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- Interference
Pham Tan Thi, Ph.D.
Department of Biomedical Engineering
Faculty of Applied Science
Ho Chi Minh University of Technology
- A Single Oscillating Wave
The formula y(x, t) = Acos(kx !t)
describes a harmonic plane wave of amplitude A moving in the +x
direction
For a wave on a string, each point on the wave oscillates in the x
direction with simple harmonic motion of angular frequency ω
2⇡ !
The wavelength = The speed/velocity v = f =
k k
The intensity is proportional to
the square of the amplitude I / A2
- Multiple Waves: Superposition
The principle of superposition states that when two or more waves
of the same type cross at a point, the resultant displacement at that
point is equal to the sum of the displacements due to each
individual wave.
For inequal intensities, the maximum and minimum intensities are:
Imax = |A1 + A2|2
Imin = |A1 - A2|2
- Multiple Waves: Superposition
Constructive “Superposition” Destructive “Superposition”
- ou added the two sinusoidal waves shown,
Superposing Sinusoidal Waves
at would the result look like?
If we added the two sinusoidal waves shown, what would the result
look like?
1 .0 0
0 .5 0
0 .0 0
1000
100
200
300
400
500
600
700
800
900
0
- 0 .5 0
- 1 .0 0
of two sines having the same frequency is another
same frequency.
ude depends on their relative phases.
2 .0 0
1 .5 0
1 .0 0
0 .5 0
- ou added the two sinusoidal waves shown,
1 .0 0
0 .5 0
Superposing Sinusoidal Waves
at would the result look like?
0 .0 0
1000
100
200
300
400
500
600
700
800
900
0
- 0 .5 0
If we added the two sinusoidal waves shown, what would the result
- 1 .0 0
look like?
1 .0 0
0 .5 0
0 .0 0
of two sines having the same frequency is another s
1000
100
200
300
400
500
600
700
800
900
0
- 0 .5 0
- 1 .0 0
same frequency.
The sum of two sines having the same frequency is another sine
ude depends
with the sameon their→ relative
frequency Its amplitudephases.
depends on their relative
phases
of two sines having the same frequency is another
2 .0 0
1 .5 0
same frequency.
1 .0 0
0 .5 0
ude depends on their relative phases.
0 .0 0
0
100
200
300
400
500
600
700
800
900
1000
- 0 .5 0
- 1 .0 0
- 1 .5 0
2 .0 0
- 2 .0 0
1 .5 0
1 .0 0
0 .5 0
- Adding Sine Waves with Different Phases
Suppose we have two sinusoidal waves with the same A1, ω and k:
y1 = A1 cos(kx !t) and y2 = A1 cos(kx !t + )
One starts at phase 𝜙 after the other
Spatial dependence
of 2 waves at t = 0
Resulting wave:
y = y1 + y 2
✓ ◆✓ ◆
↵ +↵
A1 (cos↵ + cos ) = 2A1 cos
2 2
y1 + y 2
y = 2A1 cos(( /2)
/2)cos(kx !t + /2)
y = 2A1 cos( /2)cos(kx !t + /2)
Amplitude Oscillation
- What happens when two waves are present at the same place?
Interference of Waves
Always add amplitudes (pressures or electric fields).
WhatHowever,
happens when two waves
we observe are present
intensity at the same place?
(power).
ForAlways
equal Aadd
andamplitude
ω: (e.g. pressures or electric fields)
However, we observe intensity (i.e. power)
For equal A = ω:
A and 2A1 cos(φ / 2) ⇒ I = 4I1 cos (φ / 2)
2
A = 2A1cos(𝜙/2) I = 4I1cos2(𝜙/2)
Example: Terminology:
Stereospeakers:
Stereo speakers: Listener: Constructive interference:
Terminology:
waves are “in phase”
Constructive
(φ = 0, interference:
2π, 4π, ..)
Listener: waves are “ininterference:
phase”
Destructive
waves
(𝜙 = 0, are “out of phase”
2π, 4π,…)
(φ = π,interference:
Destructive 3π, 5π, …)
waves are “out of phase”
(𝜙 = π, 3π, 5π,…)
Of course, φ can take on an infinite number
of values. We won’t use terms like “mostly
constructive” or “slightly destructive”.
