lecture 5 resonance standing waves overtones & harmonics instructor: david kirkby...
TRANSCRIPT
Physics of Music, Lecture 5, D. Kirkby 2
Review of Lecture 4We looked at wave refraction and diffraction.
We explored how waves propagate in two dimensions.
We learned how the sound from a moving source appears to change its frequency (Doppler effect).
Physics of Music, Lecture 5, D. Kirkby 3
ResonanceEvery time you add energy to a system, it gradually dissipates. This is damping (see Lecture 3).
The way in which you add energy can influence how rapidly it dissipates.
An analogy: filling up a tapered cylinder.
Energydissipatesas fast asit is added
Energybuilds upand isstored
Energydissipatesas fast asit is added
Physics of Music, Lecture 5, D. Kirkby 4
One way to add energy to a system is periodically, i.e., in small packets delivered at a constant frequency.
Resonance is a build-up of energy when it is delivered at a particular frequency.
(Frequency plays the role of the ball size in the previous tapered cylinder example).
Physics of Music, Lecture 5, D. Kirkby 5
Example: A Playground SwingHow do you get a swing going?
The usual technique is to deliver energy by rotating your body in synch with the swing’s motion.
Physics of Music, Lecture 5, D. Kirkby 6
Most people can get a swing going, but what would happen if you deliberately pumped at the wrong frequency?
Try these online demonstrations…
Pumping the swing at just the right frequency leads to a build-up of energy that gets the swing higher off the ground.
This is an example of resonance.
Physics of Music, Lecture 5, D. Kirkby 7
Resonance and DampingWhy doesn’t the swing keep getting higher and higher until you are doing circles?
An idealized resonant response builds an unlimited amount of energy.
Realistic resonant systems do not do this because of dissipation, i.e., they are damped.
Compare the motion of the swing when it is pumped at the right frequency but with different amounts of damping.
Physics of Music, Lecture 5, D. Kirkby 8
Resonant FrequenciesA physical system may have one or more frequencies at which resonances build up. These are called resonant frequencies (or natural frequencies).
The basic requirements for a system to be resonant are that:
•It have well-defined and stable boundary conditions,•That it not have excessive damping.
This means that most systems have at least one type of resonance!
Resonant frequencies are often in the audible range (about 20-20,000 Hz). Try tapping an object to hear its resonant response.
Physics of Music, Lecture 5, D. Kirkby 9
A system may have more than one resonant frequency.
We call the lowest resonant frequency the fundamental frequency. Any higher frequencies are called overtones.
The playground swing has only one resonant frequency.
Most of the systems responsible for generating musical sound have many resonances.
We will see examples of systems with overtones later in this lecture. A familiar (non-musical) example occurs when different parts of a car rattle at certain speeds.
Physics of Music, Lecture 5, D. Kirkby 10
Visualizing ResonanceA resonance curve measures how much total energy builds up when a fixed (small) amount of energy is delivered periodically.
It is described the themathematical function:
y(x) = 1/(1+x2)
http://www.2dcurves.com/cubic/cubicr.html
log(Driving Frequency)En
erg
y B
uild
up
logarithmic axis!
tooslow
toofast
just right
Physics of Music, Lecture 5, D. Kirkby 11
Sidebar on Logarithmic Graph AxesMoving one unit to the right on a normal (linear) graph axis means add a constant amount.
Moving one unit to the right on a logarithmic axis means multiply by a constant amount.
Example: the exponential decay law (e.g., from damping) results in a decrease by a fixed fraction after each time interval.
What would this look like if time is plotted on a logarithmic axis?
Physics of Music, Lecture 5, D. Kirkby 12
Musical notes (A,B,C,…,G) correspond tologarithmically-spaced frequencies.
Therefore a piano keyboard or a musicalstaff are actually logarithmic axesin disguise!
En
erg
y B
uild
up
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Damping and Resonance QualityThe amount of damping determines how long a sound takes to die away when you stop adding energy.
It also determines how sharply peaked the resonance curve is.
We measure this sharpness with a “quality factor” or“Q-value”:
Q = resonant frequency / curve width
We say that a sharply peaked resonance curve corresponds to a “High-Q” resonator and that a broad resonance curve corresponds to a “Low-Q” resonator.
Physics of Music, Lecture 5, D. Kirkby 14
Resonance Curves of Different Q
log(Driving freq. / Fundamental freq.)
