ELECTROMAGNETIC FIELDS
CONTENTS 1 LINEAR ANALYSIS LINEAR SPACES LINEAR TRANSFORMATIONS THE INVERSION PROBLEM GREEN’S FUNCTIONS RECIPROCITY GREEN’S DYADICS CONVERGENCE OF A SERIES EIGENFUNCTIONS INTEGRAL OPERATORS EIGENFUNCTION EXPANSIONS DISCRETIZATION MATRICES SOLUTION OF MATRIX EQUATIONS: STABILITY FINITE DIFFERENCES PERTURBATIONS 2 VARIATIONAL TECHNIQUES STATIONARY FUNCTIONALS A SUITABLE FUNCTIONAL FOR THE STRING PROBLEM FUNCTIONALS FOR THE GENERAL L TRANSFORMATION EULER’S EQUATIONS OF SOME IMPORTANT FUNCTIONALS DISCRETIZATION OF THE TRIAL FUNCTIONS SIMPLE FINITE ELEMENTS FOR PLANAR PROBLEMS MORE FINITE ELEMENTS DIRECT NUMERICAL SOLUTION OF MATRIX PROBLEMS ITERATIVE NUMERICAL SOLUTION OF MATRIX PROBLEMS 3 ELECTROSTATIC FIELDS IN THE PRESENCE OF DIELECTRICS VOLUME CHARGES IN VACUUM GREEN’S FUNCTION FOR IN?NITE SPACE MULTIPOLE EXPANSION POTENTIAL GENERATED BY A SINGLE LAYER OF CHARGE POTENTIAL GENERATED BY A DOUBLE LAYER OF CHARGE POTENTIAL GENERATED BY A LINEAR CHARGE SPHERICAL HARMONICS DIELECTRIC MATERIALS CAVITY FIELDS DIELECTRIC SPHERE IN AN EXTERNAL FIELD DIELECTRIC SPHEROID IN AN INCIDENT FIELD NUMERICAL METHODS 4 ELECTROSTATIC FIELDS IN THE PRESENCE OF CONDUCTORS CONDUCTIVITY POTENTIAL OUTSIDE A CHARGED CONDUCTOR CAPACITANCE MATRIX THE DIRICHLET PROBLEM THE NEUMANN PROBLEM NUMERICAL SOLUTION OF THE CHARGE DENSITY PROBLEM CONDUCTOR IN AN EXTERNAL FIELD CONDUCTORS IN THE PRESENCE OF DIELECTRICS CURRENT INJECTION INTO A CONDUCTING VOLUME CONTACT ELECTRODES CHAINS OF CONDUCTORS 5 SPECIAL GEOMETRIES FOR THE ELECTROSTATIC FIELD TWO-DIMENSIONAL POTENTIALS IN THE PLANE FIELD BEHAVIOR AT A CONDUCTING WEDGE FIELD BEHAVIOR AT A DIELECTRIC WEDGE SEPARATION OF VARIABLES IN TWO DIMENSIONS TWO-DIMENSIONAL INTEGRAL EQUATIONS FINITE METHODS IN TWO DIMENSIONS IN?NITE COMPUTATIONAL DOMAINS MORE TWO-DIMENSIONAL TECHNIQUES LAYERED MEDIA APERTURES AXISYMMETRIC GEOMETRIES CONICAL BOUNDARIES 6 MAGNETOSTATIC FIELDS MAGNETIC FIELDS IN FREE SPACE: VECTOR POTENTIAL FIELDS GENERATED BY LINEAR CURRENTS FIELDS GENERATED BY SURFACE CURRENTS FIELDS AT LARGE DISTANCES FROM THE SOURCES SCALAR POTENTIAL IN VACUUM MAGNETIC MATERIALS PERMANENT MAGNETS THE LIMIT OF IN?NITE PERMEABILITY TWO-DIMENSIONAL FIELDS IN THE PLANE NUMERICAL METHODS: INTEGRAL EQUATIONS NUMERICAL METHODS: FINITE ELEMENTS NONLINEAR MATERIALS STRONG MAGNETIC FIELDS AND FORCE-FREE CURRENTS 7 RADIATION IN FREE SPACE MAXWELL’S EQUATIONS THE WAVE EQUATION POTENTIALS SINUSOIDAL TIME DEPENDENCE: POLARIZATION PARTIALLY POLARIZED FIELDS THE RADIATION CONDITION