Shock structure for electromagnetic waves in bianisotropic, nonlinear materials



[1] Shock waves are discontinuous solutions to quasi-linear partial differential equations and can be studied through a singular perturbation known as the vanishing viscosity technique. The vanishing viscosity method is a means of smoothing the shock, which we use to study the case of electromagnetic waves in bianisotropic materials. We derive the conditions arising from this smoothing procedure for a traveling wave, and the waves are classified as fast, slow, or intermediate shock waves.

1. Introduction

[2] Electromagnetic waves propagating in an instantaneously reacting material can be modeled with a system of quasi-linear partial differential equations. It is well known that such a model can exhibit shock solutions, i.e., solutions that become discontinuous in finite time even if the initial/boundary data are smooth. This poses severe problems for numerical methods, such as finite difference schemes, which are often based on the assumption of continuous and differentiable solutions.

[3] In order to overcome this problem, we can model the material on a finer scale, which requires a denser grid and thus increases the memory demands and the computation time. Another approach is to develop more powerful numerical methods, which can handle discontinuous solutions. The development of these numerical schemes benefits from an understanding of the propagation of shock waves; for instance, Godunov's scheme is based on the solution of Riemann's problem [Godlewski and Raviart, 1996; Godunov, 1959], where the shock wave is generated by discontinuous initial data. A variation of Godunov's scheme is Glimm's scheme, which is used to show global existence of solutions to certain systems of equations [Glimm, 1965; Hörmander, 1997].

[4] The aim of this paper is to increase the understanding of electromagnetic shock waves, modeled with the Maxwell equations. Mainly using techniques from the work of Sjöberg [2000], we analyze the wave propagation in bianisotropic materials, i.e., materials with different properties for different polarizations of the waves, and a possible coupling between the electric and the magnetic field [Kong, 1986, p. 7]. This adds insight not only to the numerical treatment of electromagnetic waves in complicated materials, but also provides some physical intuition.

[5] The Maxwell equations can be considered as a hyperbolic system of conservation laws. A good introduction to the numerical approximation of such systems is given by Godlewski and Raviart [1996], which introduces the analytical theory as well as some common schemes in one and two spatial dimensions. There is presently not a good mathematical understanding of systems of conservation laws in several dimensions, but some general references are the works of Godlewski and Raviart [1996], Hörmander [1997], Dafermos [2000], Serre [2001], and Taylor [1996].

[6] Perhaps the most familiar kind of “electromagnetic” shock wave is in the field of magnetohydrodynamics, from which we give only a few references [Bekefi and Brown, 1966; Conley and Smoller, 1974; Farjami and Hesaaraki, 1998; Germain, 1960; Grabbe, 1984; Landau et al., 1984, pp. 245–253]. Electromagnetic shock waves in isotropic media have previously been treated theoretically [Bloom, 1993, and references therein; see Landau et al., 1984, pp. 388–391]. Recently, a few papers on experiments concerning electromagnetic shock waves have been published [Dolan, 1999; Brooker et al., 1999; Branch and Smith, 1996]. In continuum mechanics, Maugin has recognized the similarity between shock waves and phase transition fronts as singular sets in irreversible motion, with a dissipation related to the power expanded by a driving force on the singularity set [see Maugin, 1998, 2000].

[7] In this paper, we study when the shock waves can be defined as the limit of continuous traveling wave solutions to an approximate problem, where the discontinuity is smoothed over a small region. This is the shock structure problem, which was introduced by Gel'fand [1963], and is given an extensive treatment by Smoller [1994]. A thorough treatment of this problem in magnetohydrodynamics is given by Farjami and Hesaaraki [1998], and a recent paper deals with the structure of electromagnetic shock waves in anisotropic ferromagnetic media [Gvozdovskaya and Kulikovskii, 1999].

[8] This paper is organized as follows. In section 2 we introduce the Maxwell equations and the constitutive relations used to model the electromagnetic waves, as well as the general form of the entropy condition. In section 3 we present the vanishing viscosity method of smoothing the solutions of quasi-linear hyperbolic equations. The consequences of the vanishing viscosity method for traveling waves are studied in sections 4 and 5, where we show that there exists three kinds of electromagnetic shock waves: the fast, the slow, and the intermediate shock wave. In section 6 we also mention another form of discontinuous solutions, contact discontinuities, which cannot be analyzed with the vanishing viscosity method for traveling waves. However, they exist only under the condition of linear degeneracy, and we present this condition and its opposite, genuine nonlinearity, in section 6. The different kinds of shock waves are illustrated with phase portraits of a certain system of ordinary differential equations in section 7, and some concluding remarks are made in section 8.

