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Keywords:

  • random medium;
  • stochastic functional approach;
  • isotropy;
  • random cylindrical wave;
  • asymptotic form;
  • localization

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stochastic Symmetries in 2D Random Medium
  5. 3. Polar Spectral Representation of Random Medium
  6. 4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium
  7. 5. Asymptotic Solution for Random Cylindrical Wave
  8. References

[1] Random wave field satisfying two-dimensional (2D) wave equation with a random medium is studied using the stochastic representation in the polar coordinate system. The random medium is assumed to be a homogeneous and isotropic Gaussian random field, and its polar spectral representation is effectively employed in the method of solution. The isotropy of a random field implies the invariance under stochastic rotational transformation Dϕ, the operator operating on both angular coordinate and probability parameter, and a random wave field with an eigenvalue eiMϕ under Dϕ, M being an integer, represents an angular mode of random cylindrical wave, and M = 0 corresponding to an isotropic mode; the situation being analogous to the case of cylindrical mode in 2D free space. For a concrete approximate solution, the random medium is assumed to be an omnidirectional random grating with an isotropic narrowband spectrum centered at 2k twice as much as the wave number k, which creates the Bragg scattering in the propagation of random cylindrical waves. The asymptotic form of the random cylindrical wave is obtained by means of the stochastic spectral representation of the random wave, as well as the random medium, and is shown to have an exponential factor equation image, where the average exponent coefficient γ is given in terms of the spectral density of random medium. This exponential behavior due to the index α in the amplitude describes the localized nature of an angular mode of random wave field generated by a source or by resonance. In the case of a narrowband grating with the variance σ2 and the spectral bandwidth kΔ, γ is concretely calculated to be proportional to equation image, which should be compared with the one-dimensional (1D) case where the corresponding exponential coefficient is proportional to the spectral height σ2/Δ.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stochastic Symmetries in 2D Random Medium
  5. 3. Polar Spectral Representation of Random Medium
  6. 4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium
  7. 5. Asymptotic Solution for Random Cylindrical Wave
  8. References

[2] Although a number of works have been made so far on the localization in one-dimensional (1D) random medium, there are only a few papers done on the localization in two-dimensional (2D) random medium [Kirsh et al., 1998; Combers et al., 1999] and even so those works are mostly directed to the mathematical eigenvalue problems and not to the properties or the propagation of random waves therein. The present author has shown that in 1D random medium there exists a plane wave solution as a representation of the translation operator group associated with the homogeneity of 1D random medium [Ogura, 1975; Ogura et al., 1979], which is a stochastic analogy of the Floquet theorem, and that the solution possesses an exponential decaying or growing factor indicating the presence of localization when the wave field is generated by a source or by a source-free resonance [Ogura et al., 1992]. The same exponential factor was obtained by various 1D theories.

[3] In this paper we deal with the random wave field in 2D random medium, where we have more degrees of freedom to introduce the symmetries in connection with the angular mode of random wave field [Ogura, 2001a, 2001b]. Let the wave equation in 2D space R2; r =(x, y)D = (r, θ)P in the Descartes or polar coordinates, be expressed

  • equation image

where ε(r, ω) denote a random medium, ω ∈ Ω denoting a sample or a realization of the random medium and Ω the space of samples where the probability is defined. The probability parameter ω in ψ(r, ω) implies that it is a stochastic functional of random medium as well as a function of rR2. The parameter ω is often deleted in the following for notational simplicity. We assume ε(r, ω) be a homogeneous and isotropic random field which implies that its probabilitistic properties are invariant under the translational and rotational group of motions in R2, which we refer to as the stochastic symmetries (section 2). In such random medium, we expect to have both plane wave and cylindrical wave in random medium as a result of translational and rotational symmmetry, in much the same way as a plane wave and a cylindrical wave in the free space [Vilenkin, 1968].