TIME-HARMONIC POTENTIALS RADIATION PATTERNS EQUIVALENT SOURCES LINEAR WIRE ANTENNAS CURVED WIRE ANTENNAS: RADIATION TRANSIENT SOURCES 8 RADIATION IN A MATERIAL MEDIUM CONSTITUTIVE EQUATIONS PLANE WAVES RAY METHODS BEAMLIKE PROPAGATION EQUIVALENT CIRCUIT OF AN ANTENNA EFFECTIVE ANTENNA AREA 9 PLANE BOUNDARIES PLANE WAVE INCIDENT ON A PLANE BOUNDARY PROPAGATION THROUGH A LAYERED MEDIUM THE SOMMERFELD DIPOLE PROBLEM MULTILAYERED STRUCTURES PERIODIC STRUCTURES FIELD PENETRATION THROUGH APERTURES EDGE DIFFRACTION 10 RESONATORS EIGENVECTORS FOR AN ENCLOSED VOLUME EXCITATION OF A CAVITY DETERMINATION OF THE EIGENVECTORS RESONANCES OPEN RESONATORS: DIELECTRIC RESONANCES APERTURE COUPLING 11 SCATTERING: GENERALITIES THE SCATTERING MATRIX CROSS SECTIONS SCATTERING BY A SPHERE RESONANT SCATTERING THE SINGULARITY EXPANSION METHOD IMPEDANCE BOUNDARY CONDITIONS THIN LAYERS CHARACTERISTIC MODES 12 SCATTERING: NUMERICAL METHODS THE ELECTRIC FIELD INTEGRAL EQUATION THE MAGNETIC FIELD INTEGRAL EQUATION THE T-MATRIX NUMERICAL PROCEDURES INTEGRAL EQUATIONS FOR PENETRABLE BODIES ABSORBING BOUNDARY CONDITIONS FINITE ELEMENTS FINITE DIFFERENCES IN THE TIME DOMAIN 13 HIGH- AND LOW-FREQUENCY FIELDS PHYSICAL OPTICS GEOMETRICAL OPTICS GEOMETRIC THEORY OF DIFFRACTION EDGE CURRENTS AND EQUIVALENT CURRENTS HYBRID METHODS LOW-FREQUENCY FIELDS: THE RAYLEIGH REGION NON-CONDUCTING SCATTERERS AT LOW FREQUENCIES PERFECTLY CONDUCTING SCATTERERS AT LOW FREQUENCIES GOOD CONDUCTORS STEVENSON’S METHOD APPLIED TO GOOD CONDUCTORS CIRCUIT PARAMETERS TRANSIENT EDDY CURRENTS 14 TWO-DIMENSIONAL PROBLEMS E AND H WAVES SCATTERING BY PERFECTLY CONDUCTING CYLINDERS SCATTERING BY PENETRABLE CIRCULAR CYLINDERS SCATTERING BY ELLIPTIC CYLINDERS SCATTERING BY WEDGES INTEGRAL EQUATIONS FOR PERFECTLY CONDUCTING CYLINDERS SCATTERING BY PENETRABLE CYLINDERS LOW-FREQUENCY SCATTERING BY CYLINDERS SLOTS IN A PLANAR SCREEN MORE SLOT COUPLINGS TERMINATION OF A TRUNCATED DOMAIN LINE METHODS 15 CYLINDRICAL WAVEGUIDES FIELD EXPANSIONS IN A CLOSED WAVEGUIDE PROPAGATION IN A CLOSED WAVEGUIDE WAVEGUIDE LOSSES WAVEGUIDE NETWORKS APERTURE EXCITATION AND COUPLING GUIDED WAVES IN GENERAL MEDIA ORTHOGONALITY AND NORMALIZATION DIELECTRIC WAVEGUIDES OTHER EXAMPLES OF WAVEGUIDES 16 AXISYMMETRIC AND CONICAL BOUNDARIES FIELD EXPANSIONS FOR AXISYMMETRIC GEOMETRIES SCATTERING BY BODIES OF REVOLUTION: INTEGRAL EQUATIONS SCATTERING BY BODIES OF REVOLUTION: FINITE METHODS APERTURES IN AXISYMMETRIC SURFACES THE CONICAL WAVEGUIDE SINGULARITIES AT THE TIP OF A CONE RADIATION