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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 A wave is… a disturbance in a medium**

f(x) t0 t1 > t0 t2 > t1 t3 > t2 v x A wave is… a disturbance in a medium -propagating in space with velocity v -transporting energy -leaving the medium undisturbed

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**1D traveling wave f(x) v x 0 1 2 3**

t1 > t0 t2 > t1 t3 > t2 v x 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.

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**works for anything that moves!**

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!

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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**

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:

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**The phase of a harmonic wave**

The phase, 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.

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**Absolute phase y = A sin[k(x ± vt) + j0] A = amplitude**

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

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**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

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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 wave**

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**Complex numbers make it less complex!**

P = (x,y) x: real and y: imaginary P = x + i y = A cos(j) + i A sin(j) where

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**“one of the most remarkable formulas in mathematics”**

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 = A cos(j) + i A sin(j), can also be written: P = A exp(ij) = A eiφ where A = Amplitude j = Phase

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**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.

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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.

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**represents direction of propagation**

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(kr cosq) If we turn the propagation constant (k = 2p/l) into a vector, kr cosq = k · r Y = A sin(k · r – wt) wave disturbance defined by r propagation along the x axis

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**General case Y = Aei(k·r - wt) k · r = xkx = yky + zkz = kr cosq ks**

arbitrary direction In complex form: Y = Aei(k·r - wt) 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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**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!

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**E and B are perpendicular**

Perpendicular to each other: III. Faraday’s law: so Always!

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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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**The 1D wave equation for light waves**

where E is the light electric field We’ll use cosine- and sine-wave solutions: where: E(x,t) = B cos[k(x ± vt)] + C sin[k(x ± vt)] kx ± (kv)t E(x,t) = B cos(kx ± wt) + C sin(kx ± wt) The speed of light in vacuum, usually called “c”, is 3 x 1010 cm/s.

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**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)

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

y

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**General case— along any direction**

which has the solution: where and

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

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**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:

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**Rate of energy transport**

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

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**Poynting vector oscillates rapidly**

Take the time average: “Irradiance” (W/m2)

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Light is a vector field 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.

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**Light is a 3D vector field**

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

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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 y x

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**Evolution of electric field vector**

circular polarization y x

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Exercises You are encouraged to solve all problems in the textbook (Pedrotti3). The following may be covered in the werkcollege on 14 September 2011: Chapter 4: 3, 5, 7, 13, 14, 17, 18, 24

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