Wave Propagation 2011 Part 3 pdf

Uniform Asymptotic Physical Optics Solutions for a Set of Diffraction Problems 49 in which 2 k is the propagation constant in the double-negative metamaterial, 2 θ is the negative refraction angle, and j ii j ij j ii j kcos kcos R k cos k cos θ −θ = θ +θ  ii jj ij ii jj k cos k cos R k cos k cos ⊥ θ −θ = θ +θ (61) ij ii j ii j 2k cos T kcos kcos θ = θ +θ  ij ii ii jj 2k cos T kcos kcos ⊥ θ = θ +θ (62) In (61) and (62), 130 kkk = = , 13 θ =θ and the superscripts i and j refer to the left and right media involved in the propagation mechanism. Comparisons with COMSOL MULTIPHYSICS ® results are reported in Figs. 14 and 15 with reference to '45φ= ° . As can be seen, the UAPO diffracted field guarantees the continuity of the total field across the two discontinuities of the GO field in correspondence of the incidence and reflection shadow boundaries, and a very good agreement is attained. Accordingly, the accuracy of the UAPO-based approach is well assessed also in the case of a lossless double-negative metamaterial layer. 3.3 Anisotropic impedance layer A layer characterised by anisotropic impedance boundary conditions on the illuminated face is now considered. Such conditions are represented by an impedance tensor x' z' ˆˆ ˆˆ Z Z x'x' Z z'z'=+ having components along the two mutually orthogonal principal axes of anisotropy ˆ x' and ˆ z' . The structure is opaque so that the transmission matrix does not exist and, according to (Gennarelli et al., 1999), the elements of R can be so expressed: Fig. 14. Amplitude of E β if β = i ' E1, φ = i ' E0 and '90 β =°, ' 45 φ =°. Circular path with 0 5ρ= λ . Layer characterised by r 4 ε =− , r 1 μ =− and 0 d0.125 = λ . Wave Propagation in Materials for Modern Applications 50 Fig. 15. Amplitude of E φ if β = i ' E0, φ = i ' E1 and '90 β =°, ' 45 φ =°. Circular path with 0 5ρ= λ . Layer characterised by r 4 ε =− , r 1 μ =− and 0 d0.125 = λ . ( ) ( ) 2i 2i 11 2i 2i A1ACBcos Ccos R A1ACBcos Ccos − ++ + θ− θ = + −− θ− θ (63) () θ =− = + −− θ− θ i 12 21 2i 2i 2B cos RR A 1 AC B cos C cos (64) ( ) ( ) 2i 2i 22 2i 2i A1ACBcos Ccos R A1ACBcos Ccos + ++ θ+ θ =− + −− θ− θ (65) in which, if χ is the angle between ˆ x' and ˆ e ⊥ , = χ+ χ ζζ 22 x' z' 00 Z Z A sin cos (66) () = −χχ ζ x' z' 0 1 B Z Z sin cos (67) ⎡ ⎤ = −χ+χ ⎢ ⎥ ζζ ⎣ ⎦ 22 x' z' 00 Z Z C cos sin (68) Uniform Asymptotic Physical Optics Solutions for a Set of Diffraction Problems 51 If an isotropic impedance boundary condition is considered (i.,e., the principal axes of anisotropy does not exist and = = x' z' ZZZ), = = 12 21 RR0, whereas 11 R and 22 R reduce to the standard reflection coefficients for parallel and perpendicular polarisations. If the illuminated surface is perfectly electrically conducting, the out diagonal elements are again equal to zero, whereas = 11 R1 and = − 22 R1 since = = x' z' ZZ0. Fig. 16. Amplitude of E β if β = i ' E1, φ = i ' E0 and '90 β =°, ' 60 φ =°. Circular path with 0 5ρ= λ . Layer characterised by 0 Z j 0.5 ζ = . Fig. 17. Amplitude of E β if β = i ' E1, φ = i ' E0 and '90 β =°, ' 60 φ =°. Circular path with 0 5ρ= λ . Layer characterised by 0 Z j 0.5ζ= . Wave Propagation in Materials for Modern Applications 52 The magnitudes of the electric field β –components of the GO field and the UAPO diffracted field on a circular path with ρ =λ 0 5 are considered in Fig. 16, where also the diffracted field obtained by using the Maliuzhinets solution (Bucci & Franceschetti, 1976) is reported in the case of an isotropic impedance boundary condition. A very good agreement exists between the two diffracted fields. The accuracy of the UAPO-based approach is further confirmed by comparing the total fields shown in Fig. 17, where also the COMSOL MULTIPHYSICS ® results are shown. 