AND SCATTERING FROM CONES 17 ELECTRODYNAMICS OF MOVING BODIES FIELDS GENERATED BY A MOVING CHARGE THE LORENTZ TRANSFORMATION TRANSFORMATION OF FIELDS AND CURRENTS RADIATION FROM SOURCES: THE DOPPLER EFFECT CONSTITUTIVE EQUATIONS AND BOUNDARY CONDITIONS MATERIAL BODIES MOVING UNIFORMLY IN STATIC FIELDS MAGNETIC LEVITATION SCATTERERS IN UNIFORM MOTION MATERIAL BODIES IN NONUNIFORM MOTION ROTATING BODIES OF REVOLUTION MOTIONAL EDDY CURRENTS ACCELERATED FRAMES OF REFERENCE ROTATING COMOVING FRAMES CHAPTER LINEAR ANALYSIS THE LINEAR EQUATIONS OF MATHEMATICAL PHYSICS CAN BE SOLVED BY METHODS THAT HAVE FOUND APPLICATIONS IN MANY DISCIPLINES (E G , ELECTROSTATICS, HYDRODYNAMICS, ACOUSTICS, AND QUANTUM MECHANICS) IT IS INSTRUCTIVE, THEREFORE, TO DESCRIBE THESE METHODS IN VERY GENERAL AND ABSTRACT TERMS SUCH AN APPROACH AVOIDS TEDIOUS REPETITION OF STEPS THAT ARE ESSENTIALLY THE SAME FOR EACH NEW EQUATION THAT IS ENCOUNTERED A REALLY RIGOROUS DISCUSSION OF THE RELEVANT METHODOLOGY REQUIRES GREAT PRECISION OF LANGUAGE SUCH AN APPROACH IS BEYOND THE PALE OF THE CURRENT CHAPTER, WHOSE SOLE AMBITION IS TO GIVE A BROAD SURVEY — ONLY THE BARE ESSENTIALS — OF SOME OF THE MOST IMPORTANT TOPICS THE READER IS DIRECTED TO MORE SPECIALIZED TEXTS FOR ADDITIONAL (AND MORE RIGOROUS) INFORMATION [147, 150, 160, 168, 174, 186, 193] TO ILLUSTRATE THE BASIC ABSTRACT CONCEPTS OF LINEAR ANALYSIS, WE WILL CONSIDER TWO PARTICULARLY SIMPLE PHYSICAL SYSTEMS THE ?RST ONE IS THE ?EXIBLE STRING WHEN THE STRING IS SUBJECTED TO A UNIFORM LONGITUDINAL TENSION T AND A VERTICAL FORCE DENSITY G(X), ITS SMALL STATIC DISPLACEMENT Y(X) SATIS?ES THE DIFFERENTIAL EQUATION D2Y DX2 = ?G(X) T (1 1) TWO DIFFERENT TYPES OF BOUNDARY CONDITIONS ARE PERTINENT THEY CORRESPOND WITH (FIG 1 1) 1 THE CLAMPED STRING, WHERE THE DISPLACEMENT Y(X) VANISHES AT BOTH ENDS, X = 0 AND X = L 2 THE SLIDING STRING, WHICH IS FREE TO SLIDE VERTICALLY AT BOTH ENDS BUT IS CONSTRAINED TO KEEP ZERO SLOPE THERE A SECOND USEFUL EXAMPLE IS AFFORDED BY THE TRANSMISSION LINE (FIG 1 2) THE VOLTAGE AND CURRENT ON THE LINE SATISFY THE SYSTEM OF EQUATIONS ?V ?X = ?RI ?L ?I ?T + ?VA ?X ?I ?X = ?GV ?C ?V ?T + ?IA ?X (1 2) LINEAR SPACES THE ?ELD QUANTITIES THAT APPEAR IN A LINEAR PROBLEM POSSESS MATHEMATICAL PROPERTIES DICTATED BY THE PHYSICAL NATURE OF THE PHENOMENON UNDER INVESTIGATION THE DISPLACEMENT OF A STRING, FOR EXAMPLE, MUST BE A CONTINUOUS FUNCTION OF X THE ELECTRIC ?ELD NEAR A METALLIC EDGE MUST BE SQUARE-INTEGRABLE IN GENERAL, THE NATURE OF THE PROBLEM REQUIRES THE ?ELD QUANTITIES TO BELONG TO A LINEAR SPACE S; THAT IS, TO A COLLECTION OF ELEMENTS F FOR WHICH ADDITION AND MULTIPLICATION BY A SCALAR HAVE BEEN DE?NED IN SUCH A MANNER THAT 1 ADDITION AND MULTIPLICATION ARE COMMUTATIVE AND ASSOCIATIVE 1 1 LINEAR SPACES (1 6) THIS NORM CORRESPONDS WITH THE EUCLIDEAN CONCEPT OF LENGTH 3 THE IN?NITE NORM ?X??