Maths Makes Waves Chris Budd. Waves are a universal phenomenon in science at all scales Light pulse 500nm Electron wave 0.5nm.
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Maths Makes Waves
Chris Budd
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∇×E = −∂B
∂t− M, ∇ ×H = −
∂D
∂t+ J,
∇.D = ρ, ∇.B = 0.
Waves are a universal phenomenon in science at all scales
Light pulse 500nm
Electron wave 0.5nm
Sound 50cm
Microwave 10cm
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
“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”
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ih∂
∂tψ =−
h2
2m∇ 2ψ +V (x,y,z)ψ
Wave function:
probability distribution of states with different energies
Basic equation of quantum mechanics
Schrodinger used it to compute the spectrum of the Hydrogen atom.
Now, used everyday in the chips in your mobile phone
Greeks observed that some musical notes from a stringed instrument sound better when played together than others
The notes C and G
(a perfect 5th)
The notes C and F
(a perfect 4th)
The notes C and E
(a major 3rd)
The octave C to C
Musical sounds: the first man made waves
But .. waves, and their mathematics, have a long history!
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?
Galileo15-02-1564
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
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u(t) = Asin(ω t) = Asin(2π f t)
Amplitude Angular Frequency
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ω t
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sin(ω t)Linked to triangles!!!
C: Frequency
f = 261.6 Hz
T=3.8ms, L=1.2m
G: Frequency
f = 392 Hz
T=2.5ms, L = 0.8m
Wavelength L, Period T, Frequency f = 1/T Amplitude 2*A
Sound waves travel through the air and are sine waves in both space and time
Speed c = f L c = 320 m/s
C:GC:E
C:F E:F
Lissajous Figures: Show good chords
But why are waves sine waves?
Pendulum observed by Galileo in 1600
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ad2θ
dt 2+ b
dθ
dt+ cθ = 0
Newton gives the differential equation
Euler finds the solution
Guess what: it’s a sine wave
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θ(t) = e−αtAsin(ω t+φ)
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!
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u(t) =a0
2+ an cos
2π n t
T
⎛
⎝ ⎜
⎞
⎠ ⎟
n=1
∞
∑ + bn sin2π n t
T
⎛
⎝ ⎜
⎞
⎠ ⎟
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
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sin(t)
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sin(t) +sin(3t)
3+
sin(5t)
5
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sin(t) +sin(3t)
3+
sin(5t)
5+K +
sin(11t)
11
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sin(t) +sin(3t)
3+
sin(5t)
5+K +
sin(49t)
49
A useful application of Fourier Analysis
The tides: a global wave
h(t)
t
Height of the Bombay tides 1872
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
US Tidal predictorKelvin Tidal predictor
Kelvin made many other discoveries concerning waves
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utt =uxxWave equation: describes waves on a string and small water waves
This equation describes small waves well
Larger waves in shallow water obey a different equation (the Shallow Water Equation)
IMPORTANT to understand Tsunamis
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c =gΛ
2πspeed Wavelength
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c = ghspeed
Depth
Almost supersonic in the ocean!!!
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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
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∇×E = −∂B
∂t− M, ∇ ×H = −
∂D
∂t+ J,
∇.D = ρ, ∇.B = 0.
Maxwell and the discovery of electromagnetic waves
But waves don’t have to go down cables
Maxwell’s equations: solutions are waves in space eg. light
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!!!!
What this led to …
But … is light a wave or a particle?
De Broglie 1924
Discovery of the particle-wave duality of light and matter Confirmed by electron diffraction
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λ= h
mv
Wave aspect of matter is formalised by the wavefunction defined by the Schrodinger Equation ,
wavelength
Planks constant
Momentum
Davisson, Germer and Thomson 1927
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
Idea: electrons and quarks within an atom are made up strings. These strings oscillate, giving the particles their flavor, charge, mass and spin.
Study of waves started with wave on strings
String theory brings waves right up to date.
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.
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