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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 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 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 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 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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