= MAX|XI| (1 7) LINEAR ANALYSIS A NORMED LINEAR SPACE PROVIDES A MEASURE OF THE CLOSENESS OF TWO ELEMENTS F1 AND F2 FROM RULE 1 ABOVE, WE NOTE THAT ?F1 ?F2 ?= 0 OCCURS ONLY FOR F1 = F2 THEREFORE ?F1 ?F2 ?, THE DISTANCE BETWEEN THE ELEMENTS, IS A MEASURE OF THEIR CLOSENESS THIS REMARK LEADS DIRECTLY TO THE CONCEPT OF CONVERGENCE, ACCORDING TO WHICH A SEQUENCE OF ELEMENTS FN CONVERGES TO F IF, FOR ANY GIVEN ?, THERE EXISTS A NUMBER N SUCH THAT ?F ?FN ?< ?, WHENEVER N > N (1 8) AS A BASIS FOR SOME IMPORTANT NUMERICAL METHODS, WE MUST GO A STEP FURTHER AND INTRODUCE THE INNER PRODUCT SPACE; THAT IS, A SPACE THAT IS ENDOWED WITH AN INNER (OR SCALAR) PRODUCT ?F1, F2? A SYMMETRIC SCALAR PRODUCT SATIS?ES THE RULES 1 ?F1, F2?S = ?F2, F1?S 2 ?A1 F1 + A2 F2, F3?S = A1?F1, F3?S + A2?F2, F3?S, (1 9) WHERE THE VARIOUS F ARE ELEMENTS OF S SUCH A PRODUCT IS RELEVANT FOR THE CONCEPTS OF REACTION AND RECIPROCITY, WHICH PLAY AN IMPORTANT ROLE IN LATER CHAPTERS TO ILLUSTRATE BY AN EXAMPLE, THE SCALAR PRODUCT ?Y1, Y2?S = ? Y1(X)Y2(X) DX IS SUITABLE FOR THE STUDY OF THE REAL DISPLACEMENTS OF A STRING HOWEVER, THERE ARE OTHER POSSIBILITIES, SUCH AS ?F1, F2?= ? [GRAD F1 GRAD F2 + L?2F1 F2] DX A MOST IMPORTANT SCALAR PRODUCT IS THE HILBERT PRODUCT, WHICH IS DE?NED BY THE RULES 1 ?F1, F2?H = ?F2, F1?? H, WHERE THE STAR DENOTES COMPLEX CONJUGATION 2 ?A1 F1 + A2 F2, F3?H = A1?F1, F3?H + A2?F2, F3?H ?F3, A1 F1 + A2 F2?H = A? 1?F3, F1?H + A? 2?F3, F2?H 3 ?F , F ?H ?0, WHERE THE EQUALITY SIGN HOLDS ONLY FOR F = 0 (1 10) SUCH A PRODUCT IS ASSOCIATED WITH THE CONCEPT “POWER,” AS ILLUSTRATED BY ITS APPLICATION TO A TRANSMISSION LINE IN THE SINUSOIDAL REGIME THE RELEVANT PRODUCT IS HERE ?V, I?H = L L VI?DX (1 11) THE RIGHT-HAND TERM OF EQUATION (1 11) IS CLEARLY THE AVERAGE VALUE OF THE COMPLEX POWER VI?ALONG THE LINE PROPERTY 3 ABOVE MAKES THE HILBERT PRODUCT PARTICULARLY SUITABLE FOR THE INTRODUCTION OF THE NORM ?F ?= ?F , F ?H, (1 12) AND THEREFORE FOR THE STUDY OF CONVERGENCE AS MEASURED BY THE MAGNITUDE OF THE ERROR WITH A HILBERT TYPE OF SCALAR PRODUCT, |?F , G?H| ??F ??G?