We present several new studies on the mathematical methods f ormulation of the important problem of electromagnetic oscillations in c avities on anisotropic and axial- anisotropic media, specially on the writing of the relevant dynamical wave equations and their correct boundary conditions. This study is the physic al model for applications of a path integral method presented earlier [LCL Botelho - IJTP, 49,1396-1404-2010]. 1. Electromagnetic Oscillations Dynamical Wave Equations We start this not by writing the initial value problem for Max well equations in the presence of sources and in a compact domain Ω with boundary ∂Ω with spatially variable constitutive parameter ( ε(/vector r),µ(/vector r),σ(/vector r)) ([1], [2]) /vector∇×/vectorE+µ∂ ∂t/vectorH= 0 (1) /vector∇×/vectorH−µε∂/vectorE ∂t=µ/vectorj+σµ/vectorE (2) /vector∇·/vectorB= 0 (3) /vector∇(ε/vectorE) =ρ (4) /vectorE(/vector r,0) =/vectorE0(/vector r) (5) 1 /vectorH(/vector r,0) =/vectorH0(/vector r) (6) Let us ﬁrstly search appropriated boundary conditions to be imposed on the electric andmagneticﬁelds /vectorE(/vector r,t)and/vectorH(/vector r,t)respectivellyinordertoleadtotheproblemunicity of the above written set of PDE’s, eq(1)-eq(6). On basis of eq(1) and eq(2), we have the obvious energy balanc e equation for the PDE’s system above written /bracketleftbigg /vectorH·/parenleftbigg /vector∇×/vectorE+µ∂ ∂t/vectorH/parenrightbigg/bracketrightbigg −/bracketleftigg (/vector∇×/vectorH)·/vectorE−µε∂/vectorE ∂t·/vectorE−/vectorjµ/vectorE−σµ/vectorE·/vectorE/bracketrightigg = 0 (7) In other words, we have the following electromagnetic energ y balance equation in Ω −/parenleftbigg1 2/integraldisplay Ω∂ ∂t/parenleftbig εµ/vectorE2+µ/vectorH2/parenrightbig d3r/parenrightbigg = +/integraldisplay Ω(µ/vectorE./vectorJ)d3r+/integraldisplay Ωµσ(/vectorE)2d3x +/integraldisplay ∂ΩFlux Poynting vector/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright/parenleftbig (/vectorE×/vectorH)·/vector n/parenrightbig dA (8) It one supposes that has two diﬀerent set of solutions for Max well equations eq(1) and eq(2), namely ( E1,H1) and (E2,H2), then theirs diﬀerence satisfy the obvious inequality +1 2∂ ∂t E(t)/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright /integraldisplay Ω/parenleftbig µ∈(∆/vectorE)2+µ(∆/vectorH)2/parenrightbig d3r ≤ −/integraldisplay ∂Ω/parenleftbig (∆/vectorE×∆/vectorH)·/vector n/parenrightbig dA (9) since the loss of ohmic energy is positive-deﬁnite /integraldisplay Ωµσ(∆/vectorE)2d3x≥0 (10) Now if one chooses boundary conditions that lead to the vanish ing of the Poyinting ﬂux on eq(8), one gets the problem unicity since 1 2∂ ∂tE(t)≤0⇒ E(t)≤ E(0) = 0. (11) Let us analyze examples of boundary conditions that lead to v anishing of the Poynting ﬂux on eq(9)/parenleftbig (/vectorE2−/vectorE1)×(/vectorH2−/vectorH1)·/vector n/vextendsingle/vextendsingle ∂Ω= 0 (12) 2 Namely: a)/vectorE/vextendsingle/vextendsingle ∂Ω= 0 (13) b)/vectorH/vextendsingle/vextendsingle ∂Ω= 0 (14)  c)/vectorE·T1,2/vextendsingle/vextendsingle ∂Ω= 0⇔(/vector∇×/vectorH)·T1,2/vextendsingle/vextendsingle ∂Ω(15) d)/vectorH·T1,2/vextendsingle/vextendsingle ∂Ω= 0 (16 −a) e) (/vectorE×/vectorH)·/vector n/vextendsingle/vextendsingle ∂Ω= 0 (16 −b) where/vectorT1,2denote the tangent vectors on the domain boundary ∂Ω. Let us consider now a medium with constant electromagnetic p arameters ε=ε0;µ=µ0;σ=σ0. In this simply case we have the decoupled dynamical equation s for Electric and Mag- netic ﬁelds with the non-absorbing boundary conditions ∆/vectorE=εµ∂2/vectorE ∂t2+σµ∂/vectorE ∂t+/parenleftigg µ∂/vectorj ∂t+/vector∇(ρ)/parenrightigg (17) /vectorE(/vector r,0) =E0(/vector r) (18) −/vectorEt(/vector r,0) =/vectorJ(/vector r,0)+σ/vectorE0(/vector r)−(∇×/vectorH0(/vector r)) (19) /vectorE·T1,2/vextendsingle/vextendsingle ∂Ω= 0 (20) ∆/vectorH=εµ∂2/vectorH ∂t2+µσ∂/vectorH ∂t−(/vector∇×/vectorJ) (21) /vectorH(/vector r,0) =H0(/vector r) (22) /vectorHt(/vector r,0) =−1 µ(/vector∇×/vectorE0(r)) (23) (/vector∇×/vectorH)·/vectorT1,2/vextendsingle/vextendsingle ∂Ω= (/vectorJ·/vectorT1,2)/vextendsingle/vextendsingle ∂Ω= 0 (24) Note that the divergencelless of the magnetic ﬁeld eq(3) in th is case can be insured straightforwardly by the equation below: HTr(/vector r,t) =H(/vector r,t)−1 4π/vector∇/vector r·/braceleftigg/integraldisplay Ω/parenleftigg div(/vectorH(/vector r′,t) |/vector r−/vector r′|/parenrightigg d3r′/bracerightigg (25) Let us re-write the problem in terms of vector and scalar pote ntials with the condition of−∞< t <∞(/vectorE(·,t) and/vectorH(·,t)∈L2(R3)). 3 Since in this case one has gauge invariance so redundancy on t he Maxwell equations solutions one should choose the generalized radiation gaug e as a natural gauge ﬁxing to ﬁnd a unique solution: 1 µ/vector∇·/vectorA−ε∂φ ∂t−σφ= 0. (26) In this case the Maxwell dynamical equations eq(17)-eq(24) take the more invariant form below through the use of the electromagnetic potential s (/vectorA,φ).   ∆/vectorA−εµ∂2 ∂t2/vectorA=−µ/vectorj+σµ∂ ∂t/vectorA ∆φ−µε∂2 ∂t2φ−µσ∂φ ∂t=ρ /vectorE·/vectorT1,2/vextendsingle/vextendsingle ∂Ω=/parenleftig −∂ ∂t/vectorA+∇φ/parenrightig ·/vectorT1,2/vextendsingle/vextendsingle ∂Ω= 0 (/vector∇×/vectorH)·/vectorT1,2/vextendsingle/vextendsingle ∂Ω≡/parenleftbig ∇×(/vector∇×/vectorA)/parenrightbig ·/vectorT1,2/vextendsingle/vextendsingle ∂Ω=µ/parenleftbig/vectorJ·/vectorT1,2/vextendsingle/vextendsingle ∂Ω/parenrightbig (27) At this point we address our readers to implement numerical a pproximate procedures to solve the above written set of linear boundary value probl ems in a computer by ﬁnite- diﬀerences or ﬁnite-elements. Finally we write the associated wave equations associative to the full Maxwell equa- tions in the context of an electromagnetic spatially variab le medium namely: ∆/vectorH−/parenleftigg/vector∇ε ε/parenrightigg (/vector r)∂2H ∂t2=/braceleftbigg −/vector∇×/vectorj+σµ∂/vectorH ∂t −(/vector∇σ)×/vectorE−/vector∇/parenleftigg/vectorH·/vector∇ µ(/vector r)/parenrightigg −/parenleftigg/vector∇ε ε/parenrightigg (/vector r)×(/vector∇×/vectorH−/vectorj−σ/vectorE)/bracerightbigg (28) ∆/vectorE−(εµ)(/vector r)∂2/vectorE ∂t2=µ∂/vectorj ∂t+σµ∂/vectorE ∂t −/vector∇/parenleftigg ρ−/vectorE·→ ∇E ε/parenrightigg −/parenleftigg/vector∇µ µ×/vector∇×/vectorE/parenrightigg (29) 4 + plus boundary and initial conditions At this point we call the reader attention that solving elect romagnetic problem in cavitieswithanon-trivialtopological/homologicalclas swithpotentials, oneencounterthe severe diﬃculty of the Helmholtz-Hodge non trivial decompsot ion of the electromagnetic ﬁelds in term of the above mentioned potentials. For instanc e,/vector∇·/vectorB= 0 in Ω means that /vectorB=/vector∇×/vectorA+/vectorBtopwhere/vectorBtopis an haarmonic vector ﬁeld conﬁguration associated to the topological-homological characterization of Ω (with Ω bei ng a domain with holes inside for instance, and of diﬃcult determination from the local Ma xwell PDE’systems! In this Helmholtz-Hodge context the Maxwell equations written in ter ms of potential ( /vectorA,ϕ) are of form with constant medium electric parameters for instan ce) ∆/vectorA−εµ∂2 ∂t2/vectorA= homological current/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright (/vector∇×/vectorHtop−εµ∂ ∂t/vectorEtop +/vectorj+σ/vectorE (30-a) ∆φ−εµ∂2 ∂t2φ=ρ ε−homological charge density/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright µ∂ ∂t(/vector∇·/vectorHtop) (30-b) radiation gauge ﬁxing/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright /vector∇·/vectorA−εµ∂φ ∂t= 0 (30-c) + boundary conditions. Here the homological-topological electromagnetic ﬁeld con ﬁgurations are deﬁned by the Hodge theorem. Hodge-Helmholtz theorem/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright /vectorH=1 µ(/vector∇×/vectorA)+Htop(31-a) 0 =∇×Etop≡ −µ∂/vectorHtop dt(31-b) We note that non trivial topology on the manifold structure o f Ω appear dinamically in Maxwell equations as sources of to electric charge and ele ctric currents. It appears thus, that considering from the beginning the dyn amical equations written directly for the strenght ﬁeld ( /vectorE,/vectorH), all subtle and very diﬃculty topological-homological constraints imposed by the Hodge-Helmotz theorem are solved a nd already built on the boundary conditions imposed directly for the dynamical ele ctromagnetic ﬁeld equations. 