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Spring 2006 – Week 5 PHY 131 1
Week 5
Standing waves in Air Columns
Spring 2006 – Week 5 PHY 131 2
Standing Waves in the Bathtub
Snapshot of at a time when displacement is max.
Snapshot one half-period later (T/2 later)
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Spring 2006 – Week 5 PHY 131 3
High Pressure
Standing Waves in the Bathtub
Low Pressure
Spring 2006 – Week 5 PHY 131 4
Standing Waves in the Bathtub
Low PressureHigh Pressure
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Spring 2006 – Week 5 PHY 131 5
Both Ends Open: 1. Mode
Spring 2006 – Week 5 PHY 131 6
Both Ends Open: 1. Mode
f1 =v
2L
14T & 3
4T
12T
0 & T
phase
λ1 = 2L wavelength
frequency
4
Spring 2006 – Week 5 PHY 131 7
Both Ends Open: 2. Mode
Spring 2006 – Week 5 PHY 131 8
Both Ends Open: 2. Mode
f2 = 2 v2L
= 2 f1
14T & 3
4T
12T
0 & T
phase
λ2 = L =12
2L wavelength
frequency
5
Spring 2006 – Week 5 PHY 131 9
Both Ends Open: 3. Mode
Spring 2006 – Week 5 PHY 131 10
Both Ends Open: 3. Mode
f3 = 3 v2L
= 3 f1
14T & 3
4T
12T
0 & T
phase
λ3 =13
2L wavelength
frequency
6
Spring 2006 – Week 5 PHY 131 11
One End Closed: 1. Mode
Spring 2006 – Week 5 PHY 131 12
One End Closed: 1. Mode
f1 =v
4L
14T & 3
4T
12T
0 & T
phase
λ1 = 4L wavelength
frequency
7
Spring 2006 – Week 5 PHY 131 13
One End Closed: 2. Mode
Spring 2006 – Week 5 PHY 131 14
One End Closed: 2. Mode
f2 = 3 v4L
14T & 3
4T
12T
0 & T
phase
λ2 =43L
wavelength
frequency
8
Spring 2006 – Week 5 PHY 131 15
One End Closed: 3. Mode
Spring 2006 – Week 5 PHY 131 16
One End Closed: 3. Mode
f3 = 5 v4L
14T & 3
4T
12T
0 & T
phase
λ3 =45L
wavelength
frequency
9
Spring 2006 – Week 5 PHY 131 17
Air Column(one end closed)
Air Column(both ends open)
Thin Rod(one end fixed)
Strings(both ends fixed)
Strings, Rods, and Air Columns
λn =1n2L fn = n
v2L
(n =1,2,3...)
fn = (2n-1) v4L
(n=1,2,3...)λn =1
2n −14L
λn =1n2L fn = n
v2L
(n =1,2,3...)
fn = (2n-1) v4L
(n=1,2,3...)λn =1
2n −14L
Spring 2006 – Week 5 PHY 131 18
What do they have in common?
Loose string/rod and closed end of pipemust have an antinode
Fixed string/rod and open end of pipemust have an node
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Spring 2006 – Week 5 PHY 131 19
Strings, Rods, and Air Columns
λ = 2L, 12
2L, 13
2L, 14
2L... f = 1 v2L
, 2 v2L
, 3 v2L
, 4 v2L
...
Strings (both ends fixed) Air Column (both ends open)
Rod (one end fixed) Air Column (one end closed)
λ = 4L, 13
4L, 15
4L, 17
4L... f = 1 v4L
, 3 v4L
, 5 v4L
, 7 v4L
...
Spring 2006 – Week 5 PHY 131 20
Some Consequences…
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Spring 2006 – Week 5 PHY 131 21
Short Blades – Long Blades
The shorter the blade, the higher theresonance frequency
f1 =v
4L
Spring 2006 – Week 5 PHY 131 22
Same length (nearly) but clarinet is much lower
Clarinet vs. Flute Playoff
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Spring 2006 – Week 5 PHY 131 23
Tuning Fork Resonator
How Long?
?
Spring 2006 – Week 5 PHY 131 24
Resonators on a Marimba
Resonators are closed at one end
… and open atthe other end
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Spring 2006 – Week 5 PHY 131 25
Resonators on a MarimbaThis Resonator Tube is 63 cm long
Resonators are closed at one end
Spring 2006 – Week 5 PHY 131 26
Forcing the 3. Mode on a String