2. The Maxwell Equations, Constitutive Relations, and Entropy Condition

[9] In this paper we use a slight modification of the Heaviside–Lorentz units for our fields [Jackson, 1999, p. 781], i.e., all electromagnetic fields are scaled to units of equation image,

equation image

where E and H is the electric and magnetic field strength, respectively, and D and B is the electric and magnetic flux density, respectively. The index SI is used to indicate the field in SI units. We use the scaled time t = c0tSI, where equation image is the speed of light in vacuum, and the constants ϵ0 and μ0 are the permittivity and permeability of free space, respectively. The six-vector notation from the works of Sjöberg [2000] and Gustafsson [2000], i.e.,

equation image

enables us to write the source free Maxwell equations in the compact form

equation image

In this paper we treat the six vectors as column vectors, i.e., we write the scalar product as eTd = Σi = 16eidi. This is merely for notational convenience and does not capture the full mathematical structure, which is not needed here. On occasions, we also consider the scalar product between two three-vectors, in which case we use the standard notation E · D = Σi = 13EiDi. For more ambitious attempts to construct a six-vector notation, we refer to the works of Gustafsson [2000] and Lindell et al. [1995].

[10] The Maxwell equations must be supplemented by a constitutive relation, whose purpose is to model the interaction of the electromagnetic field with the material. When the material reacts very fast to excitation, we can model it with an instantaneous constitutive model, where the values of the electric flux density D and the magnetic flux density B are completely determined by the values of the electric field strength E and magnetic field strength H at the same point in space-time. We write this as

equation image

where d(e) is the gradient of a scalar function ϕ(e) with respect to e, i.e., in terms of thermodynamics, the field gradient of the thermodynamic potential (or the free energy density or the free enthalpy density) [Landau et al., 1984; Coleman and Dill, 1971]. We use the notation d(e) = ϕ′(e) to denote this derivative, i.e., di(e) = ∂ϕ/∂ei, i = 1, …, 6. The model is passive if we require that the symmetric 6 × 6 matrix d′(e) = ϕ″(e), where [d′(e)]ij = ∂2ϕ/∂eiej, is a positive definite matrix, which is the case if the scalar function ϕ(e) is a convex function.

[11] The initial value problem for the Maxwell equations with an instantaneously reacting constitutive model is

equation image

and since d′(e) is positive definite and symmetric, this is by definition a quasi-linear, symmetric, hyperbolic system of partial differential equations [Taylor, 1996, p. 360]. This system has been extensively studied by Sjöberg [2000], where it is shown that the equations in general support two waves, the ordinary and the extraordinary wave, each with its own refractive index.

[12] Due to the quasi-linearity, the system (5) may exhibit shock solutions, i.e., even if we give smooth initial data, the solution becomes discontinuous in finite time. This means that the derivatives cannot be classically defined everywhere, but we can make a weak formulation of the problem by requiring the equality

equation image

to hold for all six-vector test functions φ defined on R3 × [0,∞), i.e., vector-valued functions which are infinitely differentiable with compact support. One problem with this weak formulation is that we lose uniqueness, i.e., there are several weak solutions e which satisfy the above criteria.

[13] If the solution e to (5) is smooth, we can multiply the equations from the left by eT to obtain the Poynting theorem, or energy conservation law,

equation image

where S(e) = E × H is the Poynting vector, and η(e) = eTd(e) − ϕ(e) is the electromagnetic energy. When the solution e is not smooth, this inequality is no longer valid since the derivatives are not defined in the classical sense. However, as is shown by Sjöberg [2001], it is reasonable to replace it with the inequality

equation image

which is interpreted in a weak sense, i.e., for all scalar test functions φ ≥ 0 defined on R3 × [0, ∞), the inequality

equation image

holds. The inequality (8) is called an entropy inequality, and if e satisfies both (8) and (5), it is called an entropy solution. It is frequently conjectured that entropy solutions are unique [Godlewski and Raviart, 1996, p. 32], and we refer to the work of Sjöberg [2001] for a discussion of the physical interpretation of this inequality. In the following section, we show how the entropy inequality is a natural consequence of the vanishing viscosity method.

