Wave Propagation in Linear Media | re-examined

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3Wave propagation in electromagnetic transmission lines classical understanding of the phase and group velocities. The slope of the stripes yields the phase velocity, and the contour of the entire pulse (most easily traced at the edge where they fade into the grey surroundings) de nes the group velocity. Obviously, these two velocities are di erent, which empirically a rms the theoretical ndings of section 1.1: in a dispersive medium, the envelope of a pulse propagates at a distinct velocity with the train of carrier oscillations moving indepently underneath its overall shape. As for the determination of from the graph, both (3.108) and (3.109) give the correct answers. Remark (Dispersion e ects) In a dispersive medium, we should of course be able to observe a broadening of the pulse. We are, however, at a loss to do so in this case because the initial pulse is too wide and the distance too short for this e ect to become visible. But if we su ciently reduced the plotted voltage range (by some nine orders of magnitude), we would encounter an e ect similar to that in g. 3.26: namely, a small portion of the wave travelling, if at all, at very low speed. Again, this is caused by the small frequency range immediately above cuto where the group velocity is hardly larger than zero. In contrast, the e ect of the evanescent components (which are present also in this case) is not noticeable at all. In accordance with the theory, the examples show that if the spectrum of a pulse is bundled below cuto , there is no phase shift, and the pulse decays exponentially without allowing for wave propagation. The amount of attenuation is determined by the centre frequency of the pulse, !0, so that we can estimate the voltage amplitude by U(X) / e ,Xp 1, 2 : (3.110) If in a data transmission system we decided, for the sake of proper signal detection, to tolerate an attenuation of 10 ,4 (80 dB), we could calculate the admissible length of the line. For = 0:5 this would give X = 10:64, whereas with = 0:8, we could cover a normalised distance of X =15:35. The actual length of the transmission line can be obtained from the de nition of the scaled space coordinate, x = Xc=!p. In a plasma with a cuto frequency of 10 GHz, X = 10 corresponds to an actual distance of x = 4:77 cm. So if we were to use signals with superluminal group velocities (as they are sometimes called) for data transmission, we could cover but a few centimetres. This, however, makes their usefulness for practical applications rather questionable. The maximum of a band-limited pulse can be transmitted at a velocity greater than that of light, there is no doubt about it. But does this mean that information can also be transmitted superluminally? The response of a linear system to an input signal can be written as a convolution integral, u(x; t) = Z 1 ,1 of the input u(0;t) and the transfer function g(x; t) = 1 2 u(0; ) g(x; t , ) d ; (3.111) Z 1 e ,1 ,jk(!)x e j!t d! : (3.112) 70
3.9 A Gaussian pulse in plasma Note that while communication engineers talk of transfer functions (or, likewise, impulse responses), theoretical physicists frequently call g(x; t) a Green's function. In either case, it is said to be causal if it is identically zero for t
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DISSERTATION Wave Propagation in Li
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Kurzfassung Seit der Entdeckung des
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Nullum est iam dictum, quod non sit
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Preface Our popular writers and rep
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the quest for superluminality and t
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Contents Part I Wave propagation ph
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7.3.4 PartitionPoints . . . . . . .
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Part I Wave propagation phenomena S
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1.1 Phase and group velocity 1.1 Ph
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1.1 Phase and group velocity ! !c v
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1.2 A few notes on dispersion He co
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1.2 A few notes on dispersion v/c 6
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1.3 Signal velocity dipoles with a
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1.3 Signal velocity The arbitrarine
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1.4 Energy velocity For electromagn
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1.5 Other velocity de nitions For n
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Page 37 and 38: 2.1 Superluminal wave propagation 2
Page 39 and 40: 2.1 Superluminal wave propagation a
Page 41 and 42: 2.1 Superluminal wave propagation t
Page 43 and 44: 2.2 Quantum mechanical tunnelling e
Page 45 and 46: 2.2 Quantum mechanical tunnelling R
Page 47 and 48: Chapter 3 Wave propagation in elect
Page 49 and 50: 3.1 Model of a transmission line th
Page 51 and 52: 3.2 Excursion: a delay line section
Page 53 and 54: 3.3 Re ection due to termination mi
Page 55 and 56: 3.4 A simple thought experiment I0
Page 57 and 58: 3.5 A dispersive system: the lossle
Page 63 and 64: 3.6 Inhomogeneous transmission line
Page 65 and 66: 3.7 Turn-on e ects in a lossless pl
Page 77 and 78: 3.8 Turn-on e ects in a wave guide
Page 79 and 80: 3.8 Turn-on e ects in a wave guide
Page 81 and 82: 3.9 A Gaussian pulse in plasma Like
Page 83 and 84: 3.9 A Gaussian pulse in plasma 250
Page 85: 3.9 A Gaussian pulse in plasma 250
Page 89 and 90: 4.1 The potential step 4.1 The pote
Page 91 and 92: 4.1 The potential step Inside the b
Page 93 and 94: 4.2 Initial wave forms -60 -50 -40
Page 95 and 96: 4.2 Initial wave forms -60 -50 -40
Page 97 and 98: 4.3 Examples of scattering processe
Page 109 and 110: 4.4 The square barrier they have va
Page 111 and 112: 4.4 The square barrier 1 0.8 0.6 0.
Page 113 and 114: 4.5 Tunnelling time de nitions for
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Page 117 and 118: 4.6 Examples of tunnelling events P
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Page 123 and 124: 4.6 Examples of tunnelling events t
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Page 135 and 136: Interlude Wave functions in graphic
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Wave functions in graphical represe
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Part II Numerical aspects of wave e
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5.1 Univariate numerical quadrature
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5.1 Univariate numerical quadrature
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5.2 Convergence acceleration one, t
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5.2 Convergence acceleration (1) ,1
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6.1 Partitioning the integration in
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6.2 Choosing the rst partition poin
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6.3 How to compute the rst integral
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6.3 How to compute the rst integral
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6.4 Asymptotic partition consuming
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6.4 Asymptotic partition of the int
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6.5 Considerations for a Mathematic
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6.6 Controlling the accuracy of the
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Chapter 7 Mathematica implementatio
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7.1 User interface of the function
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7.3 Implementation of OscInt and re
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7.4 Auxiliary functions While[itera
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7.4 Auxiliary functions Linear appr
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7.4 Auxiliary functions For the sak
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7.4 Auxiliary functions Out[8]= 3 f
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7.5 Test of the quadrature routine
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8.1 Preparation of the wave integra
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8.2 Outline of the program structur
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8.2 Outline of the program structur
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8.3 Implementation of the quadratur
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8.3 Implementation of the quadratur
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8.4 Test of the package 8.4 Test of
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8.4 Test of the package -200 -150 -
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8.4 Test of the package 0.4 0.2 -0.
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8.4 Test of the package 8.4.2 Forma
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8.4 Test of the package Out[24]= 0
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A.1 Numerical quadrature Asymptotic
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A.1 Numerical quadrature WynnDegree
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A.1 Numerical quadrature Which[ft =
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A.2 Solutions for the step potentia
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A.3 Solutions for the square barrie
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A.3 Solutions for the square barrie
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A.4 Electromagnetic waves function
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A.5 Utilities for displaying points
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Bibliography Bibliography [1] James
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Bibliography [29] Kurt Edmund Oughs
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Bibliography [59] Ch. Spielmann, R.
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Bibliography [91] C. R. Leavens and
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Bibliography [123] T. O. Espelid an
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Index Index absorption, 7, 9 accura
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Index saddle point integration, 11,
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Curriculum vitae Dipl.-Ing. Thilo S
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