Wave Propagation 2011 Part 17 doc

Pulse Wave Propagation in Bistable Oscillator Array 487 M. Yamauchi, M. Wada, Y. Nishio and A. Ushida, “Wave propagation phenomena of phase states in oscillators coupled by inductors as a ladder”, IEICE Trans.Fundamentals, vol.E82-A, no.11, pp.2592-2598, 1999. M.Sato, B.E.Hubbard, A.J.Sievers, B.Ilic, D.A.Czaplewski and H.G.Craighead, “Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array”, Phys.Rev.Lett., vol.90, no.4 (2003) 044102. J.P. Keener, “Propagation and its failure in coupled systems of discrete excitable cells”, SIAM J. Appl. Biol., vol.47, no.3, pp.556-572, 1987. K.Shimizu, T. Endo and D.Ueyama, “Pulse Wave Propagation in a Large Number of Coupled Bistable Oscillators”, IEICE Trans. Fundamentals, vol.E91-A, no.9, pp.2540-2545, 2008. T. Endo and T. Ohta, “Multimode oscillations in a coupled oscillator system with fifthpower nonlinear characteristics”, IEEE Trans. on circuit and systems, vol.cas-27, no.4, pp.277-283, 1980. T. Yoshinaga and H. Kawakami, “Synchronized quasi-periodic oscillations in a ring of coupled oscillators with hard characteristics”, Electronics and Communications in Japan, Part III, vol.76, no.5, pp.110-120, 1993. H. Kawakami, “Bifurcation of periodic responses in forced dynamic nonlinear circuits: computation of bifurcation values of the system parameters”, IEEE Trans. Circuits Syst., vol.CAS-31, no.3, pp.248-260, 1984. T.S.Parker and L.O.Chua, Practical numerical algorithms for chaotic systems, Springer- Verlag, New York, 1989. Y. Katsuta, H. Kawakami, “Bifurcations of equilibriums and periodic solutions in nonlinear autonomous system with symmetry”, Electronics and Communications in Japan, Part III, vol.76, no.7, pp.1-14, 1993. Y.A. Kuznetsov, “Elements of applied bifurcation theory”, Springer-Verlag, New York, p.466, 1995. A. The propagating pulse wave PW2 The PW2 is a certain kind of propagating pulse wave. The mapped points of PW1 and PW2 projected onto the (x 1 , x 3 , x 5 ) phase space are shown in Fig.9(a) and (b), respectively. Comparing both cases, each flow on the phase space moves along a different orbit. In addition, for PW1 the mapped points stay for a long time on several points (which correspond to the locus of the nodes N i , i = 1, 2, … , 6.). This is one of characteristic feature of PW1 originating in the heteroclinic tangle. On the other hand, for PW2 the mapped points no longer stay the locus for a long time. Therefore, we distinguish PW2 from PW1. The existence region of such solution is shown in Fig.5. It should be noted that the starting point of PW2 is no longer close to the existence region of PS. That is, between them the existence region of W is sandwiched. For example, for β = 3.26 and ε = 0.36, PS disappears via PF bifurcation at α PF 0.083. In contrast, PW2 begins to exist for α 0 0.087. Namely, there exists a gap between them. Probably, it originates in the standing wave where two adjacent oscillators are oscillating and where other oscillators are not. This is confirmed by continuously changing the parameter β of Fig.9. Further research will be necessary to clarify the generation mechanism of PW2. Wave Propagation in Materials for Modern Applications 488 (a) Mapped points of PW1 (b) Mapped points of PW2 Fig. 9. Mapped points of PW1 and PW2 projected onto the (x 1 , x 3 , x 5 ) phase space. (a) PW1 ( α = 0.089, β = 3.25 and ε = 0.36). The initial condition is given as x 1 = 2.0, y 2 = 1.3 and all other variables are zero. (b) PW2 ( α = 0.089, β = 3.26 and ε = 0.36). The initial condition is given as x 1 = 1.7, x 2 = –2.2, x 3 = 0.9, x 4 = 0.2, x 5 = 0.1, x 6 = 0.5, y 1 = 1.8, y 2 = 0.4, y 3 = –2.3, y 4 = y 5 = 0.3 and y 6 = –0.3. 