thursday, sep. 4phy208 lecture 2 1 from last time… waves interference please pick up pack of color...

Thursday, Sep. 4 Phy208 Lecture 2 1

From last time…

Waves

Interference

Please pick up pack of color sheets


Interference of 2 speakers

cresttrough

constructive interference,loud tone

destructive interferencequit tone


A little more detail

Same distance, same phase

Path 1Path 2

Path-length difference = (Path 2) - (Path 1) = 0

d

Path 1

Path 2


Other angles?

Interference: Constructive: =

= Extra path length

d

€

/d = sinθ

δ = d sinθ

Path 1Path 2

Path-length difference 0

Destructive: = €

mλ

€

2m +1

2λ

€

=0, λ , 2λ , 3λK

€

=1

2λ ,

3

2λ ,

5

2λK


QuestionSuppose that the frequency of the sound wave increases by a factor of two. The adjacent maxima are

A) Farther apart

B) Closer together

C) Same

D) Need to know speaker spacing

Higher frequency, -> shorter wavelength

Less path length diff required to make 1 wavelength diff.


QuestionIn your room you have two speakers in different

corners. At your desk you are exactly 1 meter from each, so that there is no interference. Your roommate moved one of your speakers 0.25 m further away from your desk. At what frequency will you hear destructive interference?

A. 170 HzB. 340 HzC. 680 HzD. 1000 HzE. 1350 Hz


Math Analysis

Constructive interference:

€

=d sinθ = 0 ⋅λ

⇒ θ = 0

€

=d sinθ =1⋅λ

⇒ sinθ = λ /d =v

fd

€

=d sinθ = 2 ⋅λ

⇒ sinθ = 2λ /d = 2v

fd

= Extra path length

€

/d = sinθ

δ = d sinθ

Path 1

Path 2

d

€

=340m /s

1000 /s( ) 1m( )= 0.34

= 0.347 rad = 19.9˚


For small angles…for constructive interference

€

sinθ = λ /d

For small ,

€

sinθ ≈ θ with in radians

Small angle approximation

e.g.

€

sinθ = λ /d = 0.34

⇒ θ ≈ 0.34 radians

Exact value:

€

=0.347 radians

Approx breaks down for large


Light waves• Light is a wave just like sound. • But can propagate in vacuum

– at speed c = 3x108 m/s

• Has same wave properties, e.g.

• Can also propagate in glass, water, but slower– Speed in medium = v = c/n

– n = Index of refraction

• Has same superposition / interference properties

€

λvac =c

f

€

λmed =c

nf=λ vacn


Interference of light


L

Recording plate

Projection on screen• Light wavelength much shorter than sound• Interference fringes much closer together

Light beam

Foil with two narrow slits

y

€

y /L = tanθ


= Extra path length

€

=d sinθ

y = position on screen =

€

L tanθ

Small-angle approximation

€

sinθ ≈ θ

tanθ ≈ θ

€

~ dy

Lfor θ → 0

Extra path length =


• For bright fringes: extra path length = mλ

• For dark fringes: extra path length = (m+1/2) λ

Fringe separation & wavelength

Bright Fringe separation =

€

λLd

€

ydark =λL

dm +

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟ m = 0, ±1, ± 2 K

€

ybright =λL

dm m = 0, ±1, ± 2K


QuestionA two-slit interference pattern is observed in air

(n=1). Then the entire system is immersed in water (n=1.33). The interference fringes are

A) 1.33 times closer together

B) 1.33 times farther apart

C) 2.66 times closer together

D) 2.66 times farther apart

E) Spacing unchanged


Interference of multiple sources

Evenly-spaced slits have maxima at same location as two-slits.

Maxima become sharper, more intense

Interference of multiple sources often called diffraction


Diffraction gratings• Diffraction grating is pattern of multiple

slits.• Very narrow, very closely spaced.• Expect very narrow interference peaks.(15,000 lines/inch)x(1 inch/2.54 cm)x(100 cm/m)=5.9x105/mSo a spacing of 1/(5.9x105/m) = 1.7x10-6 m = 1700 nm.

1700 nm


QuestionSuppose white light is shined through a

diffraction grating. What would the pattern on the screen look like?

A)

B)

C)

D)

€

ybright =λL

dm

m = 0, ±1, ± 2 K


Diffraction: only one slit, but “wide”

• Huygen’s principle: each portion of the slit acts as a source of waves

• These sources interfere according to path-length difference.


Diffraction minima

• Interference-like pattern from a single slit.

How wide is wide?


Phase difference• Waves start in phase• Travel different distances (extra path length = )• No longer in phase when combined (Phase diff )

=λ =2π

=λ/2 =π

General reln:

€

φ=2πδ

λ

Constructive: =2πm

Destuctive: =2π(m+1/2)


Thin film interferenceBlack

Colors changing with thickness


Thin film interference

• If film is much thinner than wavelength, ~ no phase shift from extra path length

• But top reflection has 180˚ phase shift, bottom not• Destructive interference for all wavelengths,

film appears black

air: n=1

n>1

180˚ (π radians) phase shift

no phase shift

t

λ

λ/n


Thicker parts

• Phase difference comes from– Phase shift from

reflection (top)– Phase shift from

extra path length

• Extra path length for normal incidence = 2t

air: n=1

n>1

180˚ phase shiftfrom reflection

t

λ

λvac/nExtra path length=2t

€

−π + 2π2t

λ vac /n( ) =Phase difference =

Reflection phase shift # wavelengths in

extra path length

€

2mπ constructive

€

2m −1( )π destructive

air: n=1


Constructive interference

€

2t = m +1

2

⎛

⎝ ⎜

⎞

⎠ ⎟λ

n (m = 0,1,2K ) constructive interference

€

2t = mλ

n (m = 0,1,2K ) destructive interference

thursday, sep. 4phy208 lecture 2 1 from last time… waves interference please pick up pack of color...

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