Lec
- Quiz
Each speaker alone produces an intensity of I1 = 1 W/m2 at the
Example: Changing phase of the Source
listener:
Example: Changing phase of the Source
Each speaker alone produces an intensity of I = 1 W/m2 at the listener:
1
Each speaker alone produces an intensity of I1 = 1 W/m2 at the listener:
Example: Changing phase
I = I = of
A =the
1 W/mSource
I = I1 = A12 = 1 W/m2
1 1
2 2
I =1 IW/m
Each speaker alone produces an intensity of I1 = 1 = A1 at=the
22 1 W/m 2
listener:
Drive the speakers in phase. What is the intensity I at the listener?
DriveDrive
thethespeakers in What
speakers in phase. phase. IWhat 2 is
= I1 = AI1at
is the intensity 1the
=the W/m 2 intensity I at the listener?
listener?
I=
Drive the speakers in phase. What is the intensity
I =I at the listener? I =?
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
I=
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
Now shift phase of one speaker by 90°. What is the intensity I at the
listener? I=
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
I=
φ
φ Lecture 2, p.7
I =?
Lecture 2, p.7
I=
φ
Lecture 2, p.7
- Quiz
Each speaker alone produces an intensity of I1 = 1 W/m2 at the
Example: Changing phase of the Source
listener:
Example: Changing phase of the Source
Each speaker alone produces an intensity of I = 1 W/m2 at the listener:
1
Each speaker alone produces an intensity of I1 = 1 W/m2 at the listener:
Example: Changing phase
I = I = of
A I=the
= I1Source
1 W/m = A12 = 1 W/m2
1 1
2 2
I =1 IW/m
Each speaker alone produces an intensity of I1 = 1 = A1 at=the
22 1 W/m 2
listener:
Drive the speakers in phase. What is the intensity I at the listener?
DriveDrive
thethespeakers in What
speakers in phase. phase. IWhat 2 is
= I1 = AI1at
is the intensity 1the
=the W/m 2 intensity I at the listener?
listener?
I=
Drive the speakers in phase. What is the intensity
I = (2A1)2 = 4I1 = 4 W/m2
I =I at the listener?
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
I=
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
Now shift phase of one speaker by 90°. What is the intensity I at the
listener? I=
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
I=
φ
φ Lecture 2, p.7
I =?
Lecture 2, p.7
I=
φ
Lecture 2, p.7
- Quiz
Each speaker alone produces an intensity of I1 = 1 W/m2 at the
Example: Changing phase of the Source
listener:
Example: Changing phase of the Source
Each speaker alone produces an intensity of I = 1 W/m2 at the listener:
1
Each speaker alone produces an intensity of I1 = 1 W/m2 at the listener:
Example: Changing phase
I = I = of
A I=the
= I1Source
1 W/m = A12 = 1 W/m2
1 1
2 2
I =1 IW/m
Each speaker alone produces an intensity of I1 = 1 = A1 at=the
22 1 W/m 2
listener:
Drive the speakers in phase. What is the intensity I at the listener?
DriveDrive
thethespeakers in What
speakers in phase. phase. IWhat 2 is
= I1 = AI1at
is the intensity 1the
=the W/m 2 intensity I at the listener?
listener?
I=
Drive the speakers in phase. What is the intensity
I = (2A1)2 = 4I1 = 4 W/m2
I =I at the listener?
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
I=
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
Now shift phase of one speaker by 90°. What is the intensity I at the
listener? I=
Now shift phase of one speaker by 90o.What is the intensity I at the listener?
