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Modal theory for the two-frequency mutual coherencefunction in random media: general theory and planewave solution: IIJasmin Oz a & Ehud Heyman aa Department of Electrical Engineering - Physical Electronics , Tel Aviv University , Tel Aviv,69978, IsraelPublished online: 19 Aug 2006.

To cite this article: Jasmin Oz & Ehud Heyman (1997) Modal theory for the two-frequency mutual coherencefunction in random media: general theory and plane wave solution: II, Waves in Random Media, 7:1, 95-106, DOI:10.1088/0959-7174/7/1/006

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Waves in Random Media7 (1997) 95–106. Printed in the UK PII: S0959-7171(97)73279-5

Modal theory for the two-frequency mutual coherencefunction in random media: general theory and plane wavesolution: II

Jasmin Oz and Ehud HeymanDepartment of Electrical Engineering – Physical Electronics, Tel Aviv University, Tel Aviv69978, Israel

Received 2 April 1996, in final form 29 July 1996

Abstract. In a previous publication (part I) it has been shown that for an arbitrary statisticallyisotropic and homogeneous medium the parabolic equation for the two-frequency mutualcoherence function can be separated and thereby expressed as a superposition of modes.A parameterization based on the longitudinal part of this representation has also beentreated. This paper explores the transverse structure and parameterization of the field solutionby employing dimensional, variational and the modified WKB procedures for solving theeigenfunction/eigenvalue problem. General expressions are derived first for a general structurefunction and then specialized for a power-law structure function with emphasis on quadratic andKolmogorov media.

1. Introduction

In the first part of this series of papers, [1], a new general solution of the two-frequencymutual coherence function0(ρ1, ρ2, z, k1, k2) in a statistically isotropic random media hasbeen presented. This solution involves a summation of modes whose excitation amplitudesare determined by the projection of the initial conditions (the sources) onto the modefunctions. A complete modal expansion theorem has therefore been established. This newprocedure reduces the parabolic partial differential equation for0 in any structure function,into a solution of an ordinary differential equation for the mode functions.

As an example of the new modal procedure, we started in [1] with a detailed analysis ofthe simplest source configuration, namely a plane-wave initial condition. Other canonicalsource configurations, namely, an initial point-source field and beam-wave field will beconsidered in [2, 3].

Specifically, it has been shown in [1] for the case of an initial plane wave0|z=0 = 1,that 0 has the form

01(ρd, z, k1, k2) =∑

n

An exp

(i

2Knz

)fn(ρd) (1)

whereρd = |ρ1−ρ2| and01 is conventionally related to0 via 01 = 0 exp[18(k1−k2)

2A(0)z]whereA(0)−A(ρd) ≡ D(ρd) is the structure function of the medium. The transverse square-integrable mode functionsfn(ρd) and the propagation coefficientsKn are solutions of theeigenfunction/eigenvalue problem

d2fn

dρ2d

+ 1

ρd

dfn

dρd− i

2k2

wkdD(ρd)fn = Knkwfn (2)

0959-7171/97/010095+12$19.50c© 1997 IOP Publishing Ltd 95

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96 J Oz and E Heyman

with boundary conditions(dfn/dρd)|ρd=0 = 0 and fn|ρd→∞ → 0. We also definedk−1

w ≡ k−11 − k−1

2 andkd ≡ k1 − k2.In [1] the emphasis has been placed on the following.

(i) Derivation of the general expansion theorem forany source configuration.(ii) Basic physical parameterization implied by the new modal solution for the case of an

initial plane-wave condition, analysing in detail the properties of the solution atρd = 0.As a special case, the analytical solution for the quadratic structure function in [2] hasbeen rederived.

The motivation in this paper is the following.

(i) To extend the analysis of [1] which emphasized the axial parameterization forρd = 0and to explore the transversal two-point behaviour of0.

(ii) To examine the available perturbation techniques for the transversal mode solutionsfn

(specifically, the variational and the modified WKB methods). Some numerical resultsimplied by these approximate solutions are presented and compared with exact numericalresults. The general theory will be demonstrated for the case of a power-law structurefunction for which the explicit results will be given.