26 Circuit Analogs for Wave Propagation in Stratified Structures Daniel Sjöberg Lund University Sweden 1. Introduction To simulate or design a reasonably large system, fast and simple models are necessary. To verify the design versus the specifications, more detailed (and costly) calculations can be performed and final adjustments made. In wave propagation problems, circuit analogs provide a powerful, yet simple, means of computing the desired response of the system, such as reflection or transmission coefficients. The reason circuit analog models are good for wave propagation problems, is that they are exact for one-dimensional wave propagation, regardless of whether we consider acoustic or electromagnetic waves. Typically, wave propagation through homogeneous media is modeled as a transmission line with propagation constant β and characteristic impedance Z, whereas obstacles such as thin sheets are modeled as lumped elements. If the sheets are lossless, the circuit models contain only reactive elements such as capacitors and inductors. Modeling complex wave propagation problems with circuit analogs was to a large extent developed in conjunction with the development of radar technology during the Second World War. Many of the results from this very productive era are collected in the Radiation Laboratory Series and related literature, in particular (Collin, 1991; 1992; Marcuvitz, 1951; Schwinger & Saxon, 1968). Further development has been provided by research on frequency selective structures (Munk, 2000; 2003). In recent years, the circuit analogs have even been used in an inverse fashion: by observing that wave propagation through a material with negative refractive index could be modeled as a transmission line with distributed series capacitance and shunt inductance, i.e., the dual of the standard transmission line, the most successful realization of negative refractive index material is actually made by synthesizing this kind of transmission line using lumped elements (Caloz & Itoh, 2004; Eleftheriades et al., 2002). This chapter is organized as follows. In Section 2 we show that propagation of electromagnetic waves in any material, regardless how complicated, boils down to an eigenvalue problem which can be solved analytically for isotropic media, and numerically for arbitrary media. From this eigenvalue problem, the propagation constant and characteristic impedance can be derived, which generates a transmission line model. In Section 3, we show how sheets with or without periodic patterns can be modeled as lumped elements connected by transmission lines representing propagation in the surrounding medium. The lumped elements can be given a firm definition and physical interpretation in the low frequency limit, and in Section 4 we show how these low frequency properties Wave Propagation in Materials for Modern Applications 490 provide some useful physical limitations on scattering characteristics. The calculation of circuit analogs in the general case using an optimization approach is treated in Section 5, and examples of the use of circuit analogs in design problems are given in Section 6. Finally, conclusions are given in Section 7. 