5 Note that the set of second-order PDE’s eq(17)-eq(18) may enl arge the original set of solutions of the ﬁrst order eqs(1)-eq(2), opposite to the potential method (→ A,ϕ). So, further direct veriﬁcations of using unique solutions eq(1 7)-eq(18) for solving eq(1) and eq(2) should be done at the end of the problem solving. Let us now consider the anisotropic electromagnetic irradi ation case in R3. In this situation we have the tensorial-matricial constituve rela tionships on the medium electro- magnetic properties between the electric displacement vec tor/vectorD(/vector r,t) and the associated electric ﬁeld /vectorE(/vector r,t) (respectively between the magnetic ﬂux vector /vectorB(/vector r,t) and the mag- netic vector ﬁeld /vectorH(/vector r,t)). /vectorD(/vector r,t) =anisotropic medium permitivity/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright [ε(/vector r)]·/vectorE(/vector r,t)⇔(Di(/vector r,t) =εC0(R3)/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright εij(/vector r)Ej(/vector r,t)(/vector r)) /vectorB(/vector r,t) = [µ(/vector r)]/vectorH(/vector r,t)⇔εijC0(R3)/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright µij(/vector r)Hj(/vector r,t) =Bi(/vector r,t) (31) Since/vector∇·/vectorB= 0, we have that exists a potential /vectorA, such that Bi(/vector r,t) =εijk∂jAk(/vector r,t) (32) As a consequence of the Maxwell equations we also have that Ei(/vector r,t) =−∂ ∂tAi(/vector r,t)+∂ ∂xiφ(/vector r,t) (33) One has thus the following anisotropic wave equation as outc ome: εirs∂ ∂xr/parenleftbig [µ−1]sq(/vector r)Bq(/vector r,t)/parenrightbig −∂ ∂t([ε]ij(/vector r)Ej(/vector r,t)) =Ji(/vector r,t) (34-a) which can be re-written as of as εirs∂ ∂xr [µ−1]sgBg /bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright εgjk∂ ∂xjAk  −∂ ∂t/parenleftbigg [ε]ij/parenleftbigg −∂ ∂tAj+(/vector∇φ)j/parenrightbigg/parenrightbigg =Ji (34-b) or equivalently /parenleftbig εirsεgjk/parenrightbig/braceleftbigg/parenleftbigg [µ−1]sg∂2 ∂xr∂xjAk/parenrightbigg +/parenleftbigg∂ ∂xr[µ−1]sg/parenrightbigg∂ ∂xjAk/bracerightbigg +[ε]ij∂2Aj ∂t2=Ji+[ε]ij∂2 ∂t∂xjϕ (34-c) 6 In other words /parenleftbig εirsεgjk/parenrightbig [µ−1]sg/parenrightbig∂ ∂xr∂xjAk+[ε]ij∂2Aj ∂t2= =Ji+/braceleftbigg [ε]ij∂2 ∂t∂xjϕ−/parenleftbigg εirsεgjk/parenleftbigg∂ ∂xr[µ−1]sg/parenrightbigg∂ ∂xjAk/parenrightbigg/bracerightbigg (35) After considering the radiation anisotropic gauge [ε]ij∂2 ∂t∂xjϕ−/parenleftbigg εirsεgjk∂ ∂xj[µ−1]sg∂ ∂xjAk/parenrightbigg ≡0 (36) one has the anisotropic second order Maxwell wave equation f or the vector potential /vectorA(/vector r,t), decoupled from the scalar potential. Cmrjk(/vector r)≡/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright [ε]−1 mi/parenleftbig εirsεgjk[µ−1]sg)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright :≡Cmrkg∂2 ∂xr∂xjAk+∂2Am ∂t2= [ε]−1 miJi (37) where the anisotropic fourth-order electromagnetic mediu m tensor is explicitly given by Cmrjk(/vector r) =/bracketleftbig ε−1 (/vector r)/bracketrightbig mi(εirsεgjk[µ−1]sg(/vector r)) (38) The equation for the φ-electric potential in the choice radiation gauge eq(36) is devoid of dynamical content and given by a sort of Poisson equation t hrough Maxwell equations once known the solution