In section 2 we define the physical parameters pertaining to the transversal behaviourof the modes, namely the lateral correlation width and the criteria for the negligibility ofthe finite scale effects. As a special case, in sections 3 and 4 we then treat a general power-law medium. Section 3 is concerned with the mathematical analysis of the eigenfunctionproblem. Both the eigenfunctions and eigenvalues possess characteristics which can bededuced by dimensional arguments. In section 3.1 we show that the eigenfunctions alldepend on the dimensionless coordinatey ≡ λρ

(2+ν)/2d . This fact can prove useful when

calculating01(ρd = 0, z) for media characterized by non-quadratic structure functions, asdemonstrated on the specific example of the Kolmogorov structure function. Analyticalexpressions for the eigenvalues and the eigenfunctions are found by both the variationalmethod (section 3.2) and the WKB method (section 3.3). In section 3.4 we present thenumerical results for the eigenvalues for power-law media. In section 4.1 we evaluate thetransversal parameters introduced qualitatively in section 2. In particular, our expressionfor the transversal correlation width in the narrow-band approximation coincides with thecorrelation distanceρ0(z) given in [4]. Results for the cases of the quadratic structurefunction and the Kolmogorov structure function are then presented in sections 5.1 and 5.2,respectively. In section 5.2 we also explore the conditions under which the structure functionof the Kolmogorov medium may be regarded as a purely power-law function. It turns outthat, in the narrow-band approximation, these conditions are identical to those given in [4].Section 6 contains the summary and conclusions.

2. Transversal parameterization ofΓ1

The parameters that characterize the evolution of01|ρd=0 (i.e. at equal transversal positions)along thez axis have already been explored in [1]. Here we shall continue to investigate theproperties that emerge from the transversal structure of the modes. Although there existsno analytical solution to (2) in the general case, there are nevertheless certain features thatcharacterize the field structure. We shall consider the following two characteristic quantities.

(i) Transversal correlation width. We define the correlation width as the maximumtransversal width of the modes whose longitudinal part exp(iKnz/2) is not exponentiallynegligible with respect to01 at z = 0.

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Modal theory for the two-frequency mutual coherence function: II 97

From (2), the transverse width of the mode function|fn(ρd)| is essentially determinedby the condition

1

2ik2

wkdD(ρd) = Im (−Knkw). (3)

For a monotonically increasing structure functionD(ρd), the higher-order modes aretherefore wider. The transversal correlation width is defined as the width of the highestmode whose longitudinal decay is not yet negligible, i.e. Im(Kn)z 6 1. Recalling thatIm (Kn) increases withn, it follows that the number of high-order modes that should beretained in the summation decreases withz and, consequently, the transversal correlationwidth also decreases withz.

(ii) Inner and outer scale effects. In general, the structure function is assumed to be knownonly within the regionlm < ρd < Lm. Under certain conditions, however, the inner andouter scale effects may be neglected. Roughly speaking, the transversal width of the relevantmodes atz, i.e. those for which| exp( 1

2iKnz)| is not exponentially negligible, is requiredto be wider thanlm but narrower thanLm. From (3), this implies that the eigenvalues ofthese modes should satisfy the condition1

2k2wkdD(lm) < Im (−Knkw) < 1

2k2wkdD(Lm) (see

figure 1).

Figure 1. Conditions for neglecting the outer and inner scale effects.

As discussed above, the modal width increases withn for a monotonically increasingstructure functionD(ρd). This implies that the inner scale effect could be ignored if thelowest-order mode satisfies

−kw Im (K0) >1

2k2

wkdD(lm) (4)

while the outer scale effect can be ignored if

−kw Im (Kn) <1

2k2

wkdD(Lm) (5)

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98 J Oz and E Heyman

where n is the highest mode still relevant in the expansion, i.e. for which we haveIm (Kn)z 6 1. Further details will be considered in section 4 in connection with thegeneral power-law medium.

3. Eigenfunctions and eigenvalues for power-law media

From this section onwards we shall treat quantitatively media characterized by a power-lawstructure function of the general formD(ρd) = Cρν

d for which analytical expressions canbe derived. Equation (2) for the eigenfunction becomes

d2fn

dρ2d

+ 1

ρd

dfn

dρd− λ2ρν

dfn = Knkwfn λ2 ≡ i

2k2

wkdC. (6)

We begin by using dimensional analysis to draw certain conclusions about the parametricbehaviour of the transversal eigenfunction in a power-law medium (section 3.1). The first-order mode, which is the dominant one, is found by the variational method (section 3.2).For the higher-order modes (section 3.3) we then use a WKB-type approach to obtaina uniform mapping of the eigenfunctions onto the Airy functions. This also defines anapproximate recursive relation for the eigenvalues. Although the techniques are general,we shall present them in the context of a power-law structure function. In section 3.4 wepresent the numerical results that support the general statements.