= ?F , F ?H ?G, G?H (1 13) THIS IMPORTANT RELATIONSHIP IS THE SCHWARZ’INEQUALITY 1 2 LINEAR TRANSFORMATIONS LINEAR TRANSFORMATIONS THE BASIC PROBLEM FOR THE CLAMPED STRING IS TO DETERMINE THE DISPLACEMENT Y(X) DUE TO A GIVEN FORCING FUNCTION G(X) WE SHALL ASSUME THAT G(X) IS PIECEWISE CONTINUOUS THE STRING PROBLEM IS A PARTICULAR CASE OF A MORE GENERAL ONE, NAMELY LF = G, (1 18) LINEAR ANALYSIS WHERE L IS AN OPERATOR MAPPING THE SPACE D OF ELEMENTS F (THE DOMAIN) INTO THE SPACE R OF ELEMENTS LF (THE RANGE) THIS MAPPING IS A TRANSFORMATION IN THE CLAMPED STRING PROBLEM, THE DOMAIN CONSISTS OF FUNCTIONS THAT ARE CONTINUOUS IN (0, L), VANISH AT X = 0 AND X = L, AND HAVE PIECEWISE CONTINUOUS SECOND DERIVATIVES IN (0, L) A TRANSFORMATION IS LINEAR WHEN IT IS ADDITIVE AND HOMOGENEOUS; THAT IS, WHEN L( F1 + F2) = LF1 + LF2 AND L(AF ) = ALF THESE PROPERTIES IMPLY, ?RST, THAT THE OPERATOR IS LINEAR AND, SECOND, THAT THE DOMAIN CONTAINS ALL LINEAR COMBINATIONS OF ANY TWO OF ITS ELEMENTS SUCH A DOMAIN IS A LINEAR MANIFOLD THE TRANSFORMATION ASSOCIATED WITH THE CLAMPED STRING PROBLEM IS OBVIOUSLY LINEAR THE TRANSFORMATION ASSOCIATED WITH THE INHOMOGENEOUS BOUNDARY CONDITIONS Y = 1 AT X = 0 AND Y = 3 AT X = L IS NOT THE REASON IS CLEAR: THE SUM OF TWO POSSIBLE DISPLACEMENTS TAKES THE VALUES Y = 2 AT X = 0 AND Y = 6 AT X = L, AND THESE VALUES VIOLATE THE BOUNDARY CONDITIONS AN ELEMENT F IS SAID TO SATISFY EQUATION (1 18) IN A WEAK SENSE IF THE LEFT AND RIGHT TERMS OF EQUATION (1 18) HAVE EQUAL PROJECTIONS ON ANY FUNCTION W BELONGING TO A SPACE THAT INCLUDES R WE WRITE ?W, LF ?= ?W, G? (1 19) WEAK SOLUTIONS WILL OFTEN BE ENCOUNTERED IN FUTURE CHAPTERS THEY ARE FREQUENTLY EASIER TO CONSTRUCT THAN DIRECT (STRONG) SOLUTIONS OF THE INITIAL EQUATION (1 18) THE SOLUTION OF THE LATTER IS GREATLY FACILITATED WHEN A LINEAR OPERATOR LA, A SCALAR PRODUCT ?F , G?, AND A DOMAIN D A CAN BE FOUND SUCH THAT ?LF , H?= ?F , LAH? (1 20) WHENEVER H BELONGS TO D A THE LINEAR TRANSFORMATION DE?NED BY OPERATOR LA AND DOMAIN D A IS THE ADJOINT OF THE ORIGINAL ONE IT ALLOWS TRANSFERRING THE OPERATOR L, ACTING ON F , TO AN OPERATOR LA ACTING ON H IN THE CASE OF THE CLAMPED STRING, THE LEFT-HAND TERM OF EQUATION (1 20) CAN BE TRANSFORMED BY INTEGRATING BY PARTS ONE OBTAINS ?LF , H?