2. Wave propagation in stratified structures In this section, we show that the description of plane waves propagating through any homogeneous material at any angle of incidence, reduces to a simple eigenvalue problem from which we can compute the propagation constant and transverse wave impedance. We consider a geometry where the material parameters are constants as functions of x and y, but may depend on z, which is considered as the main propagation direction. This corresponds to a laminated structure, z being the lamination direction. Our strategy is to eliminate the x and y dependence through a spatial Fourier transform, and then eliminate the field components along the z direction. This is motivated by the fact that the remaining field components, E t = E x ˆ x + E y ˆ y and H t = H x ˆ x + H y ˆ y , are continuous across interfaces, and are thus easily matched at boundaries. The resulting equation (24) (or (25) for isotropic media) can be formulated as an algebraic eigenvalue problem by looking for solutions where the only z dependence is through a propagation factor e −j β z . The wave number β corresponds to the eigenvalue, and the wave impedance is given by the eigenvectors. 2.1 Notation We consider time harmonic waves using time convention e jωt . The material is described through the mapping from the fields [E,H] to the fields [D,B]: ⋅ +⋅ ⎧ ⎨ ⋅ +⋅ ⎩ = = D EH B ζ E μ H ε ξ (1) where the dyadics ε, ξ, ζ, and μ can be represented by 3 × 3 matrices. Other mappings for the material are possible, for instance from the fields [E,B] to [D,H]. In vacuum the relations are D = ε 0 E and B = μ 0 H, where the permittivity and permeability of vacuum are denoted by ε 0 = 8.854 · 10 −12 F/m and μ 0 = 4π · 10 −7 H/m, respectively. Materials are often classified according to the various symmetries of the material dyadics as in Table 1. When choosing a particular direction z, it is natural to introduce a decomposition as (where the index t represents the x and y components) ++++ε tttt ˆˆˆˆˆ =,= zzzz EEE z z z zzεε ε ε (2) so that the transverse components of the D and B fields are (vector equations) Type ε, μ, ξ, ζ Isotropic All ~1 Both 0 An-isotropic Some not ~1 Both 0 Bi-isotropic All ~1 Both ~1 Bi-an-isotropic All other cases Table 1. Classification of electromagnetic materials (1 denotes the unit dyadic). Circuit Analogs for Wave Propagation in Stratified Structures 491 ⋅ ++⋅+ ttttt tttt = zz EHDE Hεε ξ ξ (3) ⋅+ + ⋅ + ttttt tttt = zz EHB ζ E ζ H μμ (4) and the z components are (scalar equations) ξ ⋅ ++⋅+ε tt = z z zz z z zz z DE HEH ξ ε (5) μ ⋅+ +⋅ + tt = z z zz z z zz z B ζ EHζ EH μ (6) Since the material parameters are assumed independent of x and y, it makes sense to represent the fields through a Fourier transform in the transverse variables x and y as π ∞ −+ −∞ ∫∫ j( ) t 2 1 ()= (, )e (2 ) kx ky xy x y zdkdkEr E k (7) ∞ + −∞ ∫∫ j( ) t (, )= ()e kx ky xy zdxdyEk Er (8) where the transverse wave vector is k t = k x ˆ x + k y ˆ y . The action of the curl operator on the Fourier amplitude is shown by π ∞ −+ −∞ ∇× ∇× ∫∫ j( ) t 2 1 ()= [(, )e ] (2 ) kx ky xy x y zdkdkEr E k π ∞ −+ −∞ ∂ ⎛⎞ −+ × ⎜⎟ ∂ ⎝⎠ ∫∫ j( ) tt 2 1 ˆ =e j (,) (2 ) kx ky xy x y zdkdk z kz Ek (9) We then obtain the decomposition ∂∂ ⎛⎞ −+ × −× + × ⎜⎟ ∂∂ ⎝⎠ ttttt ˆˆ j (, )= j (, ) (, )zzz zz kz Ek kEk zEk ∂ −× −× + × ∂     tt t t t t t ˆˆ parallel to orthogonal to ˆ orthogonal to ˆˆ = j (, ) j (, ) (, ) z zEz z z zz z kE k kz k zE k (10) The result for the curl of the magnetic field is exactly the same. 2.2 Application to Maxwell’s equations We now apply the above decompositions with respect to z to Maxwell’s equations. These are ω ∇ × ()= j () H rDr (11) ω ∇× −()= j ()Er Br (12) When considering the Fourier amplitudes of the electromagnetic fields and using the constitutive relations this turns into (in the following we suppress the arguments z and k t of the fields for brevity) Wave Propagation in Materials for Modern Applications 492 () ω ∂ ⎛⎞ − +× ⋅+⋅ ⎜⎟ ∂ ⎝⎠ t ˆ j=j z zE ε ξkH H (13) () ω ∂ ⎛⎞ − +×−⋅+⋅ ⎜⎟ ∂ ⎝⎠ t ˆ j=j z zE ζ E μ kH (14) Another way to write this is by using dyadics (identifying (13) as the first row and (14) as the second row, and writing 1 for the unit dyadic) ω −× − × ⎛ ⎞⎛⎞⎛ ⎞⎛⎞ ⎛ ⎞⎛⎞ ∂ ⋅⋅−⋅ ⎜ ⎟⎜⎟⎜ ⎟⎜⎟ ⎜ ⎟⎜⎟ ×× ∂ ⎝ ⎠⎝⎠⎝ ⎠⎝⎠ ⎝ ⎠⎝⎠ t t ˆ j =j ˆ j z 01 0 1 10 1 0 zE k E E zHk Hζμ H ε ξ (15) The left hand side is orthogonal to ˆ z , and the equations for the z components are then (using that the cross product k t × E t is necessarily in the z direction since both vectors are in the xy-plane, with the scalar value ˆ z · (k t × E t ) = ( ˆ z × k t ) · E t ) ξ ωω −× ⎛⎞⎛ ⎞⎛⎞ ⎛ ⎞⎛⎞ ⎛ ⎞⎛⎞ ⋅− ⋅− ⎜⎟⎜ ⎟⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟⎜⎟ × ⎝⎠⎝ ⎠⎝⎠ ⎝ ⎠⎝⎠ ⎝ ⎠⎝⎠ ε tt t tt t ˆ 0j =jj ˆ 0j z z zz zz z zz zzzz z E ζμH 0 0 zk E E zk H ζμ H ξ ε (16) from which we solve for the z components of the fields: ξ ω ω − − − ⎡⎤ ⎛⎞ −× ⎛⎞⎛ ⎞ ⎛ ⎞⎛⎞ −⋅ ⎢⎥ ⎜⎟ ⎜⎟⎜ ⎟ ⎜ ⎟⎜⎟ ⎜⎟ × ⎢⎥ ⎝⎠⎝ ⎠ ⎝ ⎠⎝⎠ ⎝⎠ ⎣⎦ ε 1 1 t t 1 t t ˆ = ˆ zzzzz zz zzzzz zz E H ζμ 0 0 E zk ζμ H zk ξ ε (17) The transverse part of (15) is −× − × ⎛⎞⎛⎞⎛ ⎞⎛⎞ ∂ ⋅ ⎜⎟⎜⎟⎜ ⎟⎜⎟ ×× ∂ ⎝⎠⎝⎠⎝ ⎠⎝⎠ tt tt ˆˆ j = ˆˆ j z z E H z 01 0 10 0 zE kz zHkz ωω ⎛⎞⎛⎞⎛⎞⎛⎞ −⋅− ⎜⎟⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠⎝⎠ tt tt t t t tt tt t t t jj z z E H E ζμ H ζμ ξξ εε ω ωω ω − − ⎡⎤ ⎛⎞ −× ⎛⎞⎛⎞ ⎛⎞⎛⎞ −⋅+ − ⎢⎥ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎜⎟ ⎜⎟ × ⎢⎥ ⎝⎠⎝⎠ ⎝⎠⎝⎠ ⎝⎠ ⎣⎦ 1 tt tt t t t t 1 tt tt t t tt t ˆ =j j ˆ z z E H 0 0 E kz ζμ H ζμ kz ξξ εε (18) Inserting the expressions for the z components of the fields implies ωω −× ⎛ ⎞⎛⎞ ⎛ ⎞⎛⎞ ⎛⎞ ∂ ⋅− ⋅+⋅ ⎜ ⎟⎜⎟ ⎜ ⎟⎜⎟ ⎜⎟ × ∂ ⎝ ⎠⎝⎠ ⎝ ⎠⎝⎠ ⎝⎠ tttttt t tttttt t ˆ =j j ˆ z 01 A 10 zE E E zHζμ HH ξ ε (19) where A is the dyadic product 1 1 A dyadic product between two vectors ab is defined by its action on an arbitrary vector c as (ab) · c = a(b · c), i.e., a vector parallel to a with amplitude |a||b · c|. Thus, dyadic multiplication does not commute unless a is parallel to b. Circuit Analogs for Wave Propagation in Stratified Structures 493 1 1 1 tt t t 1 1 tt t t ˆˆ = ˆˆ zz zz z z zz zz z z ζμ ξ ωω ωω − − − − − ⎡⎤⎡⎤ ⎛⎞ ⎛⎞ −× −× ⎛⎞⎛ ⎞ ⎛⎞ −− ⎢⎥⎢⎥ ⎜⎟ ⎜⎟ ⎜⎟⎜ ⎟ ⎜⎟ ⎜⎟ ⎜⎟ ×× ⎢⎥⎢⎥ ⎝⎠⎝ ⎠ ⎝⎠ ⎝⎠ ⎝⎠ ⎣⎦⎣⎦ 00 A 00 ε kz zk ζμ ζμ kz zk εε ξξ (20) For an isotropic material, where ε = ε1, μ = μ1, ξ = ζ = 0, this is (writing a = −1 0 k k t × ˆ z = − −1 0 k ˆ z × k t where k 0 = ω/c 0 is the wave number in