[4] The present paper intends to show that such a random cylindrical wave does exist as an angular mode as a result of rotational symmetry [Ogura, 2001a], and that the random wave equation can be treated in the polar coordinates using the polar spectral representation of a homogeneous and isotropic random medium [Ogura, 1968] combined with the stochastic functional approach [Ogura, 1975; Ogura and Takahashi, 1996]. Its asymptotic solution at large distance from the origin is obtained by means of the spectral representation, which shows again an exponential decaying or growing factor arising from the multiple Bragg scattering in the random medium. The random plane wave as a representation of the translation operator group based on homogeneity will be discussed in a succeeding paper, in which we obtain the same exponential factor as the random cylindrical wave in this paper, and derive a concrete random plane wave by means of the stochastic functional approach [Ogura, 2001b]. Once the plane wave solutions as well as the evanescent waves are obtained we will be able to construct a cylindrical wave by integration in much the same way as the Sommerfeld representation of Bessel function or cylindrical wave in the free space.

2. Stochastic Symmetries in 2D Random Medium

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stochastic Symmetries in 2D Random Medium
  5. 3. Polar Spectral Representation of Random Medium
  6. 4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium
  7. 5. Asymptotic Solution for Random Cylindrical Wave
  8. References

[5] To deduce possible forms of random wave field we pay attention to the homogeneity and isotropy of 2D random medium, which could create some stochastic symmetries in the random wave field.

2.1. Translations and Rotations in R2

[6] We first note the transformation groups in R2; translation ga and rotation gϕ such that gar = r + a, gϕr = (r, θ + ϕ), which forms an additive group respectively and together form the (noncommutative) Euclidean motion group. Plane waves and cylindrical waves in free space are the results of the representation of these groups.

2.2. Translations and Rotations in Ω

[7] 2D random medium ε(r, ω), (rR2, ω ∈ Ω, 〈 〉: average over Ω) is a homogeneous and isotropic random field. That the probability is invariant under translation and rotations in R2, which implies that there exist the measure preserving (point) transformations Ta, Tϕ on Ω such that ω [RIGHTWARDS ARROW] Ta ω, ω [RIGHTWARDS ARROW] Tϕω. In these notations, we express the homogeneity and the isotropy of the random medium respectively as

  • equation image

2.3. Translations and Rotations in R2 × Ω

[8] For a random field Ψ(r, ω) generated by ε(r, ω), i.e., a function on R2 × Ω, we introduce the two sets of transformations by

  • equation image

which satisfy the multiplicative relations similar to those of ga, gϕ and Ta, Tϕ;

  • equation image

In terms of these operator groups, we can express the homogeneity and the isotopy as the invariance under Da and Dϕ, respectively:

  • equation image

We further note that these operators commute with the Laplacian ∇2; that is, Da2 = ∇2Da, Dϕ2 = ∇2Dϕ, and also that they keep the wave equation (1) invariant. This is the very reason that these operators play important roles in studying the form of solution satisfying the random wave equation.

2.4. Representation of Dϕ and Angular Mode With Quantum Number M

[9] In what follows we deal with the stochastic symmetry with respect to the rotational operator group Dϕ. Consider a random wave field such that

  • equation image

which indicates a 1-D representation of the rotational operator Dϕ, with eiMϕ the eigenvalue and ψ(M)(r, ω) the eigenvector, M = 0 corresponding to isotropic random field in the sense of Dϕ invariance. Because the operator Dϕ keeps invariant the random wave equation, there exists a random wave solution satisfying (6), which we may regard as an angular mode with quantum number M. Such a mode of random wave field, which exhibits the symmetry under rotational transformation, can be generated if the source has the same symmetry under gϕ or Dϕ. In the case of nonrandom, free space, the stochastic angular mode is reduced to the cylindrical wave described by the Bessel function of M-th order.

3. Polar Spectral Representation of Random Medium

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stochastic Symmetries in 2D Random Medium
  5. 3. Polar Spectral Representation of Random Medium
  6. 4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium
  7. 5. Asymptotic Solution for Random Cylindrical Wave
  8. References

3.1. Correlation Function

[10] Let the random medium ε(r, ω), with mean 0 be a homogeneous and isotropic Gaussian random field, the correlation function of which depends only on the distance between two points and represented in the following form [Ogura, 1968]

  • equation image
  • equation image

where ∣G(λ)∣2, λ = ∣λ∣ denotes an isotropic spectral density in 2D (wave number) λ-space, and σ2 the variance of ε(r). The distance between the two points r = (r, θ), r′ = (r′, θ′) is given by equation image. In particular when ∣G(λ)∣2 = 1, ε(r) is reduced to a white noise field.