4. Junctions of layers The UAPO solution for the field diffracted by the edge of a truncated planar layer as derived in Section 2 can be extended to junctions by taking into account the diffraction contributions of the layers separately. This very useful characteristic is due to the property of linearity of the PO radiation integral. Accordingly, if the junction of two illuminated semi-infinite layers as depicted in Fig. 18 is considered, the total scattered field in (1) can be so rewritten: + ⎡⎤ =− − ζ + × = ⎣⎦ ⎡⎤ =− − ζ + × + ⎢⎥ ⎣⎦ ⎡⎤ −−ζ+× =+ ⎢⎥ ⎣⎦ ∫∫ ∫∫ ∫∫ 12 11 1 22 2 sPOPO 00 sms SS PO PO 00 1 sms S PO PO s s 00 2 sms 12 S ˆˆ ˆ Ejk (IRR)JJRG(r,r')dS ˆˆ ˆ jk (I RR) J J R G(r,r') dS ˆˆ ˆ j k(IRR)JJ RG(r,r')dSEE (69) and then =+ 12 DD D, with 1 D given by (46). The diffraction matrix 2 D related to the wave phenomenon originated by the edge of the second layer forming the junction can be determined by using again the methodology described in Section 2. If the external angle of the junction is equal to n π , a ( ) n1 − π rotation of the edge-fixed coordinate system must be considered for the second layer. The incidence and observation angles with respect to the illuminated face are now equal to n ' π −φ and n π −φ, respectively, so that the UAPO solution for 2 D uses n ' π −φ instead of ' φ and n π −φ instead of φ . The results reported in (Gennarelli et al., 2000) with reference to an incidence direction normal to the junction of two resistive layers confirm the validity of the approach and, in particular, the accuracy of the solution is well assessed by resorting to a numerical technique based on the Boundary Element Method (BEM). S f ' (f = np) 2 f x y S (f = np) 2 P r (f=0) S 1 Fig. 18. Junction of two planar truncated layers. Uniform Asymptotic Physical Optics Solutions for a Set of Diffraction Problems 53 5. Conclusions and future activities UAPO solutions have been presented for a set of diffraction problems originated by plane waves impinging on edges in penetrable or opaque planar thin layers. The corresponding diffracted field has been obtained by modelling the structure as a canonical half-plane and by performing a uniform asymptotic evaluation of the radiation integral modified by the PO approximation of the involved electric and magnetic surface currents. The resulting expression is given terms of the UTD transition function and the GO response of the structure accounting for its geometric, electric and magnetic characteristics. Accordingly, the UAPO solution possesses the same ease of handling of other solutions derived in the UTD framework and has the inherent advantage of providing the diffraction coefficients from the knowledge of the reflection and transmission coefficients. It allows one to compensate the discontinuities in the GO field at