Maths Makes Waves Chris Budd. Waves are a universal phenomenon in science at all scales Light pulse 500nm Electron wave 0.5nm.

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.