of the vector potential dynamics as g iven by eq(36) ∂ ∂xi/parenleftbigg [ε]ir∂ ∂xrφ/parenrightbigg =/parenleftbigg ρ+/parenleftbigg∂ ∂xi[ε]ir/parenrightbigg∂ ∂tAr/parenrightbigg (39) We now show that it is possible to choose the anisotropic radi ation gauge eq(36). Let (A,φ) be a given ﬁxed electromagnetic potential conﬁgurtion and/parenleftbig A+∇Λ,ϕ+∂Λ ∂t/parenrightbig it is gauge transformed. We now show that it is possible to determine the gauge transfo rmaton parameter Λ(/vector r,t) with the gauge ﬁeld transformed electromagnetic potentia l conﬁguration satisfying the gauge ﬁxing analytical deﬁnition eq(34). So let us suppose that /bracketleftbigg εgjkεirs/parenleftbigg∂ ∂xr([µ−1]sq(/vector r))∂ ∂xjAk/parenrightbigg/bracketrightbigg −∂ ∂t/bracketleftbigg [ε]ij(r)∂ ∂xjφ/bracketrightbigg /negationslash= 0. (40) 7 So we need to determine Λ such that /bracketleftbigg εgjkεirs([µ−1]sq)∂ ∂xj(A+/vector∇Λ)k/bracketrightbigg −/bracketleftbigg∂ ∂t/parenleftbigg [ε]ij/parenleftbigg∂φ ∂xj+∂Λ ∂t/parenrightbigg/parenrightbigg/bracketrightbigg = 0 (41) We have thus, that the gauge parameter function Λ( /vector r,t) satisﬁes the second order PDE equation below /parenleftbigg εgjkεirs∂ ∂xr([µ−1]sg/parenrightbigg∂ ∂xj/parenleftbigg∂ ∂xkΛ/parenrightbigg −[ε]ij∂2 ∂t2Λj= =source as function of ( A,ϕ)/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright eq(40) (42) Finally, the reader should realize perturbative analytica l calculations by considering weakly anisotropy around the isotropic case ( g1,2≪1) [µ(/vector r)]ij=δij+g1µij(/vector r) [ε(/vector r)]ij=δij+g2εij(/vector r)(43) 2. Maxwell Equations in an axial anisotropic conductive mediu m Letusstartthissection2byconsideringthesetofMaxwelle quationsinamediumwith constitutive parameters depending solely on the spatial va riablez, i.e. our anisotropic medium has a permittivity E=E(z) and a media permeability µ=µ(z). Note also the supposed zdependence of the medium conductivity σ(z). Theelectricﬂuxdensity /vectorB(seeeq(31)andeq(30))arealsosupposedtobetimevariant and solely depending on the spatial variable z /vectorD=/vectorD(z,t) = (Dx,Dy,Dz)(z,t) (44-a) /vectorB=/vectorB(z,t) = (Bx,By,Bz)(z,t) (44-b) /vectorD=ε(z)/vectorE(z,t) (44-c) /vectorB=µ(z)/vectorH(z,t) (44-d) We have thus the following set of partial diﬀerential equati ons describing the electro- magnetic pulse in a such electromagnetic axial anisotropic dependent medium ( ε(z),µ(z)) 8 with a source also still depending on the time and the z-variable solely 1 ε(z)r◦t/parenleftbig/vectorD(z,t)/parenrightbig +/vector∇/parenleftbigg1 ε(z)/parenrightbigg ×/vectorD(z,t) =−∂/vectorB ∂t(z,t) (45-a) 1 µ(z)r◦t/parenleftbig/vectorD(z,t)/parenrightbig +/vector∇/parenleftbigg1 µ(z)/parenrightbigg ×/vectorB(z,t)−∂ ∂t/vectorD(z,t) =/vectorj(z,t)+σ(z) ε(z)/vectorD(z,t) (45-b) ∂ ∂zBz(z,t) = (div /vectorB)(z,t) = 0 (45-c) ∂ ∂zDz(z,t) = (div /vectorD)(z,t) = 0 (45-d) After re-writing the set of axial anisotropic Maxwell equati ons eq(45-a)-eq(45-d) in components, it yields −∂ ∂z/parenleftbigg1 ε(z)∂Dy ∂z/parenrightbigg +∂ ∂z/parenleftbiggε′(z) ε(z)Dy/parenrightbigg =−∂Bx ∂t∂z(46-a) 1 µ(z)∂Bx ∂t∂z−/parenleftbiggµ′(z) µ2(z)/parenrightbigg∂Bx ∂t−∂2 ∂t2Dy =∂ ∂tjy+σ(z) ε(z)∂ ∂tDy (46-b) In a decoupled form after some elementary algebra, one gets t he fully decoupled com- ponent wave equation /parenleftbigg −1 ε(z)∂2 ∂z2Dy+/parenleftbiggε′(z) ε2(z)/parenrightbigg∂Dy ∂z+ε′(z) ε2(z)∂Dy ∂z+/parenleftbiggε′ ε2/parenrightbigg′ (z)Dy(z)/parenrightbigg =/parenleftbigg −/parenleftbiggµ′(z) µ(z)ε(z)/parenrightbigg∂Dy ∂z+/parenleftbiggµ′ε′ µε2/parenrightbigg (z)Dy −µ(z)∂2 ∂t2Dy−µ(z)∂ ∂tjy−/parenleftigµσ ε/parenrightig (z)∂ ∂tDy/parenrightbigg (47) Similar algebric procedures give decoupled equations for Dx(z,t) and the magnetic ﬂux density /vectorB, after determining the candidate solutions for the electri c ﬂux density 9 /vectorD= (Dx,Dy). ∂ ∂z/parenleftbigg1 µ(z)∂Bx ∂z/parenrightbigg −∂ ∂z/parenleftbiggµ′(z) µ2(z)Bx/parenrightbigg = =ε(z)∂2Bx ∂t2+/braceleftbiggeﬀective source term/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright /parenleftbiggε′ ε/parenrightbigg (z)∂Dy ∂t(z,t)−∂ ∂zjy(z,t)+∂ ∂z/parenleftbiggσ(z) ε(z)Dy(z,t)/parenrightbigg/bracerightbigg (48) Formal harmonic solutions for the decoupled eq(47) are easi ly found for a harmonic source with a deﬁnite frequency ( −∞< t <∞) Dy(z,t) =eiwtΦ(z) (49-a) jy(z,t) =eiwtjy(z) (49-b) Here Φ(z) = exp/braceleftbiggα(z)≡/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright −1 2/integraldisplayz z0b(z)dz/bracerightbigg ϕ(z) (50-a) b(z)≡/bracketleftbigg/parenleftbigg2ε′ ε2+µ′ µε/parenrightbigg ·1 µ/bracketrightbigg (50-b) −d2ϕ dz2+(iwd(z)−V(z))ϕ(z) =w2ϕ−jy(z)e−α(z)(50-c) d(z)≡σ(z) ε(z)(50-d) V(z) =α′′(z)+(α′(z))2+b(z)α′(z)+c(z) (50-e) a(z)≡/parenleftbigg1 εµ/parenrightbigg (z) (50-f) c(z)≡/parenleftbigg/bracketleftbigg −/parenleftbiggε′ ε2/parenrightbigg′ −µ′ µε′ ε/bracketrightbigg ·1 µ/parenrightbigg (z) (50-g) We have thus reduced the harmonic electromagnetic wave prop agation described in a axial-anisotropic medium to the ordinary diﬀerential equa tion eq(50-c). Let us ﬁnally point out that at the limit of higher frequencies and jy(z,t)≡0, one may introduce the eﬀective spatial variable Z([5]) Z=ω γz ϕ(z) =U(Z);d(z) =/tildewided(Z),etc. (51) 10 withγbeing an expansion parameter and get the explicitly solutio ns at the asymptotic higher frequence limit U(Z) =A 4/radicalig −/tildewidec(Z)exp/braceleftbigg i/integraldisplayZ Z0/parenleftbig −/tildewidec(Z)/parenrightbig1 2dZ/bracerightbigg +B 4/radicalbig −/tildewidec(Z)exp/braceleftbigg −i/integraldisplayZ Z0/parenleftbig −/tildewidec(Z)/parenrightbig1 2dZ/bracerightbigg (52) useful for scattering problem of an electromagnetic pulse i nto a slab (or layers) (work in progress). Work on applied settings ([4]) are in progress.