Wave Propagation in Linear Media | re-examined

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7.3.1 OscInt ApproxLimGeneric AsymptoticExpand PolynomialDegree 7 Mathematica implementation of a quadrature function Approximation error control Zero computation ApproxLimQuad ApproxLimHyp QuadOffset QuadZero HypOffset HypZero Figure 7.2: The auxiliary functions of OscInt HypOffsetExact HypZeroExact Since there are two di erent list of arguments that can be passed to OscInt (see also section 7.1), we must specify two di erent rules that match these possibilities. The more general set of arguments is the one where the rst partition point is preset by the user. The case where only the lower integration limit is given can be easily extended to the general rule by setting a0 = a. Therefore only the body of the general rule needs to contain the entire code, whereas the body of the more special case simply calls OscInt again with the proper parameters, which is generally considered good programming style. OscInt[fint_, fzero_, fa_, a0_g, opts___Rule] := Module[fparttab,lowlim,prec, wp = WorkingPrecision/.foptsg/.Options[OscInt], nt = NSumTerms/. foptsg/.Options[OscInt], ne = NSumExtraTerms/. foptsg/.Options[OscInt], ac = AccuracyControl/. foptsg/.Options[OscInt], pc = PrecisionControl/.foptsg/.Options[OscInt]g, If[ac === Infinity && pc === Infinity, lowlim = If[a == a0, a, N[Max[a,a0],2 wp]]; parttab = PartitionTable[fzero,lowlim,ne+nt+1,opts]; PartInt[fint,parttab,a,opts], prec = If[pc === Infinity, Max[wp,$MachinePrecision+1], Max[wp,$MachinePrecision+1,pc+13]]; OscIntControlled[fint,fzero,a,WorkingPrecision->prec,opts]] ]; OscInt[fint_, fzero_, a_, opts___Rule] := OscInt[fint,fzero,fa,ag,opts]; Remark (Multiple de nitions) Note that the order of the two de nitions within the package is essential, because Mathematica searches the list from the top to the bottom 166
7.3 Implementation of OscInt and related functions in order to nd a rule that matches a given set of arguments. If the two de nitions were interchanged, a list fb,b0g would match the pattern a , and consequently the variable a would erroneously be assigned the value fb,b0g. The function rst extracts the values of a couple of relevant options and then checks whether error control is required. If both AccuracyControl and PrecisionControl equal Infinity, then no error control is used, and OscInt performs a straightforward computation. The lower limit of the list of subdivision points is set either to a or maxfa; a0g, the latter being calculated numerically at a higher precision, because the function Max works only with numbers but not with symbols. Remark (Options and defaults) The ltering of the options is worth a separate mention [146]. It is accomplished by a double replacement rule like for instance in wp = WorkingPrecision/.foptsg/.Options[OscInt], which works as follows: If the user speci es a value WorkingPrecision->prec for this option, then it is passed to the function in the argument opts. The statement is now evaluated from the left to the right, and the rst replacement evaluates to prec. The second substitution fails because the list of default options obtained by Options[OscInt] contains of course no rule prec->x. If, on the other hand, the function is called without any option, then the rst replacement cannot be evaluated. In this case, the second rule matches and wp is set to the default value. The number of partition points is obtained from NSumTerms plus NSumExtraTerms plus one, so that the sequence of partial sums has the correct length for extrapolation. With the list of subdivision points, PartInt is called to compute the actual integral. If error control is to be used, a reasonable value for the WorkingPrecision is calculated from the machine precision plus one, the parameter wp that might have been set by the user (or equals the default value otherwise), and the PrecisionControl parameter if applicable. This working precision, which might di er from the user's speci cation, is then used in the call of the function OscIntControlled. Note that this particular option is placed before the other options opts, which causes the new setting for WorkingPrecision to override any other speci cation for this option, because Mathematica takes the rst value it encounters and disregards all subsequent ones. 7.3.2 PartitionTable This function actually only calls PartitionOffs and PartitionPoints, but is nonetheless useful for the computation of a list of subdivision points. Its syntax is PartitionTable[zerof, a, n, opts] with the same input arguments as OscInt except for n, which denotes the number of partition points to be computed. The options are the same as for OscInt. 167
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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 131 and 132: 4.6 Examples of tunnelling events 7
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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 and 178: Chapter 7 Mathematica implementatio
Page 179 and 180: 7.1 User interface of the function
Page 181: 7.3 Implementation of OscInt and re
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Page 187 and 188: 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
Page 229 and 230: 8.4 Test of the package Out[24]= 0
Page 231 and 232: 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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