= L D2F DX2 H DX = L F D2H DX2 DX + HDF DX ?F DH DX L (1 21) IT IS SEEN THAT EQUATION (1 20) IS SATIS?ED IF ONE CHOOSES LA TO BE THE DIFFERENTIAL OPERATOR D2/DX2, AND THE DOMAIN D A TO CONSIST OF FUNCTIONS THAT ARE ZERO AT X = 0 AND X = L [WHEREBY THE BRACKETED TERM IN (1 21) VANISHES] AND POSSESS PIECEWISE-CONTINUOUS SECOND DERIVATIVES CLEARLY, THE ADJOINT OF THE CLAMPED STRING TRANSFORMATION IS THE TRANSFORMATION ITSELF, WHICH IS THEREFORE TERMED SELF-ADJOINT THE PATTERN SUGGESTED BY EQUATION (1 21) IS FREQUENTLY ENCOUNTERED IN MATHEMATICAL PHYSICS IN GENERAL, THE SCALAR PRODUCT IS AN N-DIMENSIONAL INTEGRAL THE EQUIVALENT OF EQUA- TION (1 21) IS THEN OBTAINED BY USING A SUITABLE GREEN’S THEOREM IN N-DIMENSIONAL SPACE, IN WHICH THE BRACKETED TERM IS REPLACED BY AN (N ?1)-DIMENSIONAL INTEGRAL, LINEAR IN F AND H, WHICH IS TERMED THE BILINEAR CONCOMITANT J( F , H) THE DOMAIN D A OF H IS DETERMINED BY ENFORCING THE CONDITION J( F , H) = 0 SELF-ADJOINT TRANSFORMATIONS OCCUR VERY FREQUENTLY IN MATHEMATICAL PHYSICS, BUT THEY ARE BY NO MEANS THE RULE IN ELECTROMAGNETISM, IN PARTICULAR IN THE AREA OF SCATTERING, WHERE 1 2 LINEAR TRANSFORMATIONS THE OPERATOR IS L = CURL, AND THE ELEMENTS F ARE REAL DIFFERENTIABLE VECTORS IN A VOLUME V, WHICH, IN ADDITION, ARE PERPENDICULAR TO THE BOUNDARY S FROM (A1 32): V H CURL F DV = V CURL H F DV + S (UN × F) H DS, WHERE UN IS THE UNIT VECTOR ALONG THE OUTER NORMAL TO S THIS RELATIONSHIP SHOWS THAT THE ADJOINT OPERATOR IS THE CURL, AND THAT D A CONSISTS OF DIFFERENTIABLE VECTORS H (WITHOUT ANY CONDITIONS IMPOSED ON THEIR BEHAVIOR ON S) THE SCALAR PRODUCT ?LF , F ?IS A QUADRATIC FORM IN F THIS CAN EASILY BE CHECKED FOR THE CLAMPED STRING, WHERE ?LF , F ?= L D2F DX2 F DX = F DF DX L ? L DF DX DX = ? L DX (1 22) IN THIS CASE, THE QUADRATIC FORM IS REAL THIS PROPERTY HOLDS FOR ALL SELF-ADJOINT TRANSFORMATIONS IN A HILBERT SPACE WE NOTE, INDEED, THAT THE PROPERTIES OF THE HILBERT SCALAR PRODUCT IMPLY THAT ?LF , F ?H = ?F , LF ?? H ON THE OTHER HAND, ?F , LF ?? H = ?LF , F ?? H BECAUSE OF THE SELF-ADJOINT CHARACTER OF THE TRANSFORMATION COMPARISON OF THESE TWO EQUA- TIONSSHOWSTHATTHEQUADRATICFORMISEQUALTOITSCONJUGATE,HENCETHATITISREAL THEQUADRATIC FORM OF THE CLAMPED STRING HAS THE ADDITIONAL PROPERTY, EVIDENT FROM EQUATION (1 22), THAT IT IS NEGATIVE OR ZERO THE SAME IS TRUE FOR THE QUADRATIC FORM OF THE SLIDING STRING THE CORRESPONDING TRANSFORMATIONS ARE TERMED NONPOSITIVE IN THE CASE OF THE STRING, THE VAN- ISHING OF THE QUADRATIC FORM ?LF , F ?IMPLIES THAT DF /DX = 0 AT ALL POINTS OF THE INTERVAL (0, 1) THIS, IN TURN, REQUIRES F (X) TO BE A CONSTANT FOR THE CLAMPED STRING, THIS CONSTANT MUST BE ZERO BECAUSE OF THE END CONDITIONS THE CORRESPONDING TRANSFORMATION IS TERMED NEGATIVE-DE?NITE, WHICH MEANS THAT IT IS A NONPOSITIVE TRANSFORMATION WHOSE ?LF , F ?IS LINEAR ANALYSIS ALWAYS NEGATIVE FOR NONZERO ELEMENTS F AND VANISHES FOR, AND ONLY FOR, THE ZERO ELEMENT THE TRANSFORMATION ASSOCIATED WITH THE SLIDING STRING IS NOT DE?NITE, BECAUSE ?LF , F ?