3. Vanishing Viscosity Regularization

[14] The loss of uniqueness for weak solutions is important to resolve if we want to make numerical approximations of the differential equations. This problem has been extensively studied for scalar conservation laws and systems of conservation laws in one space variable [Godlewski and Raviart, 1996; Smoller, 1994; Hörmander, 1997; Evans, 1998; Taylor, 1996], where the conservation law is typically written equation image. Our knowledge of systems of conservation laws in several space variables is limited, but a common assumption is that reasonable (physical) solutions should arise as limits to the regularized equation equation image, when δ → 0. Since the second-order derivative is often used as a model for a small viscous effect, this method is called the vanishing viscosity method. The benefit of the vanishing viscosity method is that for each δ > 0 we can usually prove that the initial value problem is well posed, with unique, differentiable solutions. We can define a unique limit u as δ → 0 if we can find a convergent sequence of solutions {uδ}. However, this limit u must also be shown to satisfy the original conservation law, which is often nontrivial. For systems of conservation laws in several dimensions, this is still an unsolved problem [Serre, 2001].

[15] We propose to use a similar method to define solutions to our quasi-linear system of equations, where we study the equations

equation image

for δ > 0. Standard PDE theory guarantees a C solution eδ to this equation for every δ > 0 for a large class of e0 [see Taylor, 1996, pp. 327–332]. An important result is that if the viscosity solution eδ converges boundedly almost everywhere in the limit δ → 0, the limit satisfies the entropy condition from the previous section. To see this, multiply (10) with eδT and observe

equation image

where η(eδ) is the electromagnetic energy in the medium and S(eδ) is the Poynting vector. Note that all the derivatives are classically defined, and we have the following scalar inequality,

equation image

It can be shown that if eδ is uniformly bounded in the supremum norm and converges almost everywhere to e as δ → 0, then this limit solution e is a weak solution to (5) and satisfies the inequality

equation image

almost everywhere [see Godlewski and Raviart, 1996, p. 27; Taylor, 1996, p. 438]. In the following sections, we study the consequences of the vanishing viscosity method in the case of traveling waves, which provides us with a more precise means of writing the entropy condition.

4. Inner Solutions and Shock Structure

[16] In this section we largely follow the ideas presented in many textbooks [e.g., Evans, 1998, pp. 600–603; Godlewski and Raviart, 1996, pp. 79–83; Smoller, 1994, pp. 508–510; Taylor, 1996, p. 431]. Dropping the index δ for brevity, we investigate the singularly perturbed Maxwell equations (10) for the existence of solutions in the form of traveling waves,

equation image

where we have chosen z to be the coordinate in the propagation direction, and v is the speed of the shock wave. We also require the derivative e′ (ζ) to disappear as ζ → ±∞, and a typical situation is depicted in Figure 1. In the language of singular perturbation theory [Kevorkian and Cole, 1996], the traveling wave corresponds to an inner solution of the problem (10), and is a means of analyzing the microscopic behavior of the solution at a scale of order δ. The microscopic properties of a number of discontinuities, which are distant at a macroscopic scale, can be treated by considering them as isolated traveling waves of the type (14). Observe that ζ → ∞ does not necessarily mean z → ∞, it is sufficient that z > vt and δ → 0.

Figure 1.

A typical traveling wave profile. The idea is that the inner solution shall provide a smooth transition between the outer solutions and the left and right constant states el and er. The solution typically arises in Riemann's problem, where the initial values are e(x, 0) = e0(x) = el for z < 0 and er for z > 0.

[17] The traveling wave solution (14) must satisfy the ordinary differential equation

equation image

where equation image denotes the unit vector in the z direction. Observe that this equation does not involve the parameter δ, reflecting the fact that we are studying properties at a certain scale. Integrating the above equation once implies

equation image

where el,r = limζ→∓∞e (ζ). Taking the opposite limit eer,l in (16) implies the Rankine–Hugoniot jump condition

equation image

where we use the notation [e] = erel and [d(e)] = d(er) − d(el) to indicate the jumps in the quantities e and d(e) over the shock. Note that the Rankine–Hugoniot condition is a vector identity, and that the jump in d(e) cannot have a component parallel to equation image, unless v = 0.

[18] We use the assumption (e′)1,r = 0 to write (16) as a system of autonomous, ordinary differential equations,

equation image

with the asymptotic boundary conditions limζ→‡∞e(ζ) = el,r. It is clear that these states are critical points for the system (18), i.e., the right-hand side is zero for these states. In the following section we investigate when the system (18) has a solution, and what conditions this infers on the speed v. The corresponding ODE for ferromagnets described by the Landau–Lifshitz constitutive equation is studied in detail by Gvozdovskaya and Kulikovskii [1999].

