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Chapter 4: Wave equations

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What is a wave?

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what is a wave? anything that moves

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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

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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

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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

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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:

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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)

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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:

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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

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

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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

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

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works for any periodic waveany ??

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

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“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

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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.

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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.

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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

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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:

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

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Cylindrical waves

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the wave nature of light

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Electromagnetic waves

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Derivation of the wave equation from Maxwell’s equations

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

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

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

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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)

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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

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

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

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

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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

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Rate of energy transport: Power (W) ctct A Power per unit area (W/m 2 ): Pointing vector Poynting Rate of energy transport

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

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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

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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:

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

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Evolution of electric field vector linear polarization x y

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Evolution of electric field vector circular polarization x y

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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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