Maths Makes Waves
Chris Budd
B
D
  E    M,   H  
 J,
t
t
.D  ,
.B  0.

Waves are a universal phenomenon in science
at all scales
Electron wave 0.5nm
Light pulse
500nm
Microwave
10cm
Sound 50cm
Sand waves
1m
Ocean wave
10m
Gravity and
Rosby waves
10-1000km
Gravitational
waves 1Gm
Aim of talk
1. To give a history of waves
2. To show how maths unites them all
3. To give examples in many applications
2011 celebrated two big wave anniversaries
Possibly the most important wave equation of
all was discovered by Schrodinger in 1926.
Erwin
Schrodinger
1887-1961
Basic equation of quantum mechanics

2
i
 
  V (x, y,z)
t
2m
2
Wave function:
probability distribution of states with different energies
“The 1926 paper has been universally celebrated as one of the most
important achievements of the twentieth century, and created a revolution
in quantum mechanics, and indeed of all physics and chemistry”
Schrodinger used it to compute the spectrum of the
Hydrogen atom.
Now, used everyday in the chips in your mobile phone
But .. waves, and their mathematics, have a long history!
Musical sounds: the first man made waves
Greeks observed that some musical notes from a stringed
instrument sound better when played together than others
The octave C to C
The notes C and G
(a perfect 5th)
The notes C and F
(a perfect 4th)
The notes C and E
(a major 3rd)
Reason was discovered by Pythagoras
Length of strings giving C and G, F and E, were in
simple fractional proportions
C:C … 2/1
C:G … 3/2
C:F … 4/3
C:E … 5/4
Gave an important hint about the underlying physics!
Pythagoras invented the Just Scale .. Sequence of notes
with frequencies in simple fractional proportions
1 9/8 5/4 4/3 3/2 5/3 15/8 2
Why does this work?
Galileo 15-02-1564
Musical notes come from waves
on the strings
Frequency (pitch) of the fundamental note is inversely
proportional to the length of the string
Simplest wave is a sine wave
u(t)  Asin( t)  Asin(2 f t)
Amplitude Angular Frequency
sin( t)
Linked to
triangles!!!
t


Sound waves travel through the air and are sine waves in both space and
time
Wavelength L, Period T, Frequency f = 1/T
Amplitude 2*A
C: Frequency
f = 261.6 Hz
T=3.8ms, L=1.2m
G: Frequency
f = 392 Hz
T=2.5ms, L = 0.8m
Speed
c=fL
c = 320 m/s
C:G
C:F
Lissajous Figures: Show good chords
C:E
E:F
But why are waves sine waves?
Pendulum observed by Galileo in 1600
Newton gives the differential equation
d
d
a 2 b
 c  0
dt
dt
2
Euler finds
the solution
Guess what: it’s a sine wave
t
(t)  e Asin( t   )
Damping

Amplitude Frequency
Phase
One wave good, many waves better!
Joseph Fourier
Any function of period T can be expressed as an infinite
sum of sine waves
Sine waves are natures building blocks!

2 n t 
2 n t 
a0
u(t)    an cos
 bn sin

 T 
 T 
2 n1

Fourier coefficients. By varying these we can change the shape of
the wave
Fourier used this idea to find the
temperature of a heated bar.
Now used EXTENSIVELY in digital
TV, radio, IPods and sound
synthesizers
Eg: Square wave
sin( t)



sin(t) 
sin(3t) sin(5t)


3
5

sin(3t) sin(5t)


3
5

sin(t) 
sin(t) 
sin(3t) sin(5t)

3
5
sin(11t)
11
sin(49t)
49
A useful application of Fourier Analysis
The tides: a global wave
Height of the Bombay tides 1872
h(t)
t
Kelvin decomposed the tidal height
into 37 independent Fourier
components
He found these out using past data and added
them up using an analogue computer
Kelvin Tidal predictor
US Tidal predictor
Kelvin made many other discoveries
concerning waves
utt  uxx
Wave equation: describes waves on a
string and small water waves
This equation describes small waves well
speed
c
g
2
Wavelength
Larger waves
 in shallow water obey a different
equation (the Shallow Water Equation)
IMPORTANT to understand Tsunamis
speed
c  gh
Depth
Almost supersonic in the ocean!!!
utt  uxx  a(x)ux  b(x)u
Helmholtz equation: describes waves in a telegraph cable
and microwave cooking
Kelvin knighted 1866 for his work on
the trans-Atlantic cable
But waves don’t have to go down cables
Maxwell and the discovery of electromagnetic waves
E 
.D  ,
B
D
 M,   H  
 J,
t
t
.B  0.
Maxwell’s
 equations: solutions are waves in space eg. light
What this led to …
Hertz: Practical demonstration of radio waves and
that they were reflected from metallic objects
Marconi: Invention of radio communication
1930 Set up of commercial radio stations
1936 First TV broadcast
1980+ Mobile phones, Wi-Fi
The modern world!!!!
But … is light a wave or a particle?
De Broglie 1924
Davisson, Germer
and Thomson 1927
Discovery of the particle-wave duality of light and matter
Confirmed by electron diffraction
Planks constant
wavelength
h

mv
Momentum
Wave aspect of matter is formalised by the wavefunction defined by
the Schrodinger Equation
The Largest Waves of all
Gravitational waves
Theoretical ripples in the curvature of spacetime
Can be caused by binary star systems composed of
pulsars or black holes
Predicted to exist by Albert Einstein in 1916 on
the basis of the theory of general relativity
Evidence from the Hulse-Taylor binary star system
Study of waves started with wave on strings
String theory brings waves right up to date.
Idea: electrons and quarks within an atom are
made up strings. These strings oscillate, giving
the particles their flavor, charge, mass and spin.
Unified theory giving a possible link
between quantum theory and relativity
… but no direct experimental evidence!
In conclusion:
Waves dominate all aspects of science
They have applications everywhere
Maths helps us to understand them.