3.2. Polar Spectral Representation

[11] When a homogeneous random field has the spectral representation in the Fourier form, but when ε(r) is homogeneous and isotropic it can be represented in the polar form [Ogura, 1968].

  • equation image

where dBm(λ, ω) denotes the Gaussian random spectral measure satisfying the orthogonality relations such that

  • equation image

Since ε(r) is real valued we have equation image and the orthogonality equation image If we multiply the spectral representation (9) at two points r, r′ to calculate 〈ε(r)ε(r′)〉 using (10) and the addition theorem for the Bessel function, then we recover the spectral representation (7) for the correlation function.

3.3. Narrowband Random Medium

[12] For concrete calculation we assume the random medium have a narrowband isotropic spectrum ∣G(λ)∣2 centered at λ = 2k with the bandwidth δ ≡ kΔ (≪2k). For example, we assume a Gaussian spectrum such that

  • equation image

This is a specific model for an omnidirectional random grating, which produces the Bragg scattering of waves propagating in arbitrary direction.

4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stochastic Symmetries in 2D Random Medium
  5. 3. Polar Spectral Representation of Random Medium
  6. 4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium
  7. 5. Asymptotic Solution for Random Cylindrical Wave
  8. References

4.1. Wave Equation in the Polar Coordinates

[13] Let the random wave field ψ(r, ω), r = (r, θ)p satisfy the 2-dimensional (2D) wave equation (1) in the polar coordinates,

  • equation image

where ε(r, ω) is represented in the polar form (9).

[14] In this paper we look for the angular mode with M = 0, such that the random wave field is invariant under the rotation operator Dϕ, meaning that the random wave field is isotropic with respect to the origin. Such an isotropic random field can be generated when the source is Dϕ-invariant; e.g., circular symmetric source or a monopole located at the origin for the simplest example.

4.2. Isotropic Angular Mode

[15] In the first place, we look for such an isotropic random wave field satisfying the wave equation, assuming that ε(r, ω) is an omnidirectional random grating. An isotropic random wave field ψ(r, θ; ω) has the spectral representation such that

  • equation image

where ψ(r, ω) ≡ ψ(r, 0; ω) and by the isotropy or Dϕ-invariance we have

  • equation image

Substituting (13) into (12) and using (14) we have a set of wave equations for ψn(r, ω) coupled by εmn(r, ω);

  • equation image

For small roughness such that kσ ≪ 1, we represent ψm(r, ω) in terms of Bessel function as follows,

  • equation image

where Cm(kr, ω), Dm(kr, ω) are supposed to vary slowly with respect to kr, satisfy

  • equation image

and are orthogonal between different m's; equation image.

5. Asymptotic Solution for Random Cylindrical Wave

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stochastic Symmetries in 2D Random Medium
  5. 3. Polar Spectral Representation of Random Medium
  6. 4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium
  7. 5. Asymptotic Solution for Random Cylindrical Wave
  8. References

5.1. Asymptotic Form of Solution

[16] Here we look for the asymptotic form of the solution. In stead of solving (15) we search for the asymptotic form of

  • equation image

using the Meissel formula for the asymptotic form of the Hankel functions modified in such a way that

  • equation image

where the parabolic approximation equation image is employed in the equation. Then the asymptotic form for the random wave field can be written

  • equation image
  • equation image
  • equation image

where the sum limit N < kr, is automatically determined during the calculation. Because of (17), C(θ, kr), D(θ, kr) are isotropic (Dϕ-invariant), so that

  • equation image

5.2. Asymptotic Form of the Narrowband Random Medium

[17] We derive similar asymptotic form for the narrowband random medium from the polar spectral representation (9):

  • equation image
  • equation image
  • equation image

where P is also determined during the calculation.