the incidence and reflection shadow boundaries, and its accuracy has been proved by making comparisons with purely numerical techniques. In addition, the time domain counterpart can be determined by applying the approach proposed in (Veruttipong, 1990), and the UAPO solution for the field diffracted by junctions can be easily obtained by considering the diffraction contributions of the layers separately. To sum up, it is possible to claim that UAPO solutions are very appealing from the engineering standpoint. Diffraction by opaque wedges has been considered in (Gennarelli et al., 2001; Gennarelli & Riccio, 2009b). By working in this context, the next step in the future research activities may be devoted to find the UAPO solution for the field diffracted by penetrable wedges (f.i., dielectric wedges). 6. Acknowledgment The author wishes to thank Claudio Gennarelli for his encouragement and helpful advice as well as Gianluca Gennarelli for his assistance. 7. References Burnside, W.D. & Burgener, K.W. (1983). High Frequency Scattering by a Thin Lossless Dielectric Slab. IEEE Transactions on Antennas and Propagation, Vol. AP-31, No. 1, January 1983, 104-110, ISSN: 0018-926X. Balanis, C.A. (1989). Advanced Engineering Electromagnetics, John Wiley & Sons, ISBN: 0-471- 62194-3, New York. Bucci, O.M. & Franceschetti, G. (1976). Electromagnetic Scattering by a Half-Plane with Two Face Impedances, Radio Science, Vol. 11, No. 1, January 1976, 49-59, ISSN: 0048-6604. Clemmow, P.C. (1950). Some Extensions of the Method of Integration by Steepest Descent. Quarterly Journal of Mechanics and Applied Mathematics, Vol. 3, No. 2, 1950, 241-256, ISSN: 0033-5614. Clemmow, P.C. (1996). The Plane Wave Spectrum Representation of Electromagnetic Fields, Oxford University Press, ISBN: 0-7803-3411-6, Oxford. Ferrara, F.; Gennarelli, C.; Pelosi, G. & Riccio, G. (2007a). TD-UAPO Solution for the Field Diffracted by a Junction of Two Highly Conducting Dielectric Slabs. Electromagnetics, Vol. 27, No. 1, January 2007, 1-7, ISSN: 0272-6343. Wave Propagation in Materials for Modern Applications 54 Ferrara, F.; Gennarelli, C.; Gennarelli, G.; Migliozzi, M. & Riccio, G. (2007b). Scattering by Truncated Lossy Layers: a UAPO Based Approach. Electromagnetics, Vol. 27, No. 7, September 2007, 443-456, ISSN: 0272-6343. Gennarelli, C.; Pelosi, G.; Pochini; C. & Riccio, G. (1999). Uniform Asymptotic PO Diffraction Coefficients for an Anisotropic Impedance Half-Plane. Journal of Electromagnetic Waves and Applications , Vol. 13, No. 7, July 1999, 963-980, ISSN: 0920-5071. Gennarelli, C.; Pelosi, G.; Riccio, G. & Toso, G., (2000). Electromagnetic Scattering by Nonplanar Junctions of Resistive Sheets. IEEE Transactions on Antennas and Propagation , Vol. 48, No. 4, April 2000, 574-580, ISSN: 0018-926X. Gennarelli, C.; Pelosi, G. & Riccio, G. (2001). Approximate Diffraction Coefficients of an Anisotropic Impedance Wedge. Electromagnetics, Vol. 21, No. 2, February 2001, 165- 180, ISSN: 0272-6343. Gennarelli, G. & Riccio, G. (2009a). A UAPO-Based Solution for the Scattering by a Lossless Double-Negative Metamaterial Slab. Progress In Electromagnetics Research M, Vol. 8, 2009, 207-220, ISSN: 1937-8726. Gennarelli, G. & Riccio, G. (2009b). Progress In Electromagnetics Research B, Vol. 17, 