Post on 17-Jan-2016
Standing Waves and Resonance
Standing Wave: “Standing waves” are formed from two or more traveling waves that collide and are “in tune” with one another in such a way that their amplitudes add or subtract in repetitive ways. This is called “Resonance”.
Wave moving right
Wave moving left
Standing Waves and ResonanceGiven a string whose length is L:
The longest standing wave possible is called the “Fundamental”, or “1st harmonic”.
L
Standing Waves and Resonance
What fraction of a sine wave is this?
L
Standing Waves and Resonance
L
What fraction of a sine wave is this? ½ wave
Therefore, using “ L”, how long is one full sine wave (l)
Standing Waves and Resonance
What fraction of a sine wave is this? ½ wave
Therefore, using “ L”, how long is one full sine wave (l) = 2L
L
Standing Waves and Resonance
1st harmonic
2nd harmonic
3rd harmonic
4th harmonic
Standing Waves and Resonance
1st Harmonic:
2nd Harmonic:
3rd Harmonic:
4th Harmonic:
L
l = 2L
l = ___?
l = ___?
l = ___?
Standing Waves and Resonance
1st Harmonic:
2nd Harmonic:
3rd Harmonic:
4th Harmonic:
L
l = 2L
l = ___?
l = ___?
l = L
Standing Waves and Resonance
1st Harmonic:
2nd Harmonic:
3rd Harmonic:
4th Harmonic:
L
l = 2L
l = ___?
l = L
l = 2L/3
Standing Waves and Resonance
1st Harmonic:
2nd Harmonic:
3rd Harmonic:
4th Harmonic:
L
l = 2L
l = L
l = 2L/3
l = L/2
Standing Waves and Resonance
What’s the pattern here?
1
22
LL
2
2LL
3
232 LL
4
2
2
LL
n
L2
,...4,3,2,1n
In general,
where:
Resonant wavelength formula
Standing Waves and ResonanceNomenclature:
Node
Antinode
Node: a region of zero amplitude
Antinode: a region of maximum amplitude
Standing Waves and ResonanceResonant Frequencies Of Harmonic Standing
Waves:
We learned l = 2L/n since v = fl,
n
Lfv
2
Standing Waves and Resonance
)2( Lfnv
n
Lfv
2
Resonant Frequencies Of Harmonic Standing Waves:
We learned l = 2L/n since v = fl,
Standing Waves and Resonance
fvL
n
2
n
Lfv
2
)2( Lfnv
Resonant Frequencies Of Harmonic Standing Waves:
We learned l = 2L/n since v = fl,
Standing Waves and Resonance
fvL
n
2
n
Lfv
2
)2( Lfnv
vL
nfn
2,...4,3,2,1nwhere:
Resonant frequency formula
Resonant Frequencies Of Harmonic Standing Waves:
We learned l = 2L/n since v = fl,
Standing Waves and ResonanceTherefore the 1st resonant frequency
corresponds to the 1st resonant wavelength…
--- 1st Harmonic
--- 2nd Harmonic
--- 3rd Harmonic
--- 4th Harmonic
1
21
L
2
22
L
3
23
L
4
24
L
vL
f
2
11
vL
f
2
22
vL
f
2
33
vL
f
2
44
Standing Waves and Resonance
Standing Waves and Resonance
What two physical characteristics play a role in determining the velocity on a string?
Standing Waves and Resonance
What two physical characteristics play a role in determining the velocity on a string?
1. Tension
2. Mass, or inertia
Standing Waves and Resonance
What two physical characteristics play a role in determining the velocity on a string?
1. Tension
2. Mass, or inertia
What do we expect the relationship with velocity to be?
Standing Waves and Resonance
What two physical characteristics play a role in determining the velocity on a string?
1. Tension
2. Mass, or inertia
We expect the relationship to be as follows:
massv
Fv
1
Standing Waves and Resonance
The mass is applied by using density; specifically linear density:
length
masslinear density
surface density
volume density
area
mass
volume
mass
Standing Waves and Resonance
Finally, the equation used to find velocity can be described:
F
v F = tension
m = linear density
Units = m/s
Standing Waves and Resonance
Wave Velocity on string:
F
v
where F = Forcestring oflength
string of mass
Problem: A force of 45 Newtons is applied to a string with a mass of 0.012 kg and a length of 1.5 meters. Find the velocity of a wave on that string:
Standing Waves and ResonanceSound Velocity on string:
F
v
where F = Force , and
string oflength
string of mass
Example: A force of 45 Newtons is applied to a string with a mass of 0.012 kg and a length of 1.5 meters. Find the velocity of a wave on that string:
m/s 75/5625008.0
4545
5.1012.0
kgNmNNF
vmkg