5.3. Coupling Equation for C, D

[18] We substitute these asymptotic forms (20)(26) into the wave equation (12) to obtain the coupling equations for C(θ, kr), D(θ, kr), neglecting 2nd order r-derivatives and higher order asymptoric terms. We also note the relations

  • equation image
  • equation image

where the differentiation C′, D′ here implies

  • equation image
  • equation image

which are of importance in the following. We note that the differentiation with respect to kr looks like bypassing the exponential term, and that the angular differentiation disappears in the final equation. Thus, putting tkr, we obtain

  • equation image
  • equation image

Since C, D are Dϕ-invariant, isotropic random field, we solve the set of coupling equations putting θ = 0, and write e.g., C(θ, kr; ω) as C(t); we can recover θ by (21), (22) whenever necessary. From the coupling equation we easily obtain the conservation formula ∣C2 − ∣D2 = const.

5.4. Reduced Coupling Equation

[19] To solve the equation we put

  • equation image

which is tantamount to putting

  • equation image

because of the meaning of differentiation. Then the coupling equation is transformed into

  • equation image
  • equation image

[20] Generally speaking u(t), v(t) ≠ 0 so that, we may include u in the exponent function Γ putting Γ [RIGHTWARDS ARROW] Γ + u′/u, v [RIGHTWARDS ARROW] v/u, u [RIGHTWARDS ARROW] 1. We note here that u′/u as well as Γ is Dϕ-invariant, isotropic with respect to θ. Therefore, in solving the equation we may assume from the onset u = 1 without losing generality. Then we have the set of equations

  • equation image
  • equation image

[21] Taking the average of the first equation (37) we get the equation

  • equation image

Since Γ = iε+v/2 is a functional higher than the second order of ε whereas v is of the first order, we can approximate the second equation (38) by the one for a linear (first-order) functional v1. Thus replacing Γ by γ = 〈Γ〉 in (38) we get the equation for v1;

  • equation image

[22] Generally speaking γ is dependent on t, but we treat γ as a constant in the following calculation, because it is verified later that average exponent coefficient γ is a constant.

5.5. Approximate Solution for v1(t)

[23] Since v1 satisfying (40) is a first-order functional of ε, we can represent it in the following form;

  • equation image
  • equation image

where (41) is given in the form of (22) and (42) is the spectral representation written in the form of (26), where Fm(Λ) is the spectral kernel to be determined. Comparing these we obtain the spectral representation for Vm;

  • equation image

By definition, d/dt on v1(t) is reduced to differentiating Vm(t), so that we have

  • equation image

Thus substituting (41), (44) into (40) we finally obtain the spectral kernel

  • equation image

5.6. Calculation of the Exponent Coefficient

[24] Now using the approximate solution for v1 we calculate the exponent coefficient γ = 〈Γ〉 as follows. In view of the nondimensional quantities t = kr and γ, we can use the unit k = 1 to simplify the following calculations (or multiply all the integrals by k2).

  • equation image
  • equation image
  • equation image

In particular, when t = kr is large enough, we can replace the sum over m by the integral over ν = m/t, dν = 1/t:

  • equation image

which is now independent of t as we expected. These equations can be regarded as the dispersion equation satisfied by γ, which is analogous to the 1D theory [Ogura, 1975] except for the double integral. These expressions for the exponent coefficient of the random cylindrical wave quite agree with the one for the random plane wave calculated by a different formulation in terms of the representation of translation operator Da [Ogura, 2001b]. It can be shown that the exponent coefficient is the same with the other angular modes with quantum number M.

[25] For the calculation of the integral, when α is sufficiently small, the imaginary part of the integral over Λ can be evaluated by ±π times the residue at the pole Λ = −ν2/4 of the integrand, and the real part can be calculated by the principal-value integral indicated by equation image

  • equation image
  • equation image

where the sign ± corresponds to the sign of α.