5. The Entropy Condition for a Traveling Wave

[19] A solution to (18) that connects its critical points el and er, where eler, is called a heteroclinic orbit [Poincaré, 1993]. Before investigating these orbits, we show that homoclinic orbits, i.e., solutions where el = er and eel,r somewhere on the orbit, cannot exist. Multiplying (18) with (e′)T we obtain

equation image

which shows there exists a scalar function ψ(e), which is nondecreasing along the orbit. Such a function must be constant on a homoclinic orbit, implying |e′|2 = 0, and thus e must be constant throughout the orbit, which degenerates to a point.

[20] The existence of a heteroclinic orbit for the system (18) requires that the unstable manifold of one critical point intersects the stable manifold of the other, where the unstable and the stable manifold is associated with the positive and the negative eigenvalues of the linearized problem, respectively. If the existence of a heteroclinic orbit is to be guaranteed even under small perturbations, then the sum of the dimensions of the stable and unstable manifold must exceed the dimension of the phase space [Smoller, 1994, p. 509]. In our case, the relevant manifolds are the unstable manifold for el and the stable manifold for er. The dimensions of these manifolds can be calculated from counting how many eigenvalues of the linearized equations that are greater/lesser than zero at each critical point. The linearized equations are

equation image

Temporarily denoting the 6 × 6 matrix d′(el,r) by A, the problem of deducing the dimension of the stable and unstable manifolds consists in counting positive and negative eigenvalues for the matrix equation image × JvA. Since A is positive definite, the signs of the eigenvalues are the same as for the problem

equation image

Using the same technique as used by Sjöberg [2000], we formulate this eigenvalue problem as

equation image

where equation image is the symmetric, positive definite square root of A, ci = v + λi, and wi = equation image. The matrix in the right-hand side is a congruence transformation of equation image, and it is well known that such a transformation does not change the signs of the eigenvalues [Goldstein, 1980, p. 251]. Since equation image has the (double) eigenvalues ±1 and 0, there are always two negative eigenvalues c3,4 < 0 and two zero eigenvalues c5,6 = 0. This implies λ3,4,5,6 ≤ −v < 0. The argument concerning the dimensions of the stable and unstable manifolds can then involve only the positive eigenvalues c1 and c2, and the corresponding λ1 and λ2. In order for the sum of the dimension of the unstable manifold (λ > 0) at el and the dimension of the stable manifold (λ < 0) at er to be larger than six (the dimension of the phase space), one of the following conditions must hold:

equation image

Observe that the dimension of the unstable manifold at el is calculated from the number of positive eigenvalues, i.e., the number of positive eigenvalues in the left column of (23). The dimension of the stable manifold at er is calculated from the number of negative eigenvalues, i.e., the number of negative eigenvalues in the right column of (23) plus four, since we deduced earlier that λ3,4,5,6 are always negative.

[21] The two positive eigenvalues c1,2 = v + λ1,2 are identified as the characteristic wave speeds in the material, which are determined from the eigenvalue problem (22) for each state el,r, as also determined by Sjöberg [2000]. The speeds are in general functions of both the state, el or er, and the propagation direction, equation image, but we choose to suppress the dependence on the propagation direction since this is constant in this paper.

[22] The conditions on λ1,2 above can be written in terms of the shock speed v and the characteristic wave speeds c1,2 as

equation image

These expressions constitute the entropy conditions for electromagnetic, plane shock waves. The nomenclature “fast shock” and “slow shock” is in accordance with the works of Gvozdovskaya and Kulikovskii [1999] and Farjami and Hesaaraki [1998] and “intermediate shock” is from the work of Farjami and Hesaaraki [1998]. Note that the fast and the slow shock are closely connected to the ordinary and extraordinary rays for anisotropic materials [see Kong, 1986, pp. 68–71; Landau et al., 1984, pp. 331–357].

[23] To conclude this section, we note that our entropy condition is analogous to the Lax entropy condition for an n-dimensional, strictly hyperbolic system of conservation laws ut + f(u)x = 0. This condition is that there should exist an index k such that

equation image

where λ1(u), …, λn(u) are the eigenvalues of the n × n matrix f′(u) and v is the shock speed [see Evans, 1998, p. 589; Godlewski and Raviart, 1996, p. 76; Hörmander, 1997, p. 61; Smoller, 1994, p. 261].