= THE INVERSION PROBLEM A VERY FUNDAMENTAL PROBLEM CONSISTS IN INVERTING A LINEAR TRANSFORMATION; THAT IS, ?NDING AN ELEMENT F OF THE DOMAIN SUCH THAT LF = G, G BEING GIVEN IN R THIS INVERSE TRANSFORMATION CAN BE REPRESENTED SYMBOLICALLY BY F = L?1G THREE QUESTIONS IMMEDIATELY ARISE: 1 IS THERE AN INVERSE? 2 IS THAT INVERSE UNIQUE? 3 IS THE SOLUTION STABLE? THE QUESTION OF UNIQUENESS CAN BE ANSWERED QUITE SIMPLY ASSUME THAT THERE ARE TWO DISTINCT SOLUTIONS, F1 AND F2 THESE SOLUTIONS SATISFY THE EQUATIONS LF1 = G AND LF2 = G 1 3 THE INVERSION PROBLEM SUBTRACTION OF CORRESPONDING MEMBERS SHOWS THAT THE DIFFERENCE F0 = F1 ?F2 MUST BE A SOLUTION OF THE HOMOGENEOUS PROBLEM LF0 = 0 (1 28) IF THIS PROBLEM DOES NOT POSSESS A NONZERO SOLUTION, F1 AND F2 MUST BE EQUAL, AND THE SOLU- TION OF THE ORIGINAL INHOMOGENEOUS PROBLEM IS UNIQUE IF, ON THE CONTRARY, THE HOMOGENEOUS PROBLEM HAS LINEARLY INDEPENDENT SOLUTIONS F01, F02, , F0N, THE SOLUTION OF LF = G IS DETER- MINED TO WITHIN AN ARBITRARY LINEAR COMBINATION OF THE F0I TERMS IT IS CLEAR THAT UNIQUENESS OBTAINS FOR A POSITIVE-DE?NITE TRANSFORMATION, AS ?LF0, F0?= 0 IMPLIES F0 = 0 FOR SUCH A CASE THE CONCEPT WELL-POSED IS MOST IMPORTANT FOR PRACTICAL NUMERICAL COMPUTATIONS IT IMPLIES THAT THE SOLUTION IS UNIQUE; A SOLUTION EXISTS FOR ANY G; AND THE SOLUTION IS STABLE; THAT IS, A SMALL VARIATION IN THE CONDITIONS OF THE PROBLEM (IN L OR G) PRODUCES ONLY A SMALL VARIATION IN THE SOLUTION F THESE POINTS ARE BELABORED FURTHER IN SECTION 1 13, WHICH IS DEVOTED TO THE SOLUTION OF MATRIX PROBLEMS NOTE THAT INSTABILITY CAN BE REMEDIED BY METHODS SUCH AS REGULARIZATION [175] TURNING NOW TO THE CLAMPED STRING, WE NOTE, FROM DIRECT INTEGRATION OF EQUATION (1 1), THAT THE HOMOGENEOUS PROBLEM ONLY HAS THE SOLUTION F0 = 0 THIS RESULT CAN BE OBTAINED IN AN INDIRECT MANNER, FREQUENTLY USED FOR NONPOSITIVE OR NONNEGATIVE TRANSFORMATIONS THE METHOD CONSISTS IN EVALUATING ?LF0, F0? FOR THE CLAMPED STRING, ?LF0, F0?= L DX L DX CLEARLY, LF0 = 0 IMPLIES THAT ?LF0, F0?= 0, WHICH IN TURN REQUIRES THE ?RST DERIVATIVE DY0/DX TO VANISH FOR A CLAMPED STRING, THIS MEANS THAT Y0 IS ZERO THE PHYSICAL INTERPRETATION IS OBVIOUS: THE CLAMPED STRING WITHOUT FORCING FUNCTION REMAINS STRETCHED ALONG THE X-AXIS FOR THE SLIDING STRING, ON THE CONTRARY, THE HOMOGENEOUS PROBLEM HAS THE NONZERO SOLUTION Y0 = CONST , WHICH MEANS THAT THE AVERAGE HEIGHT OF THE STRING IS NOT DE?NED