Maths Makes Waves

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Maths Makes Waves
Chris Budd
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Waves are a universal phenomenon in science at
all scales
Electron wave 0.5nm
Light pulse 500nm
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Microwave 10cm
Sound 50cm
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Sand waves 1m
Ocean wave 10m
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Gravity and Rosby waves 10-1000km
Gravitational waves 1Gm
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• Aim of talk
• To give a history of waves
• To show how maths unites them all
• To give examples in many applications

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2011 celebrated two big wave anniversaries
Possibly the most important wave equation of all
was discovered by Schrodinger in 1926.
Erwin Schrodinger 1887-1961
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Basic equation of quantum mechanics
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
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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)
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Reason was discovered by Pythagoras
Length of strings giving C and G, F and E, were
in simple fractional proportions
CC 2/1 CG 3/2 CF 4/3 CE
5/4
Gave an important hint about the underlying
physics!
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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
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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
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Simplest wave is a sine wave
Amplitude Angular Frequency
Linked to triangles!!!
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Sound waves travel through the air and are sine
waves in both space and time
Wavelength L, Period T, Frequency f 1/T
Amplitude 2A
C Frequency f 261.6 Hz T3.8ms, L1.2m
G Frequency f 392 Hz T2.5ms, L 0.8m
Speed c f L c 320 m/s
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Lissajous Figures Show good chords
CG
CE
CF
EF
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But why are waves sine waves?
Pendulum observed by Galileo in 1600
Newton gives the differential equation
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Euler finds the solution
Guess what its a sine wave
Damping
Amplitude Frequency Phase
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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!
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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
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Eg Square wave
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A useful application of Fourier Analysis The
tides a global wave
Height of the Bombay tides 1872
h(t)
t
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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
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US Tidal predictor
Kelvin Tidal predictor
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Kelvin made many other discoveries concerning
waves
Wave equation describes waves on a string and
small water waves
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This equation describes small waves well
Wavelength
speed
Larger waves in shallow water obey a different
equation (the Shallow Water Equation) IMPORTANT
to understand Tsunamis
speed
Depth
Almost supersonic in the ocean!!!
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Helmholtz equation describes waves in a
telegraph cable and microwave cooking
Kelvin knighted 1866 for his work on the
trans-Atlantic cable
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But waves dont have to go down cables
Maxwell and the discovery of electromagnetic waves
Maxwells equations solutions are waves in space
eg. light
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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!!!!

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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
Momentum
Wave aspect of matter is formalised by the
wavefunction defined by the Schrodinger Equation
,
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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
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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!
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In conclusion
Waves dominate all aspects of science They have
applications everywhere Maths helps us to
understand them.