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Chapter 4: Wave equations. What is a wave?

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Presentation on theme: "Chapter 4: Wave equations. What is a wave?"— Presentation transcript:

1 Chapter 4: Wave equations

2 What is a wave?

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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 0 1 2 3 Waves move t0t0 t 1 > t 0 t 2 > t 1 t 3 > t 2 x f(x) v A wave is… a disturbance in a medium -propagating in space with velocity v - transporting energy -leaving the medium undisturbed

18 To displace f(x) to the right: x  x-a, where a >0. Let a = vt, where v is velocity and t is 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. 0 1 2 3 t0t0 t 1 > t 0 t 2 > t 1 t 3 > t 2 x f(x) v 1D traveling wave

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

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 T A at a fixed time:at a fixed point: A frequency: = 1/T angular frequency:  = 2  propagation constant: k = 2  / wave number:  = 1 / ≠ v v = velocity (m/s) = frequency (1/s) v = Note:

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

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

24 How fast is the wave traveling? The phase velocity is the wavelength/period: v = / T Since = 1/T : In terms of k, k = 2  /, and the angular frequency,  = 2  / T, this is: v = v =  / 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 waveany ??

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

30 “one of the most remarkable formulas in mathematics” Euler’s formula Links the trigonometric functions and the complex exponential function e i  = cos  + i sin  so the point, P = A cos(  ) + i A sin(  ), can also be written: P = A exp(i  ) = A e iφ where A = Amplitude   = Phase

31 Harmonic waves as complex functions Using Euler’s formula, we can describe the wave as y = Ae i(kx-  t) so that y = Re(y) = A cos(kx –  t) y = Im(y) = A sin(kx –  t) ~ ~ ~ Why? Math is easier with exponential functions than with trigonometry.

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

33 Wave vector k represents direction of propagation  = A sin(kx –  t) Consider a snapshot in time, say t = 0:  = A sin(kx)  = A sin(kr cos  ) If we turn the propagation constant ( k = 2  / into a vector, kr cos  = k · r  = A sin(k · r –  t) wave disturbance defined by r propagation along the x axis

34 arbitrary direction General case k · r = xk x = yk y + zk z  kr cos   ks In complex form:  = Ae i(k·r -  t) 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 http://www.youtube.com/watch?v=YLlvGh6aEIs Derivation of the wave equation from Maxwell’s equations

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

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

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 We’ll use cosine- and sine-wave solutions: where: where E is the light electric field The speed of light in vacuum, usually called “c”, is 3 x 10 10 cm/s. The 1D wave equation for light waves E(x,t) = B cos[k(x ± vt)] + C sin[k(x ± vt)] kx ± (kv)t E(x,t) = B cos(kx ±  t) + C sin(kx ±  t)

44 The speed of an EM wave in free space is given by:  0  = permittivity of free space,    = 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 ) EM wave propagation in homogeneous media

45 x z y Simple case— plane wave with E field along x, moving along z :

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

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

48 Energy density (J/m 3 ) in an electrostatic field: Energy density (J/m 3 ) in an magnetostatic field: Energy density (J/m 3 ) in an electromagnetic wave is equally divided: Electromagnetic waves transmit energy

49 Rate of energy transport: Power (W) ctct A Power per unit area (W/m 2 ): Pointing vector Poynting Rate of energy transport

50 Take the time average: “Irradiance” (W/m 2 ) Poynting vector oscillates rapidly

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

52 Light is a 3D vector field A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space. A light wave has both electric and magnetic 3D vector fields:

53 Polarization corresponds to direction of the electric field determines of force exerted by EM wave on charged particles (Lorentz force) linear,circular, eliptical

54 Evolution of electric field vector linear polarization x y

55 Evolution of electric field vector circular polarization x y

56 You are encouraged to solve all problems in the textbook (Pedrotti 3 ). The following may be covered in the werkcollege on 14 September 2011: Chapter 4: 3, 5, 7, 13, 14, 17, 18, 24 Exercises


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