Wave Propagation in Linear Media | re-examined

More documents

Recommendations

Info

integration limit a or the rightmost extremum of f, whichever is greater, are calculated. In the second case, the first n function values of f to the right of a are sought. Note that in this case, f is supposed to return values only to the right of the rightmost extremum. In any case, f must be of the form f[...][x]." OscInt::usage = "OscInt[f,fpartition,a,(opts)] returns the univariate integral of an oscillating function f over the range (a,Infinity) by means of a partition extrapolation strategy. The function f is required to of the form f[...][x], x being the integration variable. Quadrature is carried out between successive partition points defined by the function fpartition[...][x] that either returns the zeros of the integrand or the argument of the oscillating part. The sequence of partition points is computed by the function PartitionTable, the actual integration is carried out with the function PartInt. The first partition point can be given explicitly with OscInt[f,fpartition,{a,a0},(opts)]. The options are passed to PartitionTable and PartInt." OscIntControlled::usage = "OscIntControlled[f,fpartition,a,(opts)] works like OscInt with the additional feature that the accuracy of the result may be specified in advance." FunctionType::usage = "FunctionType is an option for PartitionTable (and OscInt) that specifies how the function passed to PartitionTable is to be interpreted. FunctionType -> ZeroList means that the function returns consecutive zeros of the integrand. FunctionType -> SinArgument or CosArgument means that it is the argument function of the respective cyclic function." ZeroList::usage = "ZeroList is a value to FunctionType." SinArgument::usage = CosArgument::usage = "Argument is a value to FunctionType." AccuracyControl::usage = "AccuracyControl specifies whether the accuracy of the result of OscInt should be controlled. With AccuracyControl -> n, at most MaxIterations are performed to achieve the number of n correct digits. With AccuracyControl -> False, no checking is done. The control loop uses the built-in function Accuracy." IterationLength::usage = "IterationLength specifies the number of terms that are added to the sequence of partial sums with each iteration." ApproxLimGeneric::nounique = "There are `1` real solutions `2`, here is the largest."; ApproxLimGeneric::nosolution = "There is no positive real valued solution."; QuadZero::noquad = "Function is not a quadratic polynomial"; ApproxLimQuad::noapprox = "No approximation needed"; ApproxLimLinear::noosc = "Function not infinitely oscillating"; ZerosInBetween::outside = "Function has no zeros inside the interval"; OscInt::lowfirst = "First partition point is smaller than lower limit, only `1` points are greater. This may cause problems."; OscInt::badparam = "\"`1`\" is no valid parameter for AccuracyControl."; OscInt::accfail = "OscInt failed to achieve the desired accuracy after `1` iterations. Increase MaxIterations or lower AccuracyControl."; PartitionTable::badparam = "\"`1`\" is no valid parameter for FunctionType."; Begin["`Private`"] Options[OscInt] = {WorkingPrecision->$MachinePrecision, AccuracyGoal->Infinity, PrecisionGoal->Automatic, MinRecursion->0, MaxRecursion->20, NSumTerms->10, NSumExtraTerms->12, 216 A Mathematica packages
A.1 Numerical quadrature WynnDegree->Infinity, MaxIterations->5, AccuracyControl->False, IterationLength->2}; Options[PartInt] = Options[OscInt]; Options[PartitionTable] = {WorkingPrecision->$MachinePrecision, AccuracyGoal->Automatic, MaxRecursion->20, FunctionType->SinArgument}; PolynomialDegree[f_] := Length[CoefficientList[Series[f[x],{x,Infinity,1}],x]]-1; PolynomialDegree[f_,x_Symbol] := Length[CoefficientList[Series[f,{x,Infinity,1}],x]]-1; AsymptoticExpand[f_] := Normal[Series[f[x],{x,Infinity,PolynomialDegree[f]}]]; AsymptoticExpand[f_,x_Symbol] := Normal[Series[f,{x,Infinity,PolynomialDegree[f,x]}]]; ApproxLimGeneric[f_,goal_] := Module[{fapprox,respos,resneg,res}, fapprox = AsymptoticExpand[f]; respos = Select[ NSolve[Evaluate[ D[fapprox,x] / D[fapprox - f[x],x]] == goal,x],(Im[x/.