6. Genuine Nonlinearity and Contact Discontinuities

[24] When ci(el) = ci(er) for i = 1 and/or i = 2, one or several of the conditions (24) may not be applicable. This phenomenon occurs for a type of waves called contact discontinuities, which are characterized by

equation image

for all e ∈ γ, where γ is a smooth curve connecting el to er in R6. Differentiating the latter condition along the curve γ, implies equation image, where equation image denotes the tangential derivative of e along this curve. This means equation image is proportional to the eigenvector ei by definition. That the speed is constant on the curve γ can also be written

equation image

where Deci denotes the gradient of the speed ci with respect to the six-vector e, i.e., (Deci)k = ∂ci/∂ek. This means that the eigenvector ei must be orthogonal to Deci. We say the field ei is linearly degenerate if eiTDeci = 0, and genuinely nonlinear if eiTDeci ≠ 0 [see Godlewski and Raviart, 1996, p. 41]. One reason for using the term linearly degenerate is that contact discontinuities travel along noncrossing characteristics, just as in the linear case. An interesting feature of contact discontinuities is that their structure is not captured by the traveling wave ansatz, since the right-hand side of (18) is identically zero. In this paper, we restrict ourselves to investigating a few explicit examples.

[25] Our first example is a constitutive relation which always has one linearly degenerate field. For an instantaneously reacting, isotropic, nonmagnetic material, we have the constitutive relations

equation image

One example is the Kerr model, where F(|E|2) = 1 + α|E|2. It is not difficult to prove that the characteristic speeds are

equation image

with the corresponding eigenvectors defined by ei = (Ei, Hi)T, where equation image for i = 1, 2, and

equation image

Since the speed is independent of H, we have eiTDeci = Ei · DEci for i = 1, 2. From the explicit expressions (29) it is seen that DEc1 ∼ DEc2E, where the ∼ sign indicates proportionality. It is clear that E1 · DEc1 ≠ 0 and E2 · DEc2 = 0, i.e., the field E1 is genuinely nonlinear and E2 is linearly degenerate. We interpret a wave where the change in E is orthogonal to E, i.e., ∂tEE2, as a circularly polarized wave. This is motivated by the fact that the amplitude |E| does not change, but the vector E appears to rotate when observed as a function of time at a given point in space. Thus, we have found that circularly polarized waves in an isotropic medium are linearly degenerate, and therefore less sensitive to nonlinear effects.

[26] Our second example is a constitutive model where there are no linearly degenerate fields. The model is

equation image

which is not valid for all E, since D′(E) = (1 + C · E)I + CE + EC is not positive definite everywhere. However, it is positive definite if |C||E| < 1/3, and thus the model suffices as an approximation for E small enough. For this model, the three-vector C represents a “nonlinear axis” of the material, which is obviously anisotropic. It is straightforward to show that when both C and E are orthogonal to equation image, we have

equation image

where the upper sign corresponds to c1 and E1, and H1,2 = equation image × E1,2. The scalar product Ei · DEci from which we analyze genuine nonlinearity can be shown to be

equation image

We see that one of these quantities is zero if E is parallel or antiparallel to C, but any situation in-between means E1 · DEc1 ≠ 0 and E2 · DEc2 ≠ 0. This shows that this model usually has no linearly degenerate fields, and contact discontinuities occurs only when the electric field is parallel or antiparallel to the axis C. We conclude this example by noting the peculiarity that when the scalar product C · E is negative, the characteristic speeds c1,2 may be larger than one, which is the speed of light in vacuum in our units. This may further restrict the validity of this model.

7. Numerical Demonstration of Shock Structure for an Anisotropic Material

[27] In this section we show numerically that there exists a structure (an inner solution dissipatively connecting two states) for a nonlinear anisotropic material. In order to present a concise example, we regularize the Maxwell equations in the electric field only, i.e.,

equation image

The benefit of this approach is to reduce the phase space of the resulting system of ordinary differential equations to two dimensions, which enables us to plot the phase space easily. The approach is reasonable if we consider the Faraday law to be exact, and a similar technique is sometimes used for equations describing gasdynamics [Evans, 1998, p. 602]. The anisotropic material is described by the constitutive equation

equation image

where in this section the fields are dimensionless (see the work of Sjöberg [2000] for details on the scaling). This model has an anisotropic linear part and an isotropic nonlinear part, i.e., practically the same example material used by Sjöberg [2000].