Norm
aliz
ed
En
erg
y B
uild
up
(curves are rescaled to all gothrough this point)
Physics of Music, Lecture 5, D. Kirkby 15
Go back to the swing demonstration to see the effect of changing the amount of damping.
The famous Tacoma Narrows disaster is an example of a complicated mechanical system that had a resonance (driven by wind) of an unexpectedly high Q.
Physics of Music, Lecture 5, D. Kirkby 16
Resonance and Phase ShiftIf you are pumping a swing below its resonant frequency, the swing responds in synch (in phase) with your pumping.
What happens if you pump faster than the swing’s resonant frequency?
Go back to the swing demonstrations to find out…
At frequencies above the resonant frequency, the motion of the swing lags behind. Far above the resonance, theswing motion is the negative of the driving force. In this case, we say that the driving force and the swing motion are 180o out of phase (or just out of phase).
Physics of Music, Lecture 5, D. Kirkby 17
Back to One Dimensional RopesWe have already considered different boundary conditions at one end of a rope.
We assumed that the rope was long enough that we could ignore its other end.
What if the rope is not so long and we allow reflections from both ends? For example, one end might be fixed and the other held (which means fixed + driven).
Physics of Music, Lecture 5, D. Kirkby 18
The Rope is a ResonatorThis is just a combination of boundary conditions that we have seen before, but a fundamentally new feature emerges: resonance!
The source of periodic energy is the person wiggling one of the rope at a fixed frequency.
The buildup of energy is evident in the amplitude of the rope’s transverse motion.
The resonant response is called a standing wave.
Try this demo to see for yourself.
Physics of Music, Lecture 5, D. Kirkby 19
Nodes and Anti-NodesAs you look along a standing wave, you find two extremes of motion which have special names:
Node: rope never moves
Antinode: rope undergoesmaximum motion
Physics of Music, Lecture 5, D. Kirkby 20
Comparison of Swing and Rope Resonances
In most ways, the two resonances are identical: resonance is another example of a universal pattern that repeats throughout many physical processes.
One new feature is that therope has many resonantfrequencies. These resonantfrequencies correspond tospecial wavelengths:
n = 2 x L / n
n = 0,1,2,…L = length
2L
L
2/3 L
L/2
Physics of Music, Lecture 5, D. Kirkby 21
Harmonic SeriesThe frequencies corresponding to these special wavelengths are:
f0 = v /(2 x L) is the fundamental frequency. f1, f2, f3,…are the overtone frequencies. Overtones that follow thisparticularly simple pattern are called harmonics.
fn = v / n
= n x v
= n x f0
2 x L
v = wave propagation speed
Physics of Music, Lecture 5, D. Kirkby 22
Fundamental, Overtones, HarmonicsThe definitions of these three terms are easy to confuse.
There is only one fundamental. It is the lowest resonant frequency of a system.
Any higher resonant frequencies are called overtones (but the lowest resonant frequency is not an overtone).
If the resonant frequencies (almost) obey fn = n f0 we call them harmonics.
The first harmonic is the same as the fundamental. The second harmonic is the same as the first overtone. The numberings of harmonics and overtones are off by one.
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Harmonic vs Inharmonic Overtones
f0 frequencyf1 f2 f3 f4 f5 f6
fun
dam
en
tal
1st o
vert
on
e
2nd o
vert
on
e
3rd o
vert
on
e
1st h
arm
on
ic
4th o
vert
on
e
5th o
vert
on
e
6th o
vert
on
e
2nd h
arm
on
ic
3rd h
arm
on
ic
4th h
arm
on
ic
5th h
arm
on
ic
6th h
arm
on
ic
7th h
arm
on
ic
Harmonic
Inharmonic
Harmonics are equally spaced on a linear scale
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Most musical instruments have overtones that are at least approximately harmonic. We will soon see how our brain exploits this fact in the way it processes sound.
However, percussion instruments generally have inharmonic overtones. This fact makes it hard for us to associate a percussive sound with a particular frequency (musical note).
Example: a tam-tam
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C D E F G A B C D E F G A B C D E F G A B
Harmonic Frequencies as Musical NotesSuppose the fundamental frequency f0 of a harmonic resonator corresponds to a C on the piano. What notes do the harmonic overtones correspond to?
f0 f1 f3f2 f4 f5
Notice how the harmonics are not evenly spaced out asthey would be on a linear scale. This reflects the factthat musical notes are logarithmically scaled.
fn = n f0 (n = overtone #)
Physics of Music, Lecture 5, D. Kirkby 26
Harmonic Frequency RatiosAny two harmonics (indexed by their overtone numbers n and m) have a definite frequency ratio:
fn = n fmm
What does multiplying by a fixed amount look like on a logarithmic axis?