vacuum) μ μ −− − − −−− − ⎛⎞⎛⎞⎛⎞⎛⎞ − ⎜⎟⎜⎟⎜⎟⎜⎟ ⎜⎟⎜⎟⎜⎟⎜⎟ − ⎝⎠⎝⎠⎝⎠⎝⎠ ε ε 11 1 1 00 2 111 1 0 00 c0c 1 == c c0c 00 0 A 000 aaaa aaaa (21) Since − ˆ z × ( ˆ z × E t ) = E t for all transverse fields E t , we can write (19) as ωω ⎡⎤ −× ⎛⎞⎛ ⎞⎛⎞ ⎛ ⎞⎛⎞ ∂ ⋅− + ⋅ ⋅ ⎢⎥ ⎜⎟⎜ ⎟⎜⎟ ⎜ ⎟⎜⎟ −× × −× ∂ ⎝⎠⎝ ⎠⎝⎠ ⎝ ⎠⎝⎠ ⎣⎦ ttttt t ttttt t ˆ =jj ˆˆˆ z 01 10 A 10 0 1 Ez E zH ζμ zzH ξ ε (22) By keeping the vector product with ˆ z in the magnetic field, the vectors E t and − ˆ z × H t will be parallel to each other in isotropic media. Identifying the transverse electric and magnetic fields as vector voltage and vector current, i.e., −× tt ˆ =and =EV zHI (23) we find ωω ⎡⎤−× ⎛⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛⎞ ∂ ⋅− + ⋅ ⋅ ⎢⎥ ⎜⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜⎟ × ∂ ⎝⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝⎠ ⎣⎦ tt tt tt tt ˆ =jj ˆ z 01 10 A 10 0 1 Vz V I ζμ zI ξ ε (24) In particular, for an isotropic material this simplifies to ω ω − − ⎡⎤ ⎛⎞ ⎛⎞ ⎛ ⎞ ⎛⎞ ∂ −+ ⋅ ⎢⎥ ⎜⎟ ⎜⎟ ⎜ ⎟ ⎜⎟ ⎜⎟ ∂ ⎢⎥ ⎝⎠ ⎝ ⎠ ⎝⎠ ⎝⎠ ⎣⎦ ε ε 1 2 1 0 0 j =j c 0 μ z μ 01 10 VV bb II aa (25) where the unitless vector a = − 1 0 k k t × ˆ z defines the direction of the TE polarized transverse electric field (electric field transverse to the plane of incidence), and the unitless vector b = ˆ z × a = −1 0 k k t defines the direction of the TM polarized transverse electric field (electric field in the plane of incidence). The amplitude of both vectors is |a| = |b| = |k t |/k 0 = sinθ, where θ is the angle of incidence in vacuum. Equation (24) is recognized as a linear dynamical system for the transverse field components. If the material parameters are constant with respect to z, the solution of (24) can be written using the exponential matrix as (where V 1 = V(z 1 ) and V 2 = V(z 2 ) etc) ωω ⎧⎫ ⎡⎤ −× ⎛⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛⎞ ⎛⎞ ⎪⎪ −⋅−+⋅⋅ ⋅ ⎨⎬ ⎢⎥ ⎜⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜⎟ ⎜⎟ × ⎝⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝⎠ ⎝⎠ ⎪⎪ ⎣⎦ ⎩⎭ 1tttt22 12 12 1tttt22 ˆ =exp ( ) j j = ( , ) ˆ zz zz 01 10 AP 10 0 1 Vz VV I ζ μ zI I ξ ε (26) This formal solution reveals an important structure, which generalizes to inhomogeneous media where the material parameters may depend on z: the transverse fields at z = z 1 can be written as a dyadic P operating on the fields at z = z 2 , where z 1 and z 2 are arbitrary (although the dyadic of course depends on z 1 and z 2 ). This dyadic is called a propagator, and its Wave Propagation in Materials for Modern Applications 494 existence is guaranteed by the linearity of the problem. We write the explicit form of this dyadic for isotropic media in (34), but first we must define a few properties. 2.3 Eigenvalue problem in infinite media If the wave is propagating in a medium which is infinite in the z direction, it is natural to search for solutions on the form β − ⎛⎞⎛⎞ ⎜⎟⎜⎟ ⎝⎠⎝⎠ 0 j 0 () =e () z z zII VV (27) This implies β β − ∂ − ∂ j 00 [(),()]= j[ , ]e z zz z VI VI which makes (24) turn into an algebraic eigenvalue problem (after dividing by −jω and the exponential factor e −jβz ) β ω ⎡⎤ −× ⎛⎞⎛ ⎞⎛ ⎞ ⎛ ⎞⎛⎞ ⋅−⋅⋅ ⎢⎥ ⎜⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜⎟ × ⎝⎠⎝ ⎠⎝ ⎠ ⎝ ⎠⎝⎠ ⎣⎦ 0tttt 0 0tttt 0 ˆ = ˆ 01 10 A 10 0 1 z I ζμ zI ξ εVV (28) Thus, the propagation constant β can