OR, EQUIVALENTLY, THAT ANY EQUILIBRIUM CON?GURATION OF THE STRING CAN BE DISPLACED VERTICALLY BY A GIVEN AMOUNT AND YET REMAIN AN EQUILIBRIUM CON?GURATION A VERY IMPORTANT REMARK SHOULD BE MADE IN THIS CONNECTION FOR THE CLAMPED STRING AN EQUILIBRIUM POSITION CAN BE OBTAINED FOR ANY FORCE DISTRIBUTION G(X) FOR THE SLIDING STRING, ON THE CONTRARY, A STABLE DISPLACEMENT IS POSSIBLE ONLY IF THE NET VERTICAL FORCE VANISHES, THAT IS, IF L G(X) DX = 0 THIS OBVIOUS PHYSICAL LIMITATION IS A PARTICULAR FORM OF A GENERAL MATHEMATICAL REQUIREMENT, WHICH MAY BE FORMULATED BY CONSIDERING THE HOMOGENEOUS ADJOINT PROBLEM OF (1 18), VIZ LAH0 = (H0 IN D A) (1 29) F ? 2 = F2 ??F ? 1, F2? ?F ? 1, F ? 1?F ? ?F ? I , FN? ?F ? I , F ? I ?F ? I (1 34) 1 4 GREEN’S FUNCTIONS YG(X|X?)P(X?) X? = L CLAMPED STRING WITH UNIT FORCE T THE NOTION OF GREEN’S FUNCTION IS FUNDAMENTAL FOR THE SOLUTION OF THE GENERAL LINEAR DIFFERENTIAL PROBLEM OF (1 18) WE WRITE THIS SOLUTION IN THE FORM F (R) = L?1G(R) = V G(R|R?)G(R?) DV? (1 41) 1 4 GREEN’S FUNCTIONS THIS RELATIONSHIP SHOULD BE VALID FOR ALL POINTS R OF THE VOLUME V IN WHICH THE DIFFERENTIAL EQUATION IS SATIS?ED IN OUR SEARCH FOR G(R|R?), LET US ASSUME THAT THE ADJOINT OF THE ORIGINAL TRANSFORMATION HAS BEEN DETERMINED THIS DETERMINATION RESTS ON THE POSSIBILITY OF DERIVING A “GREEN’S THEOREM” ?LF , H?H = V (LF )H?DV = V F (LAH)?DV + S J( F , H) DS, (1 42) WHERE ONE ASSUMES THAT F , H, LF , LAH SATISFY THE NECESSARY CONTINUITY CONDITIONS IN V NOTE THAT V IS AN N-DIMENSIONAL VOLUME (SPACE AND TIME COORDINATES CAN BE INCLUDED) AND THAT S IS ITS (N ?1)-DIMENSIONAL BOUNDARY (OFTEN WRITTEN AS ?V) EQUATION (1 42) SUGGESTS THAT THE OPERATOR OF THE ADJOINT TRANSFORMATION SHOULD BE LA THE DOMAIN D A OF THE TRANSFORMATION IS DE?NED BY REQUIRING THE BILINEAR CONCOMITANT (WHICH IN THIS CASE IS THE SURFACE INTEGRAL) TO VANISH WHEN F BELONGS TO THE DOMAIN OF THE ORIGINAL TRANSFORMATION AND H BELONGS TO D A A GLANCE AT THE PARTICULAR CASE OF THE ?EXIBLE STRING WILL READILY CLARIFY THESE RATHER ABSTRACT STATEMENTS CONSIDER NOW A FUNCTION H THAT SATIS?ES THE BOUNDARY CONDITIONS ASSOCIATED WITH D A, AND ALSO LAH = FINITE VOLUME, FROM WHICH POINTS R0 AND R1 HAVE BEEN EXCLUDED J?(G, H) DS 1 5 RECIPROCITY ?2P ?T2 = G(R, T), (1 55) 4? ALL SPACE G(R?, T ?|R? ?R|/C0) |R? ?R| DV? (1 56) THERE IS NO “USUAL” FUNCTION G THAT ALLOWS REPRESENTATION OF THIS RELATIONSHIP IN THE FORM P(R, T) = +? ?? DT? ALL SPACE G(R, T|R?, T?)G(R?, T?) DV?, BUT THE DESIRED RESULT MAY BE OBTAINED BY SETTING G(R, T|R?, T?) = ?