What about on a piano keyboard?
Physics of Music, Lecture 5, D. Kirkby 27
Musical IntervalsA musical interval is a fixed frequency ratio. The harmonic frequencies contain most of the common musical intervals:
C D E F G A B C D E F G A B C D E F G A B
f0 f1 f3f2 f4 f5
Octave(1:2)
Fifth(2:3)
Fourth(3:4)
Major3rd
(4:5)
Minor3rd
(5:6)
Doubling the frequency of any note corresponds to a newnote that is one octave higher, etc.
Physics of Music, Lecture 5, D. Kirkby 28
Musical Intervals on a Stretched StringWe can reproduce the notes of the harmonic frequency series by listening to the fundamental frequency of a string whose length is varied according to:
Fundamental: L = 50cm
First Harmonic: L = 25cmOctave higher
Second Harmonic: L = 16.7cmFifth higher
Third Harmonic: L = 12.5cmFourth higher
Physics of Music, Lecture 5, D. Kirkby 29
Boundary ConditionsWe analyzed the string with both ends fixed (the end being held is considered fixed as far as reflections are concerned).
This is an example of a boundary condition, and leads to standing waves which have nodes (no motion) at each end.
What are some other possible boundary conditions?
(1) One end fixed, the other free.
(2) Both ends free (hard to do but easy to imagine!)
Physics of Music, Lecture 5, D. Kirkby 30
Try this online demonstration of a rope with one end free.
The new boundary condition at the free end is that it must be an anti-node. This has two effects on the resonant frequencies:
(1) The fundamental frequency is 2 times lower than for the rope with both ends fixed: f0 = v /(4 x L)
(2) The even harmonics are forbidden: fn = n f0
with n = 1,3,5,…
Physics of Music, Lecture 5, D. Kirkby 31
Air Columns as ResonatorsThe air contained within a pipe can resonate just like a string. What are the corresponding boundary conditions?
(1) fixed + free ends
(2) two free ends
(3) two fixed ends
……open + closed ends
……two open ends
……two closed ends (!)
Listen to the heated “hoot tube” demonstration for an example of resonance in a tube open at both ends.
Physics of Music, Lecture 5, D. Kirkby 32
Nodes and Anti-Nodes in an Air Column
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Demonstration: Singing RodA long aluminum rod can sustain two kinds of vibrations:
•Longitudinal (squeezing & stretching along its length)
•Transverse (bending transverse to its length)
Since these two resonances involve fundamentally differenttypes of motion, their fundamental frequencies have nosimple relationship.
Watch and listen to the vibrations of an aluminum rod.What were the boundary conditions?
Physics of Music, Lecture 5, D. Kirkby 34
Complex Driving ForcesThe demonstrations of singing rods, plucked strings and hoot tubes that you heard today appear to be missing one of the crucial ingredients for resonance:
That energy is provided periodically at a constant driving frequency.
We were able to excite resonances in all three cases without paying attention to the frequency at which energy was provided. Why?
Physics of Music, Lecture 5, D. Kirkby 35
Noisy Energy SourcesPlucking a string, heating the air near a metal mesh, and drawing your fingers along a rod are all examples of noisy energy sources.
Noise is the superposition of many simultaneousvibrations (of air, a string, a rod, …) covering acontinuous range of frequencies.
Since no single frequency dominates, we do not hear a definite pitch, even though all frequencies are present!
Since all frequencies are present in some range, we are guaranteed to excite any resonances present within the range.
Physics of Music, Lecture 5, D. Kirkby 36
SummaryResonance is a buildup of energy when it is delivered at the right frequency.
Many physical systems are resonant. Some have more than one kind of resonant response (eg, the singing rod).
A system may have several resonant frequencies for the same type of response.
Examples of resonance: swing, rope fixed at both end, air column, aluminum rod.
Physics of Music, Lecture 5, D. Kirkby 37
Review QuestionsWhat do logarithms have to do with piano keyboards?
What are the resonators responsiblefor the production of musical soundin each of these instruments?
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Can a string vibrate at more than one frequency at once? What frequencies are possible for an idealized string?
Do you actually need to drive a guitar string at its harmonic frequency in order to set up a standing wave that you can hear?
Why did we stop at the 5th overtone when looking at harmonics and musical intervals on the piano keyboard?