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Chapter 21
Superposition and standing waves
http://www.phy.ntnu.edu.tw/java/waveSuperposition/waveSuperposition.html
2
When waves combineTwo waves propagate in the same medium
Their amplitudes are not largeThen the total perturbation is the sum of the
ones for each wave
On a string:Wave 1: y1
Wave 2: y2
Total wave: y1+y2
To an excellent approximation: waves
just add up
3
The combination of waves can give very interesting effects
On a string:Wave 1: y1=A sin(k1x-ω1t+1)
Wave 2: y2=A sin(k2x- ω2t+ 2)
Now suppose thatk1=k+Δk k2=k- Δk
ω 1= ω + Δω ω 2= ω - Δω 1= + Δ 2= - Δ
note: 2- 1=2Δ
And usesin(a+b) + sin(a-b)=2 sin(a) cos(b)
to get
y = y1 + y2 = 2A cos(Δk x – Δω t + Δ) sin(kx – ωt + )
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Assume Δk=0 Δω =0
y = [2A cos(Δ ) ] sin(kx – ωt + )
The amplitude can be between 2A and 0, depending on Δ
When Δ =0
y = [2A ] sin(kx – ωt + )The waves add up
The phase difference in WAVELENTGHS is 0
When Δ =π/2 or 2- 1=2Δ =π
y = 0The waves cancel out
The phase difference in WAVELENTGHS is λ/2
5
More Generally
y = [2A cos(Δk x – Δω t + Δ) ] sin(kx – ωt + )
Behaves as a slowly changing amplitude
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Two waves are observed to interfere constructively. Their phase difference, in radians, could be
a. π/2.
b. π.
c. 3π.
d. 7π.
e. 6π.
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Standing wavesA pulse on a string with both ends attached
travels back and forth between the ends
If we have two waves moving in opposite directions:Wave 1: y1=A sin(kx- ωt)Wave 2: y2=A sin(kx+ ωt)
y = y1 + y2 = 2A sin(k x) cos(ωt)
If this is to survive for an attached string of length Ly(x=0)=y(x=L)=0
L
sin(k L)=0
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2 ,
2
1 ,
3
2 , ,2
4 ,3 ,2 ,
LLLLL
kL
ππππ
If λ does not have these values the back and forth motion will produce destructive interference.
For example, if λ=L/2
λ/2
applet
8
A pulse on a string with both ends attached travels back and forth between the ends
Standing wave:y = 2A sin(k x) cos(ω t)
λ=2L/n, n=1,2,3,…
1 2 3.2
v n Tf n , , ..
L
Smallest f is the fundamental frequency
The others are multiples of it or, harmonics
T
Lf
2
1fund
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14.9• Two speakers are driven in phase by the same oscillator
of frequency f. They are located a distance d from each other on a vertical pole as shown in the figure. A man walks towards the speakers in a direction perpendicular to the pole.
• a) How many times will he hear a minimum in the sound intensity
• b) How far is he from the pole at these moments? Let v represent the speed for sound and assume that the ground is carpeted and will not reflect the sound
L
h
d
h
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Exercise 14.9 This is a case of interferenceI will need Pythagoras theorem
2 21x d L
)2/ cos()cos(2
]2/(cos[])2/(cos[
)cos()cos( 21
xktkXA
txXkAtxXkA
tkxAtkxAy
ωωω
ωω
2/ 2/2
21
2121
xXxxXx
xxXxxx
Cancellations whenkx=…-5p,-3p,-p,p, 3p, 5p …
L
h
d
h2x L
2cos cos cos( ) cos( )x y x y x y
1integer
2 2 2
m m mm odd x n n
k
π π π
13
12
)2/1()/(
)2/1(2
)2/1(
2 ,1 ,0 ,)2/1(
22
22
22
n
ndL
n
ndL
nnLLd
Allowed n:all such that the right hand side is positive
λ=v/f
2 2 1integer
2x d L L n n
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• Destructive interference occurs when the path difference is
A. /2
B. C. 2D. 3E. 4
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14.13
• Two sinusoidal waves combining in a medium are described by the wave functions
y1=(3cm)sin(πx+0.6t)
y2=(3cm)sin(πx-0.6t)
where x is in cm an t in seconds. Describe the maximum displacement of the motion at
a) x=0.25 cm b) x=0.5 cm c) x=1.5 cm
d) Fine the three smallest (most negative) values of x corresponding to antinodes
nodes when x=0 at all times
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Exercise 14.13
Superposition problem
) 6.0cos() sin(6
) 6.0 sin(3) 6.0 sin(3
tx
txtxy
ππππππ
( ) 0 25 : 6sin( / 4)cos(0.6 ) 6sin( / 4) 4.24ma x . y tπ π π
m6) 6.0cos()2/sin(6 :50 )( ty.xb ππ
m6) 6.0cos()2/3sin(6 :51 )( ty.xc ππ
( ) Antinodes: sin( ) 1
3 5 7 cm, cm, cm,
2 2 2
d x
x
π
sin(a+b) + sin(a-b)=2 sin(a) cos(b)
17
Sources of Musical Sound
• Oscillating strings (guitar, piano, violin)• Oscillating membranes (drums)• Oscillating air columns (flute, oboe, organ pipe)• Oscillating wooden/steel blocks (xylophone, marimba)
• Standing Waves-Reflections & Superposition
• Dimensions restrict allowed wavelengths-Resonant Frequencies
• Initial disturbance excites various resonant frequencies http://www.falstad.com/membrane/index.html
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Standing Wave Patterns for Air Columns
Pipe Closed at Both Ends• Displacement Nodes• Pressure Anti-nodes• Same as String
fn nv
2L all n
Pipe Open at Both Ends Displacement Anti-nodes Pressure Nodes (Atmospheric) Same as String
fn nv
2L all n
3. Open One end
fn nv
4L n odd
http://www.physics.gatech.edu/academics/Classes/2211/main/demos/standing_wwaves/stlwaves.htm
2L
n
2L
n
4L
n
beats demo
v/λ=f
19
Organ Pipe-Open Both Ends
A N A
Turbulent Air flow excites large number of harmonics
“Timbre”
n = 1λ = 2L
20
Problem 9
vsound = 331 m/s + 0.6 m/s TC
TC is temp in celcius