Wave Propagation in Linear Media | re-examined

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7 Mathematica implementation of a quadrature function fint is the integrand function (x) u( (x)), which can be de ned in two ways. One is sin x to use a so-called pure function, e. g. Sin[#]/#& for x (see also the opening remark in section 7.5). If the integrand is declared explicitly, then it must be of the form f[x] or, if there are several other parameters, f[a,b,c][x]. In this case, only the name of the function together with the additional parameters (f[a,b,c]) has to be passed to OscInt. fzero is the function being used to determine the partition points fx0;x1;x2;:::g. Depending on the option FunctionType it is understood either as an explicit enumeration of the partition points x (n); n 0 or as the argument function (x) (or an approximation (x)) of the oscillating factor of the integrand, which is then used to compute the subdivision points xn numerically. In the latter case, the function must be di erentiable. In the former case, it is the user's responsibility to ensure that fzero returns points in ascending order that lie exclusively to the right of the rightmost extremum or in exion point of the argument function. It is, however, not necessary to see to it that x (0) > a, as the routine will seek the rst x (k) > a itself. Like fint, the function fzero can be given either as a pure function or in explicit notation. a is the lower integration limit. opts are options the user can set to take in uence on the computation. They are explained below. Normally, the partition points are chosen such that they lie in a region where (x) and its derivative 0 (x) are monotonic, in order to avoid the problems discussed in section 6.2. If an enumeration function is passed to OscInt, this function must be written accordingly. If the original (x) is used as fzero, then z0 is selected to be larger than the largest real part of the solutions of 0 (x) =0. In some cases, however, it might be more convenient tochoose the rst subdivision point according to a di erent criterion | particularly if we use an approximation (x). Thus there is a second way to invoke OscInt with an explicit speci cation of a lower subdivision limit a0: OscInt[fint, fzero, fa, a0g, opts]. Apart from the new parameter, the meanings of the other input values remain unchanged. The partition points are now chosen such that x (k) >a0; k K(a0), or x0 >a0, depending on the interpretation of fzero. There is a variety of options indispensable to control the evaluation of OscInt. Some of them are standard Mathematica options that are used also for other numerical functions, others have been newly introduced. WynnDegree is the standard option for the computation of the limit of the sequence of partial sums. Its default value is Infinity. 162
7.1 User interface of the function OscInt NSumTerms is taken from NSum although this function is not used within OscInt. Its meaning, however, is the same | it speci es how many elements of the sequence of partial sums are not included in the extrapolation. The default value is 10. NSumExtraTerms is used like in NSum to prescribe the number of sequence members subject to extrapolation (default value 12). Together with NSumTerms, it de nes how many elements of the sequence of partial sums have to be computed in total. MinRecursion is a standard option used for all numerical integrations that speci es how many levels of recursion are at least required before tests for convergence start. Its default value is 0. MaxRecursion sets the maximum depth for recursive algorithms and a ects both the numerical quadrature with NIntegrate and the numerical solving of equations with FindRoot. For the latter, its value is passed to the option MaxIterations. This change of name was necessary for reasons of consistency, because OscInt uses the option MaxIterations in a di erent way. The default value is MaxRecursion->20. AccuracyGoal is used to specify the absolute error limit for NIntegrate and defaults to Infinity (i. e. it is not used as a criterion for the quadrature). PrecisionGoal sets the relative error limit for the numerical quadrature and like in NIntegrate has the default value Automatic, which means that it is set to ten digits less than WorkingPrecision. PartitionAccuracy is in fact the setting for AccuracyGoal used in FindRoot to determine the partition points. With the change of name this parameter can be set independently of the AccuracyGoal of NIntegrate. The default value is PartitionAccuracy ->Automatic. AccuracyControl sets the upper limit for the absolute error in the extrapolation result in that is speci es the digits of accuracy required. Its default value is Infinity, which means that it is not used as a criterion for terminating the iteration. If it is used, care must be taken to set the working precision high enough that the desired accuracy can be reached. PrecisionControl gives the upper limit for the relative error in the extrapolation result and is recommended for controlling the iteration. With its default value Infinity, itis normally not used. If precision or accuracy control is enabled, the iterative procedure continues to increase the length of the sequence of partial sums until either the absolute or relative error goal is achieved or the iteration count reaches MaxIterations. MaxIterations speci es how many improvements of the extrapolation are tried before OscInt terminates with an error message. The default value is 6. IterationLength denotes how many elements are added to the sequence of partial sums with each iteration. It defaults to 4. 163
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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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1.5 Other velocity de nitions evide
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2.1 Superluminal wave propagation 2
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2.1 Superluminal wave propagation a
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2.1 Superluminal wave propagation t
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2.2 Quantum mechanical tunnelling e
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2.2 Quantum mechanical tunnelling R
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Chapter 3 Wave propagation in elect
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3.1 Model of a transmission line th
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3.2 Excursion: a delay line section
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3.3 Re ection due to termination mi
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3.4 A simple thought experiment I0
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3.5 A dispersive system: the lossle
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3.6 Inhomogeneous transmission line
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3.7 Turn-on e ects in a lossless pl
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3.8 Turn-on e ects in a wave guide
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3.8 Turn-on e ects in a wave guide
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3.9 A Gaussian pulse in plasma Like
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma Note
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4.1 The potential step 4.1 The pote
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4.1 The potential step Inside the b
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4.2 Initial wave forms -60 -50 -40
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4.2 Initial wave forms -60 -50 -40
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4.3 Examples of scattering processe
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4.4 The square barrier they have va
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4.4 The square barrier 1 0.8 0.6 0.
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4.5 Tunnelling time de nitions for
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4.5 Tunnelling time de nitions for
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4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events 2
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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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Page 135 and 136: Interlude Wave functions in graphic
Page 137 and 138: Wave functions in graphical represe
Page 139 and 140: Part II Numerical aspects of wave e
Page 141 and 142: 5.1 Univariate numerical quadrature
Page 143 and 144: 5.1 Univariate numerical quadrature
Page 145 and 146: 5.2 Convergence acceleration one, t
Page 147 and 148: 5.2 Convergence acceleration (1) ,1
Page 149 and 150: 6.1 Partitioning the integration in
Page 155 and 156: 6.2 Choosing the rst partition poin
Page 157 and 158: 6.3 How to compute the rst integral
Page 159 and 160: 6.3 How to compute the rst integral
Page 161 and 162: 6.4 Asymptotic partition consuming
Page 163 and 164: 6.4 Asymptotic partition of the int
Page 165 and 166: 6.5 Considerations for a Mathematic
Page 171 and 172: 6.6 Controlling the accuracy of the
Page 177: Chapter 7 Mathematica implementatio
Page 181 and 182: 7.3 Implementation of OscInt and re
Page 189 and 190: 7.4 Auxiliary functions While[itera
Page 191 and 192: 7.4 Auxiliary functions Linear appr
Page 193 and 194: 7.4 Auxiliary functions For the sak
Page 195 and 196: 7.4 Auxiliary functions Out[8]= 3 f
Page 197 and 198: 7.5 Test of the quadrature routine
Page 207 and 208: 8.1 Preparation of the wave integra
Page 213 and 214: 8.2 Outline of the program structur
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Page 217 and 218: 8.3 Implementation of the quadratur
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Page 221 and 222: 8.4 Test of the package 8.4 Test of
Page 223 and 224: 8.4 Test of the package -200 -150 -
Page 225 and 226: 8.4 Test of the package 0.4 0.2 -0.
Page 227 and 228: 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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