Traveling Waves: Superposition
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R. Field 11/14/201 3 Unive PHY 2053 Page 1 Traveling Waves: Superposition • Wave Superposition: f y A ) sin( ) sin( ) , ( 1 1 1 1 1 1 t x k A A t x y y-axis A 1 1 y 1 (x,t) f y A ) sin( ) sin( ) , ( 2 2 2 2 2 2 t x k A A t x y y-axis A 2 2 y 2 (x,t) the two waves together (superposition of wave 1 and wave 2) as follows: ) sin( ) sin( ) sin( ) sin( ) , ( ) , ( ) , ( 2 2 2 1 1 1 2 2 1 1 2 1 12 t x k A t x k A A A t x y t x y t x y
description
Traveling Waves: Superposition. Wave Superposition:. Add the two waves together (superposition of wave 1 and wave 2) as follows:. Traveling Waves: Superposition. Wave Superposition:. Wave 2. Superposition!. Wave 1. The intensity of the new wave is proportional to A 12 squared!. - PowerPoint PPT Presentation
Transcript of Traveling Waves: Superposition
R. Field 11/14/2013 University of Florida
PHY 2053 Page 1
Traveling Waves: Superposition• Wave Superposition:
f
y
A
)sin()sin(),( 111111 txkAAtxy y-axis
A1
1 y1(x,t)
f
y
A
)sin()sin(),( 222222 txkAAtxy y-axis
A2
2 y2(x,t)
Add the two waves together (superposition of wave 1 and wave 2) as follows:
)sin()sin(
)sin()sin(),(),(),(
222111
22112112
txkAtxkA
AAtxytxytxy
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PHY 2053 Page 2
Traveling Waves: Superposition
)sin()sin(
)sin()sin(),(),(),(
222111
22112112
txkAtxkA
AAtxytxytxy
• Wave Superposition:
y-axis
A1 1
2
A2
y-axis
A1 1
2 A2 A12
)sin()sin(),( 111111 txkAAtxy )sin()sin(),( 222222 txkAAtxy
Wave 1
Wave 2Superposition!
The intensity of the new wave is proportional to A12 squared!
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PHY 2053 Page 3
Traveling Waves: Superposition
)sin(cos2
)2/sin()2/sin(
),(),(),( 2112
tkxA
tkxAtkxA
txytxytxy
fff
• Wave Superposition: Consider two waves with the same amplitude, frequency, and wavelength but with an overall phase difference of = f.
y-axis
A
A
)2/sin()sin(),( 11 f tkxAAtxy)2/sin()sin(),( 22 f tkxAAtxy
New intensity!
Superposition!
f 12
)sin(),( 1212 tkxAtxy
cos2cos212 AAA f
2212
2221212
221
cos4cos4
cos4
III
AAI
AIII
ff
New amplitude!
sinA+sinB = 2sin[(A+B)/2]cos[(A-B)/2]
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PHY 2053 Page 4
Traveling Waves: Interference• Maximal Constructive Interference: Consider two waves with the same
amplitude, frequency, and wavelength but with an overall phase difference of = 2n, where n = 0, ±1, ±2,…
y-axis
A A
Max Constructive!
n212
)sin(2),(12 tkxAtxy
II 412
,2,1,0 n
• Maximal Destructive Interference: Consider two waves with the same amplitude, frequency, and wavelength but with an overall phase difference of = +2n, where n = 0, ±1, ±2,…
y-axis
A
A
n 212
0),(12 txy
012 I Max Destructive!
,2,1,0 n
III 21
III 21
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Example Problem: Superposition• Two traveling pressure waves (wave A and wave B) have the same frequency and
wavelength. The waves are superimposed upon each other. The amplitude of the resulting wave (wave C) is 13 kPa. If the amplitude of wave A is 12 kPa and the phase difference between wave B and wave A is fB – fA = 90o, what is the amplitude of wave B and the magnitude of the phase difference between wave A and wave C, respectively?
Answer: 5 kPa, 22.62o
AA 90o
AB
222CBA AAA
62.22 AC
kPakPakPaAAA ACB 5)12()13( 2222
923.013
12cos
C
AAC A
A
AA
AB
AC
AC
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PHY 2053 Page 6
Traveling Waves: Superposition
x = 0
A
= -t
tkd 11
A
1
A 2
Wave 1 distance d1
• Lateral Phase Shift: Consider two waves with the same amplitude, frequency, and wavelength that are in phase at x = 0.
x = 0
A
= -t
tkd 22
dkddk )( 1212
Wave 2 distance d2
n
k
nd
ndk
2
2
,2,1,0 n
)()2(
2
21
nk
nd
ndk
,2,1,0 n
Max Constructive
Max Destructive
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Examples: Superposition
d = max constructive
Wave Superposition
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 1 2 3 4 5 6 7 8
kx (radians)ysum = y1 + y2
Wave Superposition
-1.0
-0.5
0.0
0.5
1.0
0 1 2 3 4 5 6 7 8
kx (radians)ysum = y1 + y2
Wave Superposition
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0 1 2 3 4 5 6 7 8
kx (radians)ysum = y1 + y2
d = max destructive
d = /4
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Example Problem: Superposition
• The figure shows four isotropic point sources of sound that are uniformly spaced on the x-axis. The sources emit sound at the same wavelength and the same amplitude A, and they emit in phase. A point P is shown on the x-axis. Assume that as the sound waves travel to the point P, the decrease in their amplitude is negligible. What is the amplitude of the net wave at P if d = /4?
Answer: Zero
d d
S1 S2 S3 P S4
d x
xd
dxd
dxd
dxd
4
3
2
1
2
3 2
)4/(222 343434
dd
dk
4
3
2
1
2
)4/(222 232323
dd
dk Max Destructive
2
)4/(222 121212
dd
dk
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PHY 2053 Page 9
Example Problem: Superposition
• Sound with a 40 cm wavelength travels rightward from a source and through a tube that consists of a straight portion and a half-circle as shown in the figure. Part of the sound wave travels through the half-circle and then rejoins the rest of the wave, which goes directly through the straight portion. This rejoining results in interference. What is the smallest radius r that results in an intensity minimum at the detector?
Answer: 17.5 cm
Point A Point B
At point A the waves have the same amplitude, wavelength, and frequency and are in phase.
Wave 1 travels a distance d1 = 2r to reach the point B, while wave 2 travels a distance d2 = r to reach the point B.
)()2( 21
12 nrddd ,2,1,0 n Max Destructive
cmcmn
r 5.17)2(2
)40(
)2(2)2(
)(min
21
