INTERFERENCE PHENOMENA
Superposition
If two waves occupy the same
Learning Objectives
After you complete the homework associated with this
lecture, you should be able to:
• Analyze the superposition of two waves when they
occupy the same place at the same time, including
criteria for constructive and destructive interference;
• Describe wave reflection at boundaries;
• Use superposition of waves to determine properties of a
standing wave, e.g., in stringed instruments;
• Describe how beats are generated by two interfering
traveling waves of slightly different frequencies.
(DEMO: two struck xylophone bars)
place at the same time, their
local displacements add
vectorially to produce the net
wave. For waves oscillating
in same plane, this means that
the algebraic values just add.
CAUTION : This means that we are dealing with signed
quantities Y you can't just add magnitudes.
web source:
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c21_int_super.html
square wave
wave: http://webphysics.davidson.edu/physlet_resources/bu_semester1/c22_squarewave_sim.html
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Interference
When two or more traveling waves superimpose, we
encounter the phenomenon of interference. The result
can be simple or complicated in structure depending
upon the amplitude and phase of the waves.
Constructive: displacements are in same direction
Destructive: displacements are in opposite direction
constructive & destructive interference
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c21_interference.html
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For two waves of the same amplitude and frequency:
y1(x,t) = Ao sin(kx – Tt) y2(x,t) = Ao sin(kx – Tt + N)
moving with same direction and speed v = T/k
and same wavelength 8 = 2B/k
Using trig identity:
sin a + sin b = 2 cos[½(a – b)] sin[½(a +b)]
Resultant wave is:
y(x,t) = y1 + y2 = {2Ao cos(½N)} sin(kx – Tt + ½N)
Same v and 8, but new A={2Ao cos(½N)} & phase ½N
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Path Difference and Interference
If you have two sources of waves of the same f and 8, then
one can easily determine if waves combine for interference
at some point P. There are L/8 cycles of 2B radians by the
time a wave reaches P. The phase difference )1 of waves
when they arrive at P is:
 L   L  
   2π 2    2π 1  
    

2nπ for constructive

 2n  1 π for destructive
where L n = path length from source Sn
 L   L  
   2π 2    2π 1  
    

2nπ for constructive

 2n  1 π for destructive
where L n = path length from source Sn
Multiply by 8/2B, we have for path difference )L = L2 – L1
• Constructive (large amplitude):
)L = L2 – L1 = n 8 2 L's differ by full wavelength
• Destructive (full cancellation):
)L = L2 – L1 =(n + ½) 8 2 L's differ by half 8
| Lconstructive |  (n  1 4  1 4 )λ
destructive
web source: interference of two circular waves http://www.walter-fendt.de/ph14e/interference.htm
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Standing Waves
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Oppositely traveling waves of same 8 and f produce
Consider 2 waves traveling in opposite directions with:
y(x,t) = y1 + y2 = {2 Ao cos(Tt)} sin(kx)
a) same amplitude
b) same wavelength 8 (and thus same k)
•
unison with angular frequency T =2Bf as time t
c) same frequency f (and thus same T)
and consequently have the same speed v = 8f .
y1(x,t) = Ao sin(kx – Tt)
y2(x,t) = Ao sin(k x + Tt)
These interfere to produce a simple combined form:
y(x,t) = y1 + y2 = 2 Ao cos(Tt) sin(kx)
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The wave amplitude {2 Ao cos(Tt)} oscillates in
varies.
•
The shape sin(k x) does not move, its just an
“envelope” that changes sinusoidally as x varies in
position. This is the pattern of a standing wave.
web source:
www.phy.ntnu.edu.tw/ntnujava/index.php?topic=19.msg124#msg124 rt click, f1=f2=5, Enter
backup www.phy.ntnu.edu.tw/ntnujava/index.php?topic=19.0
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Reflection of Waves at Boundary
When a wave hits a boundary (e.g., end-point) of the
medium in which it is traveling, the motion of wave
must match the motion of the boundary. This is a
BOUNDARY-VALUE
problem or situation.
1. This will produce some kind of reflected wave that
carries energy.
2. If the boundary moves with the wave, there will
also be a transmitted wave.
Character of Reflected Wave
1. If boundary is fixed, displacement of wave must be
zero at the boundary. The reflected wave will be
"inverted" (180° phase shifted) relative to the
incident wave. http://www.physicsclassroom.com/mmedia/waves/fix.cfm fixed end
2. If boundary is "flexible" and moves, the wave must
be oscillating at that boundary. Whether or not the
wave is inverted (phase shifted or not) depends on
the properties of the medium beyond the boundary.
web source:
http://www.physicsclassroom.com/mmedia/waves/free.cfm free end
incident/reflected http://webphysics.davidson.edu/physlet_resources/bu_semester1/c21_int_reflections.html
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Standing Wave with Boundary Conditions (BCs)
Consider a possible wave on a string of length L whose ends
Using the equation for a
are fixed. Composed of two oppositely traveling waves, 180°
standing wave:
out of phase due to reflection.
y(x,t) = y1 + y2 = 2 Ao sin(k x) cos(Tt)
We see that y(x=0,t) = y(x=L,t) = 0 for any t implies
web source
source: http://www.walter-fendt.de/ph14e/stwaverefl.htm develop standing wave
If string fixed at both ends, only certain frequencies produce
composite waves with requisite nodes (N) at the end points:
y(x=0,t) = y(x=L,t) = 0
(BCs)
web source
source: http://www.physicsclassroom.com/mmedia/waves/swf.cfm
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sin(k L) = 0
2
k L = n B 2 8 = 2L/n
Since 8f = v (determined by medium), only certain
frequencies can exist:
f = ½n v /L
(standing waves)
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Beat Phenomenon
If two waves traveling in the same direction have
frequencies (f1 and f2 ) very close to one another, they
will produce beats in amplitude with a frequency equal
to their frequency difference: (f1 – f2 ). This is the
frequency at which one "hears" no sound, i.e., zero
amplitude.
web source:
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c22_beats.html
http://www.walter-fendt.de/ph14e/beats.htm
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