1. Resonance Standing Waves Overtones & Harmonics
                                                                                 
Lecture 5
Resonance
Standing Waves
Overtones & Harmonics
Instructor: David Kirkby

2. Review of Lecture 4
We 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. Resonance
Every 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.
Energy dissipates as fast as it is added
Energy builds up and is stored
Energy dissipates as fast as it is added
Physics of Music, Lecture 5, D. Kirkby

4. One way to add energy to a system is periodically, i. e
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 Swing
How 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. Try these online demonstrations…
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 Damping
Why 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 Frequencies
A 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
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 Resonance
A resonance curve measures how much total energy builds up when a fixed (small) amount of energy is delivered periodically. It is described the the mathematical function:
y(x) = 1/(1+x2)
log(Driving Frequency)
Energy Buildup
logarithmic axis!
just right
too
slow
too
fast
Physics of Music, Lecture 5, D. Kirkby

11. Sidebar on Logarithmic Graph Axes
Moving 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 to logarithmically-spaced frequencies.
Therefore a piano keyboard or a musical staff are actually logarithmic axes in disguise!
Energy Buildup
Physics of Music, Lecture 5, D. Kirkby

13. Damping and Resonance Quality
The 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
(curves are rescaled to all go through this point)
Normalized Energy Buildup
log(Driving freq. / Fundamental freq.)
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 Shift
If 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, the swing 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 Ropes
We 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 Resonator
This 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-Nodes
As you look along a standing wave, you find two extremes of motion which have special names:
Node: rope never moves
Antinode: rope undergoes maximum 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 the rope has many resonant frequencies. These resonant frequencies correspond to special wavelengths: 2L L
n = 2 x L / n n = 0,1,2,… L = length
2/3 L
L/2
Physics of Music, Lecture 5, D. Kirkby

21. Harmonic Series fn = v / n = n x v = n x f0 2 x L
The frequencies corresponding to these special wavelengths are:
fn = v / n
= n x v = n x f0
2 x L
v = wave propagation speed
f0 = v /(2 x L) is the fundamental frequency. f1, f2, f3,… are the overtone frequencies. Overtones that follow this particularly simple pattern are called harmonics.
Physics of Music, Lecture 5, D. Kirkby

22. Fundamental, Overtones, Harmonics
The 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.
Physics of Music, Lecture 5, D. Kirkby

23. Harmonic vs Inharmonic Overtonesf0
frequency f1 f2 f3 f4 f5 f6
fundamental
1st overtone
2nd overtone
3rd overtone
1st harmonic
4th overtone
5th overtone
6th overtone
2nd harmonic
3rd harmonic
4th harmonic
5th harmonic
6th harmonic
7th harmonic
Harmonic
Inharmonic
Harmonics are equally spaced on a linear scale
Physics of Music, Lecture 5, D. Kirkby

24. 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
Physics of Music, Lecture 5, D. Kirkby

25. Harmonic Frequencies as Musical Notes
Suppose the fundamental frequency f0 of a harmonic resonator corresponds to a C on the piano. What notes do the harmonic overtones correspond to?
fn = n f0
(n = overtone #) C D E F G A B f1 f2 f3 f4 f5 f0
Notice how the harmonics are not evenly spaced out as they would be on a linear scale. This reflects the fact that musical notes are logarithmically scaled.
Physics of Music, Lecture 5, D. Kirkby

26. Harmonic Frequency Ratios
Any two harmonics (indexed by their overtone numbers n and m) have a definite frequency ratio:
fn = n fm m
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 Intervals
A musical interval is a fixed frequency ratio. The harmonic frequencies contain most of the common musical intervals: C D E F G A B f1 f2 f3 f4 f5 f0
Octave (1:2)
Fifth (2:3)
Fourth (3:4)
Major3rd (4:5)
Minor3rd (5:6)
Doubling the frequency of any note corresponds to a new note that is one octave higher, etc.
Physics of Music, Lecture 5, D. Kirkby

28. Musical Intervals on a Stretched String
We 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 = 25cm
Octave higher
Second Harmonic: L = 16.7cm
Fifth higher
Third Harmonic: L = 12.5cm
Fourth higher
Physics of Music, Lecture 5, D. Kirkby

29. Boundary Conditions
We 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 f with n = 1,3,5,…
Physics of Music, Lecture 5, D. Kirkby

31. Air Columns as Resonators
The 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
Physics of Music, Lecture 5, D. Kirkby

33. Demonstration: Singing Rod
A 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 different types of motion, their fundamental frequencies have no simple 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 Forces
The 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 Sources
Plucking 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 simultaneous vibrations (of air, a string, a rod, …) covering a continuous 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. Summary
Resonance 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 Questions What do logarithms have to do with piano keyboards?
What are the resonators responsible for the production of musical sound in each of these instruments?
Physics of Music, Lecture 5, D. Kirkby

38. Can a string vibrate at more than one frequency at once
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?
Physics of Music, Lecture 5, D. Kirkby