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Maths Makes Waves Chris Budd

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Waves are a universal phenomenon in science at all scales Light pulse 500nm Electron wave 0.5nm

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Sound 50cm Microwave 10cm

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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 1.To give a history of waves 2. To show how maths unites them all 3. 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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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 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

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Greeks observed that some musical notes from a stringed instrument sound better when played together than others The notes C and G (a perfect 5 th ) The notes C and F (a perfect 4th) The notes C and E (a major 3 rd ) The octave C to C Musical sounds: the first man made waves But.. waves, and their mathematics, have a long history!

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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!

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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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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

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C:G C:E C:FE:F Lissajous Figures: Show good chords

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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 h(t) t Height of the Bombay tides 1872

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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 predictorKelvin 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 Larger waves in shallow water obey a different equation (the Shallow Water Equation) IMPORTANT to understand Tsunamis speed Wavelength 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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Maxwell and the discovery of electromagnetic waves But waves dont have to go down cables Maxwells equations: solutions are waves in space eg. light

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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 …

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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 Wave aspect of matter is formalised by the wavefunction defined by the Schrodinger Equation, wavelength Planks constant Momentum Davisson, Germer and Thomson 1927

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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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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!

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In conclusion: Waves dominate all aspects of science They have applications everywhere Maths helps us to understand them.

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