#] == 0)&]; resneg = Select[ NSolve[Evaluate[ D[fapprox,x] / D[fapprox - f[x],x]] == -goal,x],(Im[x/.#] == 0)&]; res = Select[Join[respos,resneg],Positive[x/.#]&]; If[Length[res] == 0, Message[ApproxLimGeneric::nosolution], If[Length[res] > 1, Message[ApproxLimGeneric::nounique,Length[res],res]]; x/.Last[res]] ] ApproxLimQuad[a_,b_,c_,d_,goal_] := Which[ a != 0 && c != 0 && d != 0, x/.Flatten[Last[ NSolve[(2 a x + b + c) / (c (1-x/Sqrt[x^2+d])) == Sign[a c d] goal,x]]], a == 0 && c != 0 && d != 0, ApproxLimLinear[b,c,d,goal], True, Message[ApproxLimQuad::noapprox];0] ApproxLimLinear[a_,b_,c_,goal_] := Which[ a + b == 0, Message[ApproxLimLinear::noosc];0, Abs[(a + b) / b] > goal, 0, True, x/.Flatten[ NSolve[(a + b) / (b (1-x/Sqrt[x^2+c])) == Sign[(a+b) b c] goal,x]]]; QuadOffset[a_,b_,c_] := Which[ a == 0, Ceiling[c/Pi], a > 0,-Floor[(b^2 - 4 a c)/(4 a Pi)], True, -Ceiling[(b^2 - 4 a c)/(4 a Pi)]]; QuadZero[a_,b_,c_,o_][k_] := Which[ a > 0, (-b+Sqrt[b^2-4 a (c-( k+o) Pi)])/(2 a), a < 0, (-b-Sqrt[b^2-4 a (c-(-k+o) Pi)])/(2 a), a == 0 && b > 0, (( k+o) Pi - c)/b, a == 0 && b < 0, ((-k+o) Pi - c)/b, True, k]; HypApproxError[b_,c_,goal_] := x/.Last[Solve[b (Sqrt[x^2 + c]-x) == Sign[c b] goal,x]]; ZerosInBetween[zerof_,a_,b_] := Module[{min=Min[a,b], max=Max[a,b]}, Which[N[zerof[0]] < min, Ceiling[i/.FindRoot[zerof[i] == max,{i,0,1}]] - Ceiling[i/.FindRoot[zerof[i] == min,{i,0,1}]], 217
Page 1 and 2:
DISSERTATION Wave Propagation in Li
Page 3:
Kurzfassung Seit der Entdeckung des
Page 7 and 8:
Nullum est iam dictum, quod non sit
Page 9 and 10:
Preface Our popular writers and rep
Page 11 and 12:
the quest for superluminality and t
Page 13 and 14:
Contents Part I Wave propagation ph
Page 15 and 16:
7.3.4 PartitionPoints . . . . . . .
Page 17 and 18:
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 59 and 60:
Page 61 and 62:
Page 63 and 64:
3.6 Inhomogeneous transmission line
Page 65 and 66:
3.7 Turn-on e ects in a lossless pl
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
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 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
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
Page 115 and 116:
4.5 Tunnelling time de nitions for
Page 117 and 118:
4.6 Examples of tunnelling events P
Page 119 and 120:
4.6 Examples of tunnelling events 2
Page 121 and 122:
4.6 Examples of tunnelling events -
Page 123 and 124:
4.6 Examples of tunnelling events t
Page 125 and 126:
4.6 Examples of tunnelling events P
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
4.6 Examples of tunnelling events T
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 151 and 152:
Page 153 and 154:
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 167 and 168:
Page 169 and 170:
Page 171 and 172:
6.6 Controlling the accuracy of the
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Chapter 7 Mathematica implementatio
Page 179 and 180:
7.1 User interface of the function
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
Page 215 and 216: 8.2 Outline of the program structur
Page 217 and 218: 8.3 Implementation of the quadratur
Page 219 and 220: 8.3 Implementation of the quadratur
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: A.1 Numerical quadrature Asymptotic
Page 235 and 236: A.1 Numerical quadrature Which[ft =
Page 237 and 238: A.2 Solutions for the step potentia
Page 245 and 246: A.3 Solutions for the square barrie
Page 247 and 248: A.3 Solutions for the square barrie
Page 249 and 250: A.4 Electromagnetic waves function
Page 251 and 252: A.5 Utilities for displaying points
Page 253 and 254: Bibliography Bibliography [1] James
Page 255 and 256: Bibliography [29] Kurt Edmund Oughs
Page 257 and 258: Bibliography [59] Ch. Spielmann, R.
Page 259 and 260: Bibliography [91] C. R. Leavens and
Page 261 and 262: Bibliography [123] T. O. Espelid an
Page 263 and 264: Index Index absorption, 7, 9 accura
Page 265 and 266: Index saddle point integration, 11,
Page 267: Curriculum vitae Dipl.-Ing. Thilo S
show all

Wave Propagation in Linear Media | re-examined

Create successful ePaper yourself

Delete template?

Save as template?