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MTU GE 4250 - COMPLEX WAVE LECTURE NOTES

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What is a wave?The one-dimensional wave equationProof that f (x ± vt) solves the wave equationWhat about a harmonic wave?Slide 5The Phase VelocityThe Group VelocityDispersion: phase/group velocity depends on frequencySlide 9Normal dispersion of visible lightComplex numbersEuler’s FormulaWaves using complex numbersAmplitude and Absolute phaseSlide 18Waves using complex amplitudesComplex numbers simplify waves!Vector fieldsThe 3D wave equation for the electric field and its solutionSlide 23EM propagation in homogeneous materialsSlide 30Slide 31Absorption of EM radiationSlide 33Slide 34Absorption coefficient and skin depthSlide 36Refractive index (n) – the dispersion equationRefractive index (n) of water and icePenetration depth of water and ice (also called absorption depth or skin depth)What is a wave?A wave is anything that moves.To displace any function f(x) to the right, just change its argument from x to x-a, where a is a positive number.If we let a = v t, where v is positive and t is time, then the displacement will increase with time.So represents a rightward, or forward, propagating wave.Similarly, represents a leftward, or backward, propagating wave, where v is the velocity of the wave.f(x)f(x-3)f(x-2)f(x-1)x0 1 2 3f(x - v t)f(x + v t)For an EM wave, we could have E = f(x ± vt)The one-dimensional wave equation2 22 2 210vf fx t� �- =� �The one-dimensional wave equation for scalar (i.e., non-vector) functions, f:where v will be the velocity of the wave.( , ) ( v )f x t f x t= �The wave equation has the simple solution:where f (u) can be any twice-differentiable function.Proof that f (x ± vt) solves the wave equationWrite f (x ± vt) as f (u), where u = x ± vt. So and Now, use the chain rule: So  and  Substituting into the wave equation:1ux�=�vut�=��f f ux u x� � �=� � �f f ut u t� � �=� � �f fx u� �=� �vf ft u� �=�� �2 222 2vf ft u� �=� �2 22 2f fx u� �=� �2 2 2 222 2 2 2 2 21 1v 0v vf f f fx t u u� �� � � �- = - =� �� � � ��What about a harmonic wave?€ E = E0cos k(x − ct)E0 = wave amplitude (related to the energy carried by the wave).= angular wavenumber(λ = wavelength; = wavenumber = 1/λ)Alternatively:Where ω = kc = 2πc/λ = 2πf = angular frequency (f = frequency)€ k =2πλ= 2π˜ ν € ˜ ν € E = E0cos(kx −ωt)The argument of the cosine function represents the phase of the wave, ϕ, or the fraction of a complete cycle of the wave.What about a harmonic wave?€ E = E0cos k(x − ct); φ= k(x − ct)In-phase wavesOut-of-phase wavesLine of equal phase = wavefront = contours of maximum fieldThe Phase VelocityHow fast is the wave traveling? Velocity is a reference distancedivided by a reference time.The phase velocity is the wavelength / period: v =  /  Since f = 1/:In terms of k, k = 2/ , and the angular frequency,  = 2/ , this is: v = f v =  / kThe Group VelocityThis is the velocity at which the overall shape of the wave’s amplitudes, or the wave ‘envelope’, propagates. (= signal velocity)Here, phase velocity = group velocity (the medium is non-dispersive)Dispersion: phase/group velocity depends on frequencyBlack dot moves at phase velocity. Red dot moves at group velocity.This is normal dispersion (refractive index decreases with increasing λ)Dispersion: phase/group velocity depends on frequencyBlack dot moves at group velocity. Red dot moves at phase velocity.This is anomalous dispersion (refractive index increases with increasing λ)Normal dispersion of visible lightShorter (blue) wavelengths refracted more than long (red) wavelengths.Refractive index of blue light > red light.Complex numbersSo, instead of using an ordered pair, (x,y), we write:P = x + i y = A cos() + i A sin()where i = √(-1) Consider a point,P = (x,y), on a 2D Cartesian grid.Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number.…or sometimes j = √(-1)Euler’s Formula Links the trigonometric functions and the complex exponential functionexp(i) = cos() + i sin()so the point, P = A cos() + i A sin(), can also be written: P = A exp(i) = A eiφwhere A = Amplitude  = PhaseVoted the ‘Most Beautiful Mathematical Formula Ever’ in 1988The argument of the cosine function represents the phase of the wave, ϕ, or the fraction of a complete cycle of the wave.Using complex numbers, we can write the harmonic wave equation as:i.e., E = E0 cos() + i E0 sin(), where the ‘real’ part of the expression actually represents the wave.We also need to specify the displacement E at x = 0 and t = 0, i.e., the ‘initial’ displacement.Waves using complex numbers€ E = E0cos k(x − ct); φ= k(x − ct)€ E = E0eik(x −ct )= E0ei(kx−ωt )Amplitude and Absolute phaseE(x,t) = A cos[(k x – t ) – ] A = Amplitude = Absolute phase (or initial, constant phase) at x = 0, t =0kxSo the electric field of an EM wave can be written:E(x,t) = E0 cos(kx – t – )Since exp(i) = cos() + i sin(), E(x,t) can also be written: E(x,t) = Re { E0 exp[i(kx – t – )] }Recall that the energy transferred by a wave (flux density) is proportional to the square of the amplitude, i.e., E02. Only the interaction of the wave with matter can alter the energy of the propagating wave.Remote sensing exploits this modulation of energy.Waves using complex numbersWaves using complex amplitudesWe can let the amplitude be complex:Where the constant stuff is separated from the rapidly changing stuff. The resulting "complex amplitude”: is constant in this case (as E0 and θ are constant), which implies that the medium in which the wave is propagating is nonabsorbing.What happens to the wave amplitude upon interaction with matter?€ E(x,t) = E0exp[i(kx − ωt −θ)]E(x,t) = E0exp(−iθ)[ ]exp[i(kx −ωt)]€ E0exp(−iθ)[ ]Complex numbers simplify waves!This isn't so obvious using trigonometric functions, but it's easywith complex exponentials:1 2 31 2 3( , ) exp ( ) exp ( ) exp ( ) ( )exp ( )totE x t E i kx t E i kx t E i kx tE E E i kx tw w ww= - + - + -= + + -% % % %% % %where all initial phases are lumped into E1, E2, and E3.Adding waves of the same frequency, but different initial


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