[28] The system of ordinary differential equations corresponding to (18) becomes

equation image

With Ezl,r = Hzl,r = 0 we have Ez = Hz = 0 throughout the shock, and |E|2 = Ex2 + Ey2. By eliminating the magnetic field and the z components, we obtain the following 2 × 2 system of ordinary differential equations,

equation image

which contains all the qualitative information we need. We remark that this system can be integrated exactly for certain values of Exl,r and Eyl,r, but we refrain from exploiting this possibility in this paper. Phase portraits, i.e., plots of the vector fields on the right-hand side of the equations above, are found in Figures 2, 3, and 4 for a fast shock, an intermediate shock, and a slow shock, respectively. Figure 5 depicts the phase portrait for a shock with mixed polarization, and Table 1 lists the relevant numbers used in each phase portrait. It is clearly seen from the figures that there exists a path connecting the critical points.

Figure 2.

Phase portrait of a fast shock wave structure problem. The critical points are (Exl, Eyl) = (0.000, 0.000) and (Exr, Eyr) = (0.500, 0.000).

Figure 3.

Phase portrait of an intermediate shock wave structure problem. The critical points are (Exl, Eyl) = (0.200, 0.000) and (Exr, Eyr) = (1.000, 0.000).

Figure 4.

Phase portrait of a slow shock wave structure problem. The critical points are (Exl, Eyl) = (0.788, 0.000) and (Exr, Eyr) = (1.000, 0.000).

Figure 5.

Phase portrait of a slow shock wave structure problem with mixed polarizations. The critical points are (Exl, Eyl) = (0.617, 0.472) and (Exr, Eyr) = (1.000, 1.000).

Table 1. Relevant Values for the Phase Portraits
  1. a

    The last column is the entropy difference Δ = equation image · (S(er) − S(el)) − v(η(er) − η(el)), and since all the numbers in the column are negative, we see that all the waves satisfy the original entropy condition (8).

Figure 20.0000.0000.5000.0000.6670.5770.7070.5550.603−0.011
Figure 30.2000.0001.0000.0000.5560.5730.6870.4470.500−0.087
Figure 40.7880.0001.0000.0000.4760.5090.5250.4470.500−0.002
Figure 50.6170.4721.0001.0000.4000.4780.5740.3420.475−0.059

8. Discussion and Conclusions

[29] By studying a parabolic regularization of the quasi-linear Maxwell equations, we have proposed a classification of electromagnetic shock waves into three categories: slow, fast, and intermediate. This classification depends on how the shock speed relates to the characteristic speeds in the material, which in turn depend on the field strengths on both sides of the shock. These shock conditions can probably be improved with the help of Conley's index theory, as in the works of Smoller [1994] and Farjami and Hesaaraki [1998].

[30] There also exists an additional kind of discontinuity, the contact discontinuity, which only occurs for linearly degenerate fields. In particular, we have showed that circularly polarized waves in isotropic, nonlinear media, exhibits contact discontinuities. The further study of contact discontinuities is beyond the scope of this paper, but it is seen from the analysis in section 6 that it is important to understand which constitutive relations that permit a linearly degenerate field.

[31] We consider the parabolic regularization term ∇2e merely as a mathematical technique used in order to obtain a well-posed problem and do not require it to have a physical interpretation. Though, it is noteworthy that it may arise as a consequence of a multiple scale analysis of a more detailed constitutive relation, for instance when temporal and/or spatial dispersion is taken into account. The dispersion can be modeled with a convolution, for instance d = χ1 * e + χ2 * e * e, where * denotes temporal and/or spatial convolution. Introducing a microscopic and a macroscopic time or space variable and performing a formal multiple scale expansion, it is found that the leading order term of the solution should satisfy ∇ × Je + ∂td(e) = δD2e, where D2 is a second-order differential operator in time and/or space. In the case of D2 = ∂t2, i.e., “temporal viscosity,” we note that even though we obtain exactly the same analysis for a traveling wave profile as for the term ∇2e used in this paper, this version of the Maxwell equations is noncausal, and very difficult to treat in more than one spatial dimension. A similar system of equations in one dimension is studied as a boundary value problem by Kristensson [1999], and the influence of the noncausality is found to be small when δ is small.


[32] The work reported in this paper is partially supported by a grant from the Swedish Research Council for Engineering Sciences and its support is gratefully acknowledged. This work was largely conducted during visits to the Department of Mathematics and Statistics of the University of Canterbury, Christchurch, New Zealand, and the Department of Mathematics and Computer Science of the University of Akron, Ohio. Their warm hospitality is most appreciated. The author also thanks the Royal Physiographical Society of Lund (Kungliga fysiografiska sällskapet i Lund) for a grant enabling these visits. The author thanks Gerhard Kristensson and Mats Gustafsson for many valuable discussions on this paper.