2009, 101- 116, ISSN: 1937-6472. Keller, J.B. (1962). Geometrical Theory of Diffraction. Journal of Optical Society of America, Vol. 52, No. 2, February 1962, 116-130, ISSN: 0030-3941. Kouyoumjian, R.G. & Pathak, P.H. (1974). A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface. Proceedings of the IEEE, Vol. 62, No. 11, November 1974, 1448-1461, ISSN: 0018-9219. Luebbers, R.J. (1984). Finite Conductivity Uniform UTD versus Knife Diffraction Prediction of Propagation Path Loss. IEEE Transactions on Antennas and Propagation, Vol. AP- 32, No. 1, January 1984, 70-76, ISSN: 0018-926X. Maliuzhinets, G.D. (1958). Inversion Formula for the Sommerfeld Integral. Soviet Physics Doklady, Vol. 3, 1958, 52-56. Senior, T.B.A. & Volakis, J.L. (1995). Approximate Boundary Conditions In Electromagnetics. The Institution of Electrical Engineers, ISBN: 0-85296-849-3, Stevenage. Veruttipong, T.W. (1990). Time Domain Version of the Uniform GTD. IEEE Transactions on Antennas and Propagation , Vol. 38, No. 11, November 1990, 1757-1764, ISSN: 0018- 926X. 3 Differential Electromagnetic Forms in Rotating Frames Pierre Hillion Institut Henri Poincaré, 86 Bis Route de Croissy, 78110 Le Vésinet, France 1. Introduction Differential forms are completely antisymmetric homogeneous r-tensors on a differentiable n-manifold 0 ≤ r ≤ n belonging to the Grasssman algebra [1] and endowed by Cartan [2] with an exterior calculus. These differential forms found an immediate application in geometry and mechanics; introduced by Deschamps [3,4] in electromagnetism, they have known in parallel with the expansion of computers, an increasing interest [5-9] because Maxwell’s equations and the constitutive relations are put in a manifestly independent coordinate form. In the Newton (3+1) space-time, with the euclidean metric ds 2 = dx 2 + dy 2 + dz 2 the conventional Maxwell equations in which the E, B, D, H fields are 3-vectors have, in absence of charge and current, the Gibbs representation ∇.B = 0, ∇∧Ε + 1/c∂ t B = 0 (1a) ∇.D = 0, ∇∧H − 1/c∂ t D = 0 (1b) and, they also have the differential form representation (∂ τ =1/c∂ t ) [5,7] d ∧ E +∂ τ B = 0, d ∧ B = 0 (2a) d ∧ H − ∂ τ D = 0, d ∧ D = 0 (2b) d = dx∂ x + dy∂ y +dz∂ z is the exterior derivative, E , H the differential 1-forms E = E x dx + E y dy + E z dz, H = H x dx + H y dy + H z dz (3a) and B , D the differential 2-forms B = B x (dy∧dz) + B y (dz∧dx) +B z (dx∧dy) , D = −[ D x (dy∧dz) + D y (dz∧dx) +D z (dx∧dy)] (3b) We are interested here, for reasons to be discussed in Sec.(6) in a Frenet-Serret frame rotating around oz with a constant angular velocity requiring a relativistic processing, We shall prove that this situation leads to an Einstein space-time with a riemannian metric. As an introduction to this problem, we give a succcinct presentation of differential electromagnetic forms in a Minkowski space-time with the metric ds 2 = dx 2 + dy 2 + dz 2 −c −2 ∂ t 2 . Wave Propagation in Materials for Modern Applications 56 2. Differential forms in Minkowski space-time [7] In absence of charge and current, the Maxwell equations have the tensor representaation [10, 11] ∂ σ F μν +∂ μ F νσ + ∂ ν F σμ = 0 a) ∂ ν F μν = 0 b) (4) the greek (resp.latin) indices take the values 1,2,3,4 (resp.1,2,3) with x 1 = x, x 2 = y, x 3 = z, x 