3.1. The parametric dependence of the eigenfunctions

Based on dimensional analysis, it has been found in [1] that the eigenvalues of (6) have thegeneral form

Knkw = −εnλ4/(2+ν) (7)

where εn are positive dimensionless numbers. These numbers will be determined,approximately, in sections 3.2 and 3.3 below, and numerically in section 3.4. In this sectionwe shall proceed further with the dimensional analysis in order to extract an approximateparametric dependence of the eigenfunctions.

Defining the complex dimensionless variabley ≡ λρ(2+ν)/2d and insertingKnkw from

(7), equation (6) may be recast into the standard form(2 + ν

2

)2 d2fn

dy2+

(2 + ν

2

)2 1

y

dfn

dy+ y−2ν/(2+ν)εnfn − fn = 0. (8)

Since (8) consists only of numerical coefficients, the eigenfunctionfn(ρd) depends onyonly, i.e. the eigenfunctions are representable as some functionsfn of the dimensionlessvariabley. It also follows from (8) that asy → 0 fn behaves likeJ0(

√εny

2/(2+ν)) whileasy → ∞ it behaves like exp[−2/(2 + ν)y].

For ν = 2 the well known normalized eigenfunctions of (8) arefn(y) =Ln(y) exp(−y/2) (see e.g. [5]). Regarding the problem mainly as a dimensional one, weexpect the eigenfunctions forν close to 2 to be approximately given by the eigenfunctionsof the ν = 2 case, in terms of the correctly dimensionalizedy. This fact can be used,for example, in the computation of the expansion coefficientsAn for the plane-wave initialconditions (01|z=0 = 1), which are calculated via (see equation (19) of [1])

An =∫

fn(y)y(2−ν)/(2+ν) dy. (9)

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Modal theory for the two-frequency mutual coherence function: II 99

Thus, using the above arguments,An for ν close to 2 could be approximately calculatedfrom (9) usingfn(y) ' Ln(y) exp(−y/2).

To substantiate this claim, we first calculated the first three expansion coefficientsA0,A1, A2 for the Kolmogorov caseν = 5/3 numerically, giving

A0 = 2.0766 A1 = −2.2416 A2 = 2.226 57. (10)

The calculation of the same coefficients usingfn ' Ln(λρ11/6d ) exp(− 1

2λρ11/6d ) and (9) yields

A0 = 2.0817 A1 = −2.3466 A2 = 2.325 (11)

which are indeed close to the results in (10).One should, however, be careful when usingLn(y) exp(−y/2) as the approximate

eigenfunctions for solution of01(ρd, z) for largeρd, because of the phase error accumulationin each mode. In particular, this approximated solution does not possess the correctasymptotic behaviour aty → 0.

3.2. Variational calculation ofε0 and the corresponding eigenfunction

Following the variational procedure outlined in [6], this involves choosing a trial functionwhich has the right asymptotic behaviour atρd = 0 and atρd → ∞. In view of theasymptotic behaviour of (8) at these limits and observing that the number of zeros offn isn, we are led to the following ansatz for the lowest-order eigenfunction

f var0 (ρd) = G exp

{−Bρ

(2+ν)/2d

}(12)

(compare with the discussion after (8)), whereB is the variational parameter. The factorG

is determined by the normalization condition∫ ∞

0 dρdρd[f var0 (ρd)]2 = 1, yielding

G2 = 2 + ν

2

(2B)4/(2+ν)

0(4/(2 + ν)). (13)

The next step is to evaluate the inner product(f var0 , Hνf

var0 ) where

Hν ≡ d2

dρ2d

+ 1

ρd

d

dρd− λ2ρν

d .

The minimum value of this inner product with respect to the variational parameterB

corresponds to the lowest eigenvalueK0kw for the approximated eigenfunction in (12).A straightforward calculation yields

(f var0 , Hνf

var0 ) = 1

0(4/(2 + ν))

[(2 + ν

4

)2

(2B)4/(2+ν) − (2B)−2ν/(2+ν)λ2

]. (14)

This expression has a minimum for

Bmin = λ√

2ν

2 + ν. (15)

Inserting this result into (12) and evaluating (14) yields the lowest eigenvalueK0kw. It hasthe form of (7) with the dimensionless modal parameter

ε0 =(

2 + ν

2

)2ν/(2+ν) [ 14(2ν)2/(2+ν) + (2ν)−ν/(2+ν)]