be found from the eigenvalue problem (28), which can easily be solved numerically once the material model is specified along with the transverse wave vector k t (which occurs only in A). In addition, the field amplitudes [V 0 , I 0 ] are the eigenvectors of the same dyadic and can be determined up to a multiplicative constant. Independent of the normalization, the eigenvectors always provide a mapping between the transverse components of the electric and magnetic fields, i.e., V 0 = Z · I 0 (29) where the dyadic Z is the transverse wave impedance of the wave. For isotropic media corresponding to (25), we have μ β ω μ −− −− ⎛⎞ − ⎛⎞ ⎛⎞ ⋅ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ − ⎝⎠ ⎝⎠ ⎝⎠ ε ε 21 00 0 21 00 0 0c = c0 bb II aa VV (30) This implies () β μμμθ ωμμ μ −− − − ⎛⎞⎛⎞⎛ ⎞ ⎛⎞ −⋅−⋅ −+⋅ − ⎜⎟⎜⎟⎜ ⎟ ⎜⎟ ⎝⎠ ⎝⎠⎝⎠⎝ ⎠ εεε εε ε 2 22 2 2 2 00 0 00000 cc c ==()=c sin 11 1VaabbV aabbV V (31) where we used a · b = 0 and |a| = |b| = sinθ. The final result for the wave number is then β ωεμ θ − − 2 2 0 =c sin (32) and the wave impedance, defined by the relation V 0 = Z · I 0 , is ωωμβ μ βεμ β ωε − ⎛⎞ −+ ⎜⎟ ⎝⎠ 2 0 22 c == || || Z1 aa bb bb ab (33) Circuit Analogs for Wave Propagation in Stratified Structures 495 In vacuum, we have ωμ/β = η 0 /cos θ and β/(ωε) = η 0 cos θ, where ημ ε 000 =/ is the intrinsic wave impedance of vacuum. Finally, the propagator dyadic for a slab of length ℓ of isotropic media is ββ ββ − ⎛⎞ ⎜⎟ ⎝⎠ AA AA 1 cos( ) j sin( ) = j sin( ) cos( ) 1Z P Z1 (34) In microwave theory, this is recognized as the ABCD-matrix of a transmission line with propagation constant β and characteristic impedance Z (Pozar, 2005, p. 185). Note however that we have generalized it to include both TE and TM polarization, through the dyadic character of Z. The important thing about the propagator dyadic is that since tangential electric and magnetic fields are continuous, we can find the total propagator dyadic for a layered structure by cascading: ⎛⎞ ⎛⎞⎛⎞ ⋅⋅ ⎜⎟ ⎜⎟⎜⎟ ⎝⎠ ⎝⎠⎝⎠ " 12 tot 1 2 tot 12 ==,where= N AB PPPP P CD VV II (35) The dyadic P tot maps the total fields from one side of the layered structure to the other. Outside the structure, the total fields can be expressed in terms of the incident field amplitude V inc using reflection and transmission dyadics r and t as (where we assume the same medium on both sides, with the characteristic impedance Z 0 , and use the fact that waves propagating along the positive z direction satisfy V + = Z 0 · I + , whereas waves propagating in the negative z direction satisfy V − = −Z 0 · I − ) −− +⋅ ⋅ ⎛⎞⎛ ⎞ ⎛⎞⎛ ⎞ ⎜⎟⎜ ⎟ ⎜⎟⎜ ⎟ ⋅−⋅ ⋅⋅ ⎝⎠⎝ ⎠ ⎝⎠⎝ ⎠ 1 inc 2 inc 11 10 inc 20inc () =and= () 1r t Z1r Zt VVVV IVIV (36) Solving for the reflection and transmission dyadics imply − −− −− +⋅ −⋅−⋅⋅ ⋅+⋅ +⋅+⋅⋅ 11111 00 00 00 00 =( ) ( )r ABZ ZCZDZ ABZ ZCZDZ (37) − −− +⋅ + ⋅+ ⋅ ⋅ 111 00 0 0 =2( )tABZZCZDZ (38) Thus, the concept of propagator dyadics enables a straight-forward analysis of layered structures, although the final results in terms of reflection and transmission coefficients may be complicated. In addition, thin sheets which are inhomogeneous in the xy-plane can also be modeled with corresponding propagator dyadics. This is explored in the next sections. 