheren2 0 for air and free space

crown prism spectroscope


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Topic 4: Waves

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Snells law% total internal reflection

When a wave moving in an optically dense region

arrives at a boundary with a less dense medium it is

possible to have all of the light reflected % trapping it

inside. This is the principle behind fiber optics.

The critical anglecis that

incident angle at which the angle

of refraction refris GB:

(nell)s law gives us a relationship

between the indices of refraction

and the critical angle:

n0sin c= n1sin GB2 n1306.

sinc = n1/ n0 (nell)s law for critical angle

n0

n1


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FYI

(ince the sine of anything is always less than or e/ualto 0! you can always tell whether a critical angle exists.

Snells law% total internal reflection

Topic 4: Waves

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#'"CT-C8: " ray of light is trapped

in piece of crown glass.

3a6 What is its critical angle if the

glass is in air7

3b6 What is its critical angle if the

glass is in water7

(;=>T-;N: >se sinc = n1/ n0:

3a6 sinc = n1/ n02 0.BB/ 0.F1 2 B.FIG c 2 $0.

3b6 sinc = n1/ n02 0.EE/ 0.F1 2 B.IFBB c 2 0.

n0

n1


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FYI *iffraction is caused by ob,ects within the medium

that interact with the wave. -t is not caused by twomediums and their boundary.

Diffraction through a single-slit and around objects

-f a wave meets a hole in a wall that is of comparable

size to its wavelength! the wave will be bent through a

process called diffraction.

-f the aperture 3hole!opening! etc.6 is much

larger than the wavelength!

diffraction will be minimal

to nonexistent.

Topic 4: Waves

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-NC-*8NT

W"J8

*-'"CT8*

W"J8

'8=8CT8*

W"J8


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Topic 4: Waves

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-f dis too small the bat)s sound waves will not even be

disturbed enough for the bat to detect the insect.


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Christian Huygens explained the behavior of diffraction

through his famous principle:

K8very point on a wavefront

emits a spherical waveletof the same velocity and

wavelength as the

original wave.L

Note that it isbecause of

Huygen)s

principle the

waves can turncorners.

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FYI The aperture siMe must be

of the order of a wavelength in

order for diffraction to occur.


The reason waves can turn corners is that the

incoming wave transmits a disturbance by causing the

medium to vibrate.

"nd wherever the medium vibrates it becomes thecenter of a new wave front as

illustrated to the right.

Note that the smaller the

aperture bthe more pronouncedthe diffraction effect.

Topic 4: Waves

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b2 01b2

b2 1

b b b


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Topic 4: Waves

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S0 S1

P0

=02$

=12

1

=02

$

=1

2E

Path difference

This animation showing two co!erent3in+phase and

same fre/uency6 wave sources (0and (1will show the

following:

path difference2 n

condition for constructiveinterference nis an integer


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Topic 4: Waves

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S0 S1

P1

=02

E

=12

1.F

=0

2E.F

=12

$

Path difference

This animation showing two co!erentwave sources

(0and (1will show the following:

path difference2 3n 4 56

condition for destructive

interferencenis an integer


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#'"CT-C8: Two identical wave sources in a ripple

tank produce waves having

a wavelength of . The

interference pattern isshown. our reference

lines are labeled

" through *.

3a6 Which reference line orlines represent constructive interference7

3b6 Which reference line or lines represent destructive

interference7

Topic 4: Waves

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Double-source interference

D

C

B

A

" and @

C and *


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Topic 4: Waves

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#'"CT-C8: Two identical wave sources in a ripple

tank produce waves having

a wavelength of . The

interference pattern isshown. our reference

lines are labeled

" through *.

3c6 Which reference line orlines represent a path difference of 1.F7

1.Fis a condition for destructive interference.

These two represent the smallest difference B.F.

C represents 0.Fdifference! and * is thus 1.F.

D

C

B

A

*

Double-source interference


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Diffraction through a single-slit

Huygen)s wavelets

not only allow the

wave to turn

corners! theyalso interfere

with each other.

Topic 4: Waves

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onstructive

interference

'

8="T-J8

-NT8N(

-T?

Destructive

interference


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8A"9#=8: -f light is diffracted by a circular hole! a

planar cross+section of the interference looks like the

picture below. What will the light look like head+on7

(;=>T-;N:

#icture the waveform of the previous slide in 1*

Diffraction through a single-slit

Topic 4: Waves

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*erive the formula 2 / bfor the

position of the first minimum of the

diffraction pattern of a single slit.

Consider a single slit of width b.

rom Huygen we know that every point

within the slit acts as a new wavelet.

"t the central maximum we see that

the distance traveled by all the waveletsis about e/ual! and thus has constructive interference.

Consider slit pointsxat one edge and yat the center:

"t the 0stminimum! the difference in distance 3blueand

reddashed6 must be / 1. Why7

Topic 4: Waves

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b

0stmin

!"

*estructive condition.

Diffraction through a single-slit # optional derivation


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We will choose the midpoint of the

slit 3"6 as our reference.

"nd we will call the angle between

the reference and the first minimum .

We construct a right triangle as

follows:

Why does the

side opposite e/ual / 17

Topic 4: Waves

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b

1"

!

1

@ecause of the condition for

destructive interference.

Diffraction through a single-slit # optional derivation


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rom the right triangle we see that

sin2 3/ 16/ 3b / 16

sin2

/b.

#erhaps you recall that

if is very small 3and

in radians6 then

sin 3

in radians6.

inally'C8: http:PPwww.math.uio.noPQkarstentPwavesPindexen.html

%ave behavior

topic 4.4 - wave behavior

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