4 = ct, ∂ j = ∂/∂x j , ∂ 4 = 1/c∂/∂ t and the summation convention is used. The components of the ten- sors F μν and F μν are with the 3D Levi-Civita tensor ε ijk B i = ½ ε ijk F jk , E i = − F i4 , H i = ½ ε ijk F jk , D i = − F i4 (5) and in vacuum D = ε 0 E, H = µ 0 −1 B , (ε 0 μ 0 ) 1/2 = 1/c (5a) Let d be the exterior derivative operator d = (∂ x dx +∂ y dy +∂ z dz +∂ t dt )∧ (6) and F =E + B be the two-form in which : E = E x (dx∧cdt) + E y (dy∧cdt) + E z (dz∧cdt) B = B x (dy∧dz) + B y (dz∧dx) + B z (dx∧dy) (7) Then the Maxwell equations (4a) have the differential 3-form representation d F = 0. Similarly for G =D + H with : H = H x (dx∧cdt) + H y (dy∧cdt) + H z (dz∧cdt) D = −[D x (dy∧dz) + D y (dz∧dx) + D z (dx∧dy)] (8) the differential 3-form representation of Maxwell’s equations (4b) is d G = 0. To manage the constituive relations (5a) the Hodge star operator [6,9] is introduced * (dx∧cdt) = c −1 (dy∧dz) , * (dy∧dz) = c (dx∧cdt) * (dy∧cdt) = c −1 (dz∧dx) , * (dz∧dx) = c (dy∧cdt) * (dz∧cdt) = c −1 (dx∧dy) , * (dx∧dy) = c (dz∧cdt) (9) Applying the Hodge star operator to F gives *F = *E + *B and one checks easily the re- lation G = λ 0 *F with λ 0 =(ε 0 /μ 0 ) 1/2 so that the Maxwell equations in the Minkowski vacu- um, have the diffrerential 3-form representation dF = 0, d *F = 0 (10) 3. Electromagnetidsm in a Frenet-Serret rotating frame We consider a frame rotating with a constant angular velocity Ω around oz. Then, using the Trocheris-Takeno relativistic description of rotation [12, 13], the relations between the Differential Electromagnetic Forms in Rotating Frames 57 cylindrical coordinates R,Φ,Z,T and r,φ,z,t in the natural (fixed) and rotating frames are with ß = ΩR /c R= r, Φ = φ coshβ − ct/r sinhβ Z = z , cT = ct coshβ − rφ sinhβ (11) and a simple calculation gives the metric ds 2 in the rotating frame ds 2 = c 2 dt 2 − dz 2 − r 2 dφ 2 − (1+B 2 −A 2 ) dr 2 − 2(A sinhß + B coshß) cdt dr − 2(A coshß + B sinhß) r dr dφ (12) A = ß sinhß ct/r + ß coshß φ + sinhß φ, B = ß sinhß φ +ß coshß ct/r − sinhß ct/r (12a) Using the notations x 4 = ct, x 3 = z, x 2 = φ, x 1 = r, we get from (12) ds 2 = g μν dx μ dx ν with g 44 = 1, g 33 = −1, g 22 = −r 2 , g 11 = −(1+ B 2 −A 2 ) g 14 = g 41 = 2(A sinhß + B coshß), g 12 = g 21 = 2(A coshß + B sinhß) (13) The determinant g of g μν is g = g 33 [ g 11 g 22 g 44 − g 12 2 g 44 − g 14 2 g 22 ] = R [g 11 − g 12 2 r −2 − g 14 2 ] (14) but g 12 2 r −2 + g 14 2 = 4(A 2 − B 2 ) (14a) and, taking into account the expression (13) of g 11 , we get finally g = r 2 [5(A 2 −B 2 ) − 1], A 2 −B 2 = (φ 2 − c 2 t 2 /r 2 )(ß 2 + sinh 2 ß + 2ß sinhß coshß) (15) So, the rotating Frenet-Serret frame defines an Einstein space-time with the riemannian metric ds 2 = g μν dx μ dx ν , and in this Einstein space-time the Maxwell equations have the tensor representation[14, 15] ∂ σ G μν +∂ μ G νσ + ∂ ν G σμ = 0 a) ∂ ν (|g| 1/2 G μν ) = 0 b) (16) in which, using the cylindrical coordinates r,φ.z ,t with x 1 = r, x 2 = φ, x 3 = z, x 4 = ct ; ∂ 1 = ∂ r , ∂ 2 = ∂ φ , ∂ 3 = ∂ z , ∂ 4 =1/c ∂ t , the components of the electromagnetic tensors are G 12 = rB z , G 13 = −Β φ , G 23 = rB r ; G 14 =−E z , G 24 = −rE φ , G 34 = −E z G 12 =H z /r, G 13 = −H φ , G 23 = H r /r; G 14 D z , G 2 4 = D φ /r, G 34 = D z (17) To work with