Maths Makes Waves Chris Budd

1. Maths Makes Waves Chris Budd

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

3. Microwave 10cm Sound 50cm

4. Sand waves 1m Ocean wave 10m

5. Gravity and Rosby waves 10-1000km Gravitational waves 1Gm

9. Aim of talk To give a history of waves To show how maths unites them all To give examples in many applications

10. 2011 celebrated two big wave anniversaries Possibly the most important wave equation of all was discovered by Schrodinger in 1926. Erwin Schrodinger 1887-1961

11. 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

12. 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)

13. 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!

14. 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

15. 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

16. Simplest wave is a sine wave Amplitude Angular Frequency Linked to triangles!!!

17. 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 = f L c = 320 m/s

18. Lissajous Figures: Show good chords C:G C:E C:F E:F

19. But why are waves sine waves? Pendulum observed by Galileo in 1600 Newton gives the differential equation

20. Euler finds the solution Guess what: it’s a sine wave Damping Amplitude Frequency Phase

21. 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!

22. 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

23. Eg: Square wave

24. A useful application of Fourier Analysis The tides: a global wave Height of the Bombay tides 1872 h(t) t

25. Kelvin decomposed the tidal height into 37 independent Fourier components He found these out using past data andadded them up using an analogue computer

26. Kelvin Tidal predictor US Tidal predictor

27. Kelvin made many other discoveries concerning waves Wave equation: describes waves on a string and small water waves

28. 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!!!

29. Helmholtz equation: describes waves in a telegraph cable and microwave cooking Kelvin knighted 1866 for his work on the trans-Atlantic cable

30. But waves don’t have to go down cables Maxwell and the discovery of electromagnetic waves Maxwell’s equations: solutions are waves in space eg. light

32. 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!!!!

33. 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 ,

34. The Largest Waves of all Gravitational waves Theoretical ripples in the curvature of spacetime Can becaused bybinary 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

35. Study of waves started with wave on strings String theory bringswaves 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!

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