I=
φ
φ Lecture 2, p.7
I = 4I1cos2(45°) = 2 I1 = 2 W/m2
Lecture 2, p.7
I=
φ
Lecture 2, p.7
- ACT 1:
Noise-cancelling
Noise Cancelling Headphones
Headphones
Noise-canceling
Noise-canceling headphones
headphones workingwork using
using
interference.
interference. A microphone
A microphone on theon the
earpiece
earpiece
monitors monitors the instantaneous
the instantaneous amplitude of the
amplitude
external of the and
sound wave, external soundon
a speaker wave,
the inside
and a speaker on the inside of the
of the earpiece produces a sound wave cancel it.
earpiece produces a sound wave to
cancel it.
1. must
1. What Whatbe
must
the be the phase
phase of the signal
of the signal from thefrom the speaker
speaker relativerelative
to the to the
externalexternal
noise? noise?
a. 0a. 0 b. 90˚ c. πc. π
b. 90 d. d.
180-180˚ e.e.2π2π
2. What must the intensity Is of the signal from the speaker relative to the
external noise?
2. What must be the intensity Is of the signal from the speaker relative to the
a. Iexternal
s = In b. Is < IInn? c. Is > In
noise
a. Is = In b. Is < In c. Is > In
Lecture 2, p.10
- ACT 1:
Noise-cancelling
Noise Cancelling Headphones
Headphones
Noise-canceling
Noise-canceling headphones
headphones workingwork using
using
interference.
interference. A microphone
A microphone on theon the
earpiece
earpiece
monitors monitors the instantaneous
the instantaneous amplitude of the
amplitude
external of the and
sound wave, external soundon
a speaker wave,
the inside
and a speaker on the inside of the
of the earpiece produces a sound wave cancel it.
earpiece produces a sound wave to
cancel it.
1. must
1. What Whatbe
must
the be the phase
phase of the signal
of the signal from thefrom the speaker
speaker relativerelative
to the to the
externalexternal
noise? noise?
a. 0a. 0 b. 90˚ c. πc. π
b. 90° d. d.
180°
-180˚ e.e.2π2π
Destructive interference occurs when the waves are ±180° out of phase (=π radians)
2. What must the intensity Is of the signal from the speaker relative to the
external noise?
2. What must be the intensity Is of the signal from the speaker relative to the
a. Iexternal
s = In b. Is < IInn? c. Is > In
noise
a. Is = We
In want
b. IAs =< AIns - Anc.
= I0. >
s
Note
In that I is never negative.
Lecture 2, p.10
- Interference of Light
- Review: Lights of Different Colors
Long wavelength
(Low frequency)
Short wavelength
(High frequency)
Violet light has the highest energy in visible region
- Interference of Light
Interference is a phenomenon in which two coherent light waves
superpose to form a resultant wave of greater or lower amplitude
(a) Constructive interference: If a crest of one wave meets a crest
of another wave, the resultant intensity increases.
(a) Destructive interference: If a crest of one wave meets a trough
of another wave, the resultant intensity decreases.
- Light Polarization
- Conditions for Interference
✓ Two interfering waves should be coherent, i.e. the phase difference
between them must remain constant with time.
✓ Two waves should have same frequency.
✓ If interfering waves are polarized, they must be in same state of
polarization.
✓ The separation between the light sources should be as small as
possible.
✓ The distance of the screen from the sources should be quite large.
✓ The amplitude of the interfering waves should be equal or at least
very nearly equal.
✓ The two sources should be narrow.
✓ The two sources should give monochromatic or very nearly
monochromatic.
- Young’s Double Split Experiment
• In 1801, Young admitted the sunlight through a single pinhole and
then directed the emerging light onto two pinholes.
• The spherical waves emerging from the pinholes interfered with
each other and a few colored fringes were observed on the screen.
- Calculation of Optical Path Difference between Two Waves
As slits (S1 and S2) are equidistant from source (S), the phase of the wave at S1 will be
same as the phase of the wave at S2 and therefore S1, S2 act as coherent sources. The
waves leaving from S1 and S interfere and produce alternative bright and dark bands on
the screen.
2d is the distance between S1 and S2
θ is the angle between MO and MP
x is the distance between O and P
Let P is an arbitrary point on screen,
which is at a distance D from source S1N is the normal on to the line S2P
S1M = S2M = GO = OH = d
nguon tai.lieu . vn