the differential forms, we introduce the exterior derivative d = (∂ r dr+∂ φ dφ+∂ z dz+∂ t dt) ∧ (18) (underlined expressions mean that they are defined with the cylindrical coordinates r, φ,z,t) and the two-forms F =E + B with Wave Propagation in Materials for Modern Applications 58 E = E r (dr∧cdt) + E φ (rdφ∧cdt) + E z (dz∧cdt) B = B r (rdφ∧dz) + B φ (dz∧dr) + B z (dr∧rdφ) (19a) and writing |g| 1/2 = rq, q = [5(A 2 -B 2 )-1] 1/2 the two-form G =D + H D =− q [D r (rdφ∧dz) + D φ (dz∧dr) + D z (dr∧rdφ)] H = q[H r (dr∧cdt) + H φ (rdφ∧cdt) + H z (dz∧cdt)] (19b) Then, the Maxwell equatios have the 3-form representation dF = 0 , d G = 0 (20) A simple calculation gives d F = [(∂ r (rB r ) +∂ φ B φ +∂ z (rB z )] (dr∧ dφ∧ dz) + [∂ t (rΒ r ) + c{∂ φ Ε z −∂ z (rΕ φ )}] (dφ∧ dz∧ dt) + [∂ t Β φ + c(∂ z Ε r −∂ r Ε z )] (dz∧ dr∧ dt) + [∂ t (rΒ z )+ c{∂ r (rΕ φ )−∂ φ Ε r }] (dr∧ dφ∧ dt) (21a) dG = − [∂ r (qrD r ) +∂ φ (qD φ ) +∂ z (qD z )] (dr∧ dφ∧ dz) + [−∂ t (qrD r ) + c {∂ φ (qH z ) −∂ z (qrH φ )}] (dφ∧ dz∧ dt) + [−∂ t (qD φ ) + c {∂ z (qH r ) −∂ r (qH z )}] (dz∧ dr∧ dt) + [−∂ t (qrD z ) + c {∂ r (qrH φ −∂ φ (qH r )}] (dr∧ dφ∧ dt) (21b) The Hodge star operator needed to take into account the constitutive relations (5a) in vacuum is defined by the relation * (dr∧cdt) = −qc −1 (rdφ∧dz), * (rdφ∧dz) = q −1 c (dr∧cdt) * (rdφ∧cdt) = −qc −1 (dz∧dr) , * (dz∧dr) = q −1 c (rdφ∧cdt) * (dz∧cdt) = −qc −1 (dr∧rdφ), * (dr∧rdφ) = q −1 c (dz∧cdt) (22) Applying (22) to F gives *F = *E + *B and it is easily checked that G = λ 0 *F with λ 0 =(ε 0 /μ 0 ) 1/2 so that in vacuum dF = 0 , d *F = 0. 4. Wave equations in vacuum 4.1 Minkowski space-time The wave equations satisfied by the electromagnetic field (in absence of charges and cur- rents) are obtained from differential forms with the help of the Laplace-De Rham operator [6,8] L = (d* d* + * d* d)∧ (23) [...]... (19a) with dx1∧dx4 = dr∧cdt, dx2∧dx4 = dφ∧cdt, dx3∧dx4 = dz∧cdt (32 ) is written E = Er (dx1∧dx4) + rEφ (dx2∧dx4) + Ez (dx3∧dx4) (33 ) but Er, rEφ, Ez are the Gi4 components of the Gμν tensor (17) so that leaving aside a minus sign E = G14 (dx1∧dx4) + G24 (dx2∧dx4) + G34 (dx3∧dx4) (34 ) L E = ½( (gαβ ∇α∇βGi4 − 2R i4αρGρα + R iρGρ4 − R4ρGρi)) (dxi∧dx4) (35 ) and we get so that the components of the electric... simulating the propagation of a variety of waves, such as sound waves, light waves and water waves, we discuss a novel numerical scheme to solve the wave equation with time dependent diffusion coefficients, see [21] We deal with a second-order linearly time dependent partial differential equation It arises in fields such as acoustics, electromagnetics and fluid dynamics, see [5] For example, when wave propagation. .. gαβΓβ,μν (B.1) 63 Differential Electromagnetic Forms in Rotating Frames In these expressions, the greek indices take the values 1,2 ,3, 4 corresponding in a cylindrical frame to the coordinates x1 = r, x2 = φ, x3 = z, x4 = ct, while ∂1 = ∂r, ∂2 = 1/r ∂φ, 3 = ∂z, ∂4 = 1/c ∂t The relations ( 13) give the components gμν of the metric tensor for a Frenet-Serret rotating frame and : g44 = 1, g 33 = −1, g22 =... 691-704 64 Wave Propagation in Materials for Modern Applications [10] J.D.Jackson, Classical Electrodynamics (Wiley, New York, 1975) [11] D.S.Jones, Acoustic and Electromagnetic waves.