The Physics of Music - Physics 15 University of California, Irvine ...

The Physics of Music - Physics 15 

University of California, Irvine 

Instructor: David Kirkby 

dkirkby@uci.edu 

Review of Lecture 4 

We looked at wave refraction and diffraction. 

Lecture 5 

• Resonance 

• Standing Waves 

• Overtones & Harmonics 

We explored how waves propagate in two dimensions. 

We learned how the sound from a moving source appears 

to change its frequency (Doppler effect). 

Instructor: David Kirkby (dkirkby @uci.edu) 

Physics of Music, Lecture 5, D. Kirkby 2 

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. 

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). 

Energy 

dissipates 

as fast as 

it is added 

Energy 

builds up 

and is 

stored 

Energy 

dissipates 

as fast as 

it is added 



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. 

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. 



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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. 

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. 



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. 

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+x 2 ) 

Energy Buildup 

too 

slow 

just right 

log(Driving Frequency) 

logarithmic axis! 

too 

fast 

http://www.2dcurves.com/cubic/cubicr.html 



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 

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 



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Damping and Resonance Quality 

The amount of damping determines how long a sound takes 

to die away when you stop adding energy. 

Resonance Curves of Different Q 

(curves are rescaled to all go 

through this point) 

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. 

Normalized Energy Buildup 

log(Driving freq. / Fundamental freq.) 



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. 

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 180 o out of phase (or just out of phase). 

Physics of Music, Lecture 5, D. Kirkby 15 


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). 

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. 



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Nodes and Anti-Nodes 

As you look along a standing wave, you find two extremes 

of motion which have special names: 

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. 

Node: rope never moves 

Antinode: rope undergoes 

maximum motion 

One new feature is that the 

rope has many resonant 

frequencies. These resonant 

frequencies correspond to 

special wavelengths: 

l n 

= 2 x L / n 

n = 0,1,2,… 

L = length 

2 / 3 L 

L/2 

L 

2L 



Harmonic Series 

The frequencies corresponding to these special 

wavelengths are: 

f n 

= v / l n 

= n x v 

2 x L 

= n x f 0 

v = wave 

propagation 

speed 

f 0 = v /(2 x L) is the fundamental frequency. f 1 , f 2 , f 3 ,… 

are the overtone frequencies. Overtones that follow this 

particularly simple pattern are called harmonics. 

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 f n = n f 0 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. 



Inharmonic 

Harmonic vs Inharmonic Overtones 

f 0 

Harmonic 

fundamental 

1 st overtone 

2 nd overtone 

3 rd overtone 

4 th overtone 

5 th overtone 

6 th overtone 

f 1 f 2 f 3 f 4 f 5 f 6 frequency 

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). 

1 st harmonic 

2 nd harmonic 

3 rd harmonic 

4 th harmonic 

5 th harmonic 

6 th harmonic 

7 th harmonic 

Example: a tam-tam 

Harmonics are equally spaced on a linear scale 



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Harmonic Frequencies as Musical Notes 

Suppose the fundamental frequency f 0 of a harmonic 

resonator corresponds to a C on the piano. What notes do 

the harmonic overtones correspond to 

f n = n f 0 (n = overtone #) 

Harmonic Frequency Ratios 

Any two harmonics (indexed by their overtone numbers n 

and m) have a definite frequency ratio: 

f n = n f m 

m 

What does multiplying by a fixed amount look like on a 

logarithmic axis 

C D E F G A B C D E F G A B C D E F G A B 

f 0 f 1 f 2 f 3 f 4 f 5 

What about on a piano keyboard 

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. 



Musical Intervals 

A musical interval is a fixed frequency ratio. The harmonic 

frequencies contain most of the common musical intervals: 

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: 

C D E F G A B C D E F G A B C D E F G A B 

f 0 f 1 f 2 f 3 f 4 f 5 

Octave 

(1:2) 

Fifth 

(2:3) 

Fourth Minor3 rd 

(3:4) (5:6) 

Major3 rd 

(4:5) 

Doubling the frequency of any note corresponds to a new 

note that is one octave higher, etc. 


Fundamental: L = 50cm 

First Harmonic: L = 25cm 

Octave higher 

Second Harmonic: L = 16.7cm 

Fifth higher 

Third Harmonic: L = 12.5cm 

Fourth higher 


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!) 

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: f 0 = v /(4 x L) 

(2) The even harmonics are forbidden: f n = n f 0 

with n = 1,3,5,… 



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Air Columns as Resonators 

The air contained within a pipe can resonate just like a 

string. What are the corresponding boundary conditions 

Nodes and Anti-Nodes in an Air Column 

(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. 



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) 

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. 

Since these two resonances involve fundamentally different 

types of motion, their fundamental frequencies have no 

simple relationship. 

We were able to excite resonances in all three cases 

without paying attention to the frequency at which 

energy was provided. Why 

Watch and listen to the vibrations of an aluminum rod. 

What were the boundary conditions 



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. 

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. 



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Review Questions 

What do logarithms have to do with piano keyboards 

Can a string vibrate at more than one frequency at once 

What frequencies are possible for an idealized string 

What are the resonators responsible 

for the production of musical sound 

in each of these instruments 

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 



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The Physics of Music - Physics 15 University of California, Irvine ...

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