0(4/(2 + ν)). (16)

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100 J Oz and E Heyman

The variational approximation for the normalized eigenfunction becomes

f var0 (ρd) =

[(2 + ν

2

)(ν−2)/(ν+2)(λ

√2ν)4/(2+ν)

0(4/(2 + ν))

]1/2

exp

{−

√2ν

2 + νλρ

(2+ν)/2d

}. (17)

Finally, the first coefficient in the series for01 is found to be

A0 = 24/(2+ν)/G. (18)

3.3. WKB expressions for the higher eigenvalues and eigenfunctions

For n � 1 the WKB approximation becomes valid and (at least formal) expressions forf WKB

n (ρd) andεn can be given. It is convenient to define

αn(ρd) ≡ −Knkw − i

2k2

wkdD(ρd) (19)

and also to define the complex turning pointρtn via αn(ρtn ) = 0, i.e.

i

2k2

wkdD(ρtn ) = −Knkw. (20)

For the special case of a power-law structure function where12ik2

wkdD(ρd) = λ2ρνd we obtain

ρνtn = εnλ

−2ν/(2+ν) with argρtn = −π/2(2 + ν).The uniform asymptotic solution of (6) is found by mapping onto the Airy equation in

order to ensure a uniform asymptotic behaviour through the turning point. Using Olver’sprocedure (see e.g. [7, 8]), we obtain the following expression for the final result

f WKBn (ρd) ' Qn

√2

ρd

[ 32ψn(ρd)]1/6

[αn(ρd)]1/4Ai

[−

(3

2ψn(ρd)

)2/3]

(21)

whereQn is some constant to be determined. Here

ψn(ρd) =∫ ρtn

ρd

√αn(ρ) dρ (22)

is the phase integral normalized with respect toρtn . It is convenient to express it withrespect to the originρd = 0 as

ψn(ρd) = ψn(0) − ψn(ρd) ψn(ρd) =∫ ρd

0

√αn(ρ) dρ. (23)

Note that hereψn(0) is a constant and that the integration with respect toρ is along thereal positive axis.

Below, we shall define the Riemann sheets of the various complex functions in (21)for a power-law structure function whereD(ρd) = Cρν

d. They will be defined so that (21)reduces correctly to the known asymptotic limits atρd → ∞ andρd → 0.

Figure 2(a) schematizes the trajectory ofαn as a function ofρd, starting at−Knkw forρd = 0 (recall thatKn lies in the first quadrant andkw may be taken to be negative). Notingthat the integration forψn in (23) involves only real values ofρ, we therefore define thebranch cut of

√αn along the negative real axis so that the integrand is a continuous function

of ρ asρ varies from 0 to∞. Furthermore, as will be demonstrated in (27) below,ψn(0)

is positive. Consequently,ψn(ρd) behaves as schematized in figure 2(b). Thus, definingthe argument ofψn to lie between 0 and−2π , rendersψ2/3

n a continuous function for allreal ρd with −5π/6 < arg(ψ2/3

n ) < 0. The resulting trajectory of the argument of the Airyfunction in (22) as a function ofρd is depicted in figure 2(c). It follows that with these

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Modal theory for the two-frequency mutual coherence function: II 101

(a) (b)

(c)

Figure 2. The complex functions in (19) as functionsof the real parameterρd: (a) αn(ρd). (b) ψn(ρd). (c)−ψn(ρd)

2/3 (i.e. the argument of the Ai function in(21)). Shaded zone: exponentially decaying solution.Cross marks: zeros of the Airy function in theoscillatory regime.

definitions, the solution in (21) is a continuous function for 0< ρd < ∞. Furthermore, forsmall ρd, the argument of the Airy function is essentially negative, causing an oscillatorybehaviour in this region. For largeρd, on the other hand, the argument of the Airy functiontends asymptotically to the directionπ/6 (see figure 2(c)), ensuring an exponential decaythere.

The solution proposed in (21) is not regular asρd → 0. In this range, equation (6)tends to the limit

d2fn

dρ2d

+ 1

ρd

dfn

dρd= Knkwfn

with the solution

fn(ρd) = QnJ0(√

−Knkwρd) (24)

which is regular atρd = 0. HereQn is a constant whose relation toQn in (21) will bedetermined in (27).

Finally, the quantization of the constantKn is obtained by matching the solutions in (21)and (24) in the common region where they overlap. Using the large-argument approximationof J0, the solution in (24) for|√Knkwρd| > 1 reