Wave Propagation in Linear Media | re-examined

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3Wave propagation in electromagnetic transmission lines the wave front are dominated by the plasma frequency, which again is not surprising. This becomes most clear by inspection of the voltage at the source at low signal frequencies, where the in nitely high source impedance permits U to oscillate freely. It shows therefore a superposition of the high resonance frequency and a component determined by the signal ( gs. 3.13 and 3.14). Remark (Dispersion e ects) That the plasma frequency ultimately dominates the free oscillations is not at all unexpected. The resonance frequency has the smallest group velocity vg = d!=d (note that we may safely use this de nition here since the wave number k = + j is purely real in the non-evanescent mode). The fastest components, on the other hand, are the high-frequent ones that constitute the sharp wave front. They rush through, and the slow components around !p remain. That the local frequency equals exactly the plasma frequency holds of course only in the limit t !1. The local wavelength exhibits a complementary e ect. If we take a snapshot of the wave along the line at some given time, we nd that a long way behind the wave front, the oscillation gradually fades away. Here, it is the wavenumber that tends to zero as ! ! !p, and thus the local wavelength grows in nitely. The explanations given so far have been of a rather mathematical nature. There is, however, also a physical interpretation why the wave front travels undistorted through the medium. The electrons in the plasma have a nite mass and consequently a certain amount of inertia. Hence they cannot react to the arriving wave front instantaneously, but only with a short delay, and the wave front at the very instant t = x=c remains una ected. This explanation was already given by Sommerfeld, who in turn owed it to Voigt [2, 12]. Remark (Confession of a graphical retouch) In the gures, the values of the waves at X = T have been set to unity. This is, admittedly, not the correct numerical result and has been introduced deliberatly to illustrate the actual height of the wave front. The true value, as dictated by Fourier, naturally is 0:5 , and 1 is only the left-sided limit, as will be demonstrated below. The space-time evolution diagrams have a sampling grid that has been chosen su ciently coarse to keep the computational e ort at a reasonable value. This impaired particularly the area about the wave front, which we examine more closely now. A re ned computation of the wave front shows that it remains indeed unaltered even for large values of X and T , as depicted in g. 3.16 for the current and g. 3.17 for the voltage. The graphs show the spatial evolution of the wave for di erent instances in time and reveal two remarkable features. First, voltage and current are in phase for some time after X = T . This is plausible if we consider that the high frequency components at the wave front see a real-valued characteristic impedance. Thus any phase shift is impossible. Second, the trailing edge of the wave front becomes steeper as it travels down the line. The left point of the time interval in each of the gures indicates to a good approximation a constant value of the wave function. It can be calculated in dependence on the X-value as 54
3.7 Turn-on e ects in a lossless plasma 30 30 0 20 0 20 10 X 10 X T 10 T 10 20 20 30 1 0.5 0 -0.5 -1 0 30 1 0 0 -0.5 I/I0 0.5 U/U0 Figure 3.14: Evolution of the current (3.82) and voltage (3.83) along a one-dimensional transmission line with a lossless plasma for a slowly oscillating input current ( =0:2). 55
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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
Page 19 and 20: 1.1 Phase and group velocity 1.1 Ph
Page 21 and 22: 1.1 Phase and group velocity ! !c v
Page 23 and 24: 1.2 A few notes on dispersion He co
Page 25 and 26: 1.2 A few notes on dispersion v/c 6
Page 27 and 28: 1.3 Signal velocity dipoles with a
Page 29 and 30: 1.3 Signal velocity The arbitrarine
Page 31 and 32: 1.4 Energy velocity For electromagn
Page 33 and 34: 1.5 Other velocity de nitions For n
Page 35 and 36: 1.5 Other velocity de nitions evide
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
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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 and 86: 3.9 A Gaussian pulse in plasma 250
Page 87 and 88: 3.9 A Gaussian pulse in plasma Note
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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4.6 Examples of tunnelling events -
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4.6 Examples of tunnelling events t
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4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events 8
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4.6 Examples of tunnelling events T
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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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