(Clarendon, Oxford, 1956) [12] M.G.Trocheris, Electrodynamics in a rotating frame of reference, Philo.Mag 7 (1949) 11 43- 1155 [ 13] H.Takeno, On relativistic theory of rotating disk, Prog Theor Phys 7 (1952) 36 7 -37 1 [14]... (dxi∧dxj) the B magnetic two-form (7) so that LB = −1/4(gαβ∇α∇βFij − 2RijαρFρα + RiρFρj − RjρFρi) (dxi∧dxj) (30 a) Summing (30 ) and (30 a) gives LF = ½ [gαβ∇α∇βFμν − 2RμναρFρα + RμρFρν − RνρFρμ] (dxμ∧dxν) In the Minkowski cartesian frame where gij = δij , g44 = −1, Eq. (31 ) reduces to (25) (31 ) 60 Wave Propagation in Materials for Modern Applications 4.2.2 Frenet-Serret frame In the Frenet-Serret frame, the... 0 0⎤⎥ 0 0 0⎥⎥ I 0 I ⎥⎥ ⎥ 0 B 0⎥ ⎦ (33 ) Then, using the notations (33 ), the relations (32 ) can be written as: ∂tt Ei (t ) = AEi (t ) + Fi (t ), t ∈ (t n , t n +1 ], n Ei (t ) = 0 (34 ) Due to our assumptions, A is a generator of the one-parameter C0 semi-group (expAt )t≥0 Hence, using the variations of constants formula, the solution to the abstract Cauchy problem (34 ) with homogeneous initial conditions... due to combined propagations in three dimensions, time dependent equations of such dynamical models become the starting point of the analysis, see [3] We concentrate on wave propagation models to obtain physically related results for time dependent diffusion parameters, see [5] For the sake of completion we incorporate the constant case, too 2.1 Wave equations In this section we present wave equations... Algorithms: Theory and Application to Constant and Time-Dependent Wave Equations 73 holds with numbers K ≥ 0 and ω ∈ R, cf [6] The estimations (37 ) and (38 ) result in Ei ∞ =K B 2 τn ei −1 +O(τn ), (39 ) where ei−1 = max{ e1,i−1 , e2,i−1 } Taking into account the definition of Ei and the norm ei = K · ∞, we obtain 2 B τn ei −1 +O(τn ), (40) and hence 2 3 ei +1 = K 1τn ei −1 +O(τn ), (41) which proves our statement... Rham operator L = d*d* + *d*d applied to the two form (7) written E =F14(dx1∧dx4) + F24(dx2∧dx4) + F34(dx3∧dx4) (29) in which dx1 = dx, dx2 = dy, dx3 = dz, dx4 = cdt gives for the component Fi4 L E = ½(gαβ∇α∇βFi4 − 2R i4αρFρα + R iρFρ4 − R4ρFρi)( (dxi∧dx4) (30 ) L E = 0 gives the 2-form representation of the wave equation in the Einstein space-time with cartesian coordinates A similar result is obtained... equations with constant and time dependent diffusion coefficients 66 Wave Propagation in Materials for Modern Applications Wave equation with constant diffusion coefficients First we deal with a wave equation that represents a simple model of a Maxwell equation which is needed for the simulation of electro-magnetic fields We have a linear wave equation with constant coefficients given by: ∂2c ∂2c ∂ 2c = . Mathematics, Vol. 3, No. 2, 1950, 241-256, ISSN: 0 033 -5614. Clemmow, P.C. (1996). The Plane Wave Spectrum Representation of Electromagnetic Fields, Oxford University Press, ISBN: 0-78 03- 3411-6, Oxford Motivated by simulating the propagation of a variety of waves, such as sound waves, light waves and water waves, we discuss a novel numerical scheme to solve the wave equation with time dependent. (dx 3 ∧dx 4 ) (33 ) but E r , rE φ , E z are the G i4 components of the G μν tensor (17) so that leaving aside a minus sign E = G 14 (dx 1 ∧dx 4 ) + G 24 (dx 2 ∧dx 4 ) + G 34 (dx 3 ∧dx 4 )