3. Lumped element models of scatterers In real applications, relatively thick homogeneous slabs are often interlaced with thinner sheets, which may also be inhomogeneous in the transverse plane. Such scatterers can be modeled as lumped elements, the simplest of which corresponds to homogeneous, thin sheets. We are thus led to study the limit of the ABCD-matrix for a slab when its thickness ℓ becomes small. Denote the thickness of the sheet by t. Considering the factors in the propagator dyadic (34) and keeping factors to first order in βt, we find β ββ →→cos( ) 1 and sin( )ttt (39) Wave Propagation in Materials for Modern Applications 496 Thus, to first order in βt the ABCD-matrix is (using ωμ β βω + ε 22 = || || Z aa bb ab ) β ωμ ωμ ββ ββ β ωε ωμ − ⎛⎞ ⎛⎞ + ⎜⎟ ⎜⎟ ⎛⎞ ⎝⎠ ⎜⎟ → ⎜⎟ ⎜⎟ ⎛⎞ ⎝⎠ ⎜⎟ + ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ε ε 2 22 2 1 2 222 j || || cos( ) jsin( ) jsin( ) cos( ) j || || t tt tt t 1 1Z Z1 1 aa bb ab aa bb ab (40) In order to treat the sheet as a lumped element, the reference planes T and T ′ in Figure 1 should coincide. This corresponds to back propagating the fields at T ′ by multiplying the dyadic above by the inverse of the corresponding dyadic for the background medium (denoted by index 0), or to first order in βt, subtracting the corresponding phase change in the off-diagonal elements. For instance, the upper right element should be replaced by (using β 2 = εμ − − 2 0 c sin 2 θ and − 2 0 c= ε 0 μ 0 ) ωμ θ ωμ θ μμ −− ⎡ ⎤ ⎡⎤ +− − +− ⎢ ⎥ ⎢⎥ ⎣⎦ ⎣ ⎦ εε 22 22 00 0 222 2 00 cc j(1)j (1) sin sin || || || || tt aa bb aa bb aba b ωμ μ ωμ θ ωμ θ μ − ⎛⎞ ⎛⎞ −++− + ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ε 2 22 0 00 22 2 c =j ( ) j j sin sin || || || ttt aa bb bb ab b ωμ μ ωμ θ ⎛⎞ −+ − ⎜⎟ ⎝⎠ ε ε 2 0 00 2 =j ( ) j 1 sin || tt1 bb b (41) In the final step we used that a and b are orthogonal and span the xy-plane, i.e., + 22 || || aa bb ab = 1. To first order, the result is then (using sin 2 θ = 2 ||a = 2 ||b ) Fig. 1. Transmission line model of an isotropic slab. Fig. 2. Definition of ABCD matrix parameters for a general twoport network. [...]... discrimination Circuit Analogs for Wave Propagation in Stratified Structures 507 7 Conclusions We have shown how relatively complex wave propagation problems can be efficiently modeled and designed using circuit analogs Propagation of plane waves in any bianisotropic material can be modeled as propagation of voltage and current in two transmission lines, one for each polarization The wave number and characteristic... Wave Propagation 27 Field Experiments on Wave Propagation and Vibration Isolation by Using Wave Barriers Seyhan Fırat1, Erkan Çelebi2, Günay Beyhan3, İlyas Çankaya4, Osman Kırtel2 and İsa Vural1 1Sakarya University, Technical Education Faculty, Construction Department University, Engineering Faculty, Civil Engineering Department 3Sakarya University, Engineering Faculty, Geophysical Engineering Department... are given in Table 2 Field Experiments on Wave Propagation and Vibration Isolation by Using Wave Barriers 513 Soil Dynamic Parameters Parameters P (Compression) Wave Velocity S (Shear) Wave Velocity Layer Thickness Density Maximum Shear Module Elasticity Module Poisson Ratio Soil Vibration Period Symbol Unit 1 Layer 2 Layer 3 Layer 4 Layer CP m/s 370 580 1012 173 9 CS h m/s m kN/m3 kN/m2 kN/m2 s 133 0.7... analysis of wave propagation problems