5.7. Calculation of α, β for Narrowband Gaussian Spectrum

[26] For the narrowband Gaussian spectrum (11) we can concretely calculate (50), (51) using the integral representation for the gamma function as follows:

  • equation image
  • equation image

To calculate β we put

  • equation image

where the principal-value integral IP(ν) can be expanded in the following manner

  • equation image
  • equation image

making use of the expansion

  • equation image

to calculate the principal-value integrals. Hence integrating (54) termwise we finally get

  • equation image
  • equation image

5.8. Comparison With the 1D Case

[27] In the 1D homogeneous random medium with power spectrum ∣G(λ)∣2 the random plane wave has the corresponding exponent coefficient γ = α + iβ [Ogura, 1975];

  • equation image
  • equation image

where ∣G(2k)∣2 equals the spectral height at λ = 2k. If we apply these equations to a narrowband 1D random grating with Gaussian spectrum:

  • equation image

we have

  • equation image

That β = 0 is due to the fact that the symmetric spectrum is exactly centered at λ = 2k.

[28] The decay constant α, the well known 1D localization index, is proportional to the spectrum height at λ = 2k as given by (60), which is roughly equal to σ2/Δ for the narrowband spectrum. However, in 2D random medium, interaction takes place in the 2D spectral domain between outward and inward going cylindrical waves via narrowband spectrum. The interacting zones described by the parabolic approximation of propagating waves is dependent on the spectral width Δ, as shown by the 2D integral in (48) or (49). Roughly speaking, in the 2D case, γ is proportional to the spectral height σ2/Δ times equation image where the factor equation image comes from the overlapping area of the narrowband spectrum of the medium and that of the cylindrical wave.

5.9. Cylindrical Waves With Angular Quantum Number M

[29] So far we have studied the isotropic angular mode with M = 0. However, M-th angular mode satisfying the relation (6) can be similarly represented in the form

  • equation image

where Cm, Dm satisfy (17m), and its asymptotic form can be written

  • equation image
  • equation image
  • equation image

where C(M), D(M) also satisfy (6). In this manner we can establish the similar equations to solve them approximately. It is shown that the exponent function Γ and its average constant γ is the same with the isotropic mode, but we omit their details here with the other statistical properties of Γ, v(t), which will be reported in a succeeding paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stochastic Symmetries in 2D Random Medium
  5. 3. Polar Spectral Representation of Random Medium
  6. 4. Random Cylindrical Waves in Homogeneous and Isotropic Random Medium
  7. 5. Asymptotic Solution for Random Cylindrical Wave
  8. References
  • Combes, J. M., P. D. Hislop, and A. Tip, Band edge localization and the density of states for acoustic and electromagnetic waves in random media, Ann. Inst. Henri Poincare, Sect. A, 70, 381428, 1999.
  • Kirsch, W., P. Stollmann, and G. Stoltz, Anderson localization for random Schoedinger operators with long range interactions, Commun. Math. Phys., 195, 495507, 1998.
  • Ogura, H., Polar spectral representation of a homogeneous and isotropic random field, J. Phys. Soc. Jpn., 21, 13701380, 1968.
  • Ogura, H., Theory of waves in a homogeneous random medium, Phys. Rev. A, 11, 942956, 1975.
  • Ogura, H., Stochastic theory of wave localization in 2-dimensional random medium, paper presented at URSI International Symposium on EMT, Int. Union of Radio Sci., Victoria, Can., 2001a.
  • Ogura, H., Random plane waves in 2-dimensional homogeneous and isotropic random medium, Rep. EMT 01-119, Inst. of Electr. Eng., London, 2001b.
  • Ogura, H., and N. Takahashi, Scattering, radiation and propagation over two-dimensional random surface—Stochastic functional approach, in PIER 14, Progress in Electromagnetics Research, Electromagnetic Scattering by Rough Surfaces and Random Media, edited by M. Tateiba, and L. Tsang, pp. 89189, EMW Publishing, Cambridge, Mass., 1996.
  • Ogura, H., T. Aoki, and Y. Yoshida, Computer simulation of the wave propagation in a homogeneous random medium, Phys. Rev. A, 13, 349356, 1979.
  • Ogura, H., M. Kitano, N. Takahashi, T. Kawanishi, Stochastic theory of localized waves in random media, paper presented at URSI International Symposium on EMT, Int. Union of Radio Sci., Sydney, Australia, 1992.
  • Vilenkin, N. Y., Special Functions and Theory of Group Representations, Chaps. 2 and 4, AMS, Providence, R. I., 1968.