Chapter 4: Wave equations

1. Chapter 4: Wave equations

2. What is a wave?

15. what is a wave? anything that moves

16. This idea has guided my research: for matter, just as much as for radiation, in particular light, we must introduce at one and the same time the corpuscle concept and wave concept. Louis de Broglie

17. Waves move f(x) t1 > t0 t2 > t1 t3 > t2 t0 v x 0 1 2 3 • A wave is… • a disturbance in a medium • -propagating in space with velocity v • -transporting energy • -leaving the medium undisturbed

18. 1D traveling wave f(x) t1 > t0 t2 > t1 t3 > t2 t0 v x 0 1 2 3 To displace f(x) to the right: xx-a, where a >0. Leta = vt, where v is velocityand tis time -displacement increases with time, and -pulse maintains its shape. So f(x -vt)represents a rightward, or forward, propagating wave. • f(x+vt)represents a leftward, or backward, propagating wave.

19. The wave equation • Let y =f(x'), where x' = x ± vt. So and Now, use the chain rule: So Þ and Þ Combine to get the 1D differential wave equation: works for anything that moves!

20. Harmonic waves • periodic (smooth patterns that repeat endlessly) • generated by undamped oscillators undergoing harmonic motion • characterized by sine or cosine functions • -for example, • -complete set (linear combination of sine or cosine functions • can represent any periodic waveform)

21. Snapshots of harmonic waves at a fixed time: at a fixed point: T A A propagation constant: k = 2p/l wave number:k = 1/l frequency: n = 1/T angular frequency:w = 2pn n≠ v v = velocity (m/s) n = frequency (1/s) v = nl Note:

22. The phase of a harmonic wave Thephase, j, is everything inside the sine or cosine (the argument). y = A sin[k(x ± vt)] j= k(x ± vt) For constant phase, dj = 0 = k(dx ± vdt) which confirms that v is the wave velocity.

23. p Absolute phase • y = A sin[k(x ± vt) + j0] A = amplitude j0 = initial phase angle (or absolute phase) at x = 0 and t =0

24. How fast is the wave traveling? The phase velocity is the wavelength/period: v = l/T Since n = 1/T: In terms of k, k = 2p/l, and the angular frequency, w = 2p/T, this is: v = ln v = w/k

25. Phase velocity vs. Group velocity Here, phase velocity = group velocity (the medium is non-dispersive). In a dispersive medium, the phase velocity ≠ group velocity.

26. works for any periodic wave any ??

29. Complex numbers make it less complex! P = (x,y) x: real andy: imaginary P = x + i y • = A cos(j) + i A sin(j) where

30. Euler’s formula “one of the most remarkable formulas in mathematics” Links the trigonometric functions and the complex exponential function eij = cosj + i sinj so the point,P = Acos(j) + iA sin(j), can also be written: P = A exp(ij) = Aeiφ where A = Amplitude j = Phase

31. Harmonic waves as complex functions Using Euler’s formula, we can describe the wave as y = Aei(kx-wt) so that y = Re(y) = A cos(kx – wt) • y = Im(y) = A sin(kx – wt) ~ ~ ~ Why? Math is easier with exponential functions than with trigonometry.

32. Plane waves A plane wave’s contours of maximum field, called wavefronts, are planes. They extend over all space. The wavefronts of a wave sweep along at the speed of light. • equally spaced • separated by one wavelength apart • perpendicular to direction or propagation Usually we just draw lines; it’s easier.

33. Wave vector k represents direction of propagation • Y = A sin(kx – wt) Consider a snapshot in time, say t = 0: • Y = A sin(kx) • Y = A sin(krcosq) • If we turn the propagation constant (k = 2p/l) into a vector, krcosq = k · r • wave disturbance defined by r • propagation along the x axis • Y = A sin(k · r – wt)

34. General case • k · r = xkx = yky + zkz • = krcosq •  ks arbitrary direction In complex form: Y = Aei(k·r - wt) Substituting the Laplacian operator:

35. Spherical waves • harmonic waves emanating from a point source • wavefronts travel at equal rates in all directions

36. Cylindrical waves

37. the wave nature of light

38. Electromagnetic waves

39. Derivation of the wave equation from Maxwell’s equations http://www.youtube.com/watch?v=YLlvGh6aEIs

40. E and B are perpendicular Perpendicular to the direction of propagation: I. Gauss’ law (in vacuum, no charges): so II. No monopoles: Always! so Always!

41. E and B are perpendicular Perpendicular to each other: III. Faraday’s law: so Always!

42. E and B are harmonic Also, at any specified point in time and space, where c is the velocity of the propagating wave,

43. EM wave propagation in homogeneous media • The speed of an EM wave in free space is given by: • 0= permittivity of free space, 0= magnetic permeability of free space • To describe EM wave propagation in other media, two properties of the medium are important, its electric permittivity ε and magnetic permeability μ. These are also complex parameters. •  = 0(1+ ) + i= complex permittivity • = electric conductivity • = electric susceptibility (to polarization under the influence of an external field) • Note that ε and μ also depend on frequency (ω)

44. x z y Simple case— planewave with E field along x, moving along z:

45. General case— along any direction which has the solution: where and and

46. = wave vector Plane of constant , constant E Arbitrary direction x z y

47. Electromagnetic waves transmit energy Energy density (J/m3) in an electrostatic field: Energy density (J/m3) in an magnetostatic field: Energy density (J/m3) in an electromagnetic wave is equally divided:

48. Rate of energy transport Rate of energy transport: Power (W) A cDt Power per unit area (W/m2): Poynting Pointing vector

49. Poynting vector oscillates rapidly Take the time average: “Irradiance” (W/m2)

50. Light is a vector field A 2D vector field assigns a 2D vector (i.e., anarrow of unit length having a unique direction) to each point in the 2D map.