in elastic medium with emphasis on soil-structure interaction due to moving loads [9-16] The reduction of the structural response may be accomplished as: a) by adjusting the frequency contents of the excitation, b) by changing the location and direction of the vibratory source, c) by modifying the wave dissipation characteristics of the soil deposit, 510 Wave Propagation. .. linear-to-circular polarizer abs(Ev/Eh) [dB] arg(Ev/Eh) [degrees] 1.5 110 105 1 100 0.5 95 0 90 0.5 1 11 85 12 13 Co 14 15 16 17 18 19 80 11 12 13 14 polarization return loss [dB] 15 16 17 18 19 16 17 18 19 XPD [dB] 0 15 10 20 20 25 30 30 40 35 Ev Eh 50 40 60 70 11 45 12 13 14 15 16 17 18 19 50 11 12 13 14 15 Fig 13 Results for the layered meander structure (the x-axes in all subfigures are frequency in... physical 501 Circuit Analogs for Wave Propagation in Stratified Structures Fig 4 Geometry and equivalent circuit for capacitive strips (TM polarization) Fig 5 Geometry and equivalent circuit for inductive strips (TE polarization) interpretations of the circuit analogs, but analogs are still valuable as a modeling tool even for higher frequencies, in particular for structures of subwavelength size The general... source by constructing a suitable wave barrier in the path of the propagating waves between the dynamic load and the affected structures to be protected When the wave barrier is located nearby the vibratory source, such application is known as active (near field) isolation If the barrier is situated away from the source but around the structure to be protected from incoming waves, such far field isolation... swelling soil problems [27] Properties of the local soil conditions should be determined to investigate isolation effect of the wave barriers accurately Boring logs are conducted on the site for ground exploration Field Experiments on Wave Propagation and Vibration Isolation by Using Wave Barriers 511 and soil strata definition Borings were located where site refraction tests indicated possible anomalies,... conductor surfaces, Microwaves, Antennas & Propagation, IET 1(1): 255–260 Caloz, C & Itoh, T (2004) Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line, IEEE Trans Antennas Propagat 52(5): 1159–1166 Collin, R E (1991) Field Theory of Guided Waves, second edn, IEEE Press, New York Collin, R E (1992) Foundations for Microwave Engineering,... 72098 20 4174 0.44 341 1.8 17. 5 199075 5 7172 7 0.35 570 2.3 18.9 625615 1319592 0.30 ρ Gmax E ν T0 0.6 Table 2 Soil dynamic parameters in the test site 3 Measurements for vibration screening performance Evaluation of the screening effectiveness precisely depends on the barrier material stiffness Hence, a series of experiments are necessary to understand the propagating characteristics of the waves The . good for wave propagation problems, is that they are exact for one-dimensional wave propagation, regardless of whether we consider acoustic or electromagnetic waves. Typically, wave propagation. Pulse Wave Propagation in Bistable Oscillator Array 487 M. Yamauchi, M. Wada, Y. Nishio and A. Ushida, Wave propagation phenomena of phase states in. through a propagation factor e −j β z . The wave number β corresponds to the eigenvalue, and the wave impedance is given by the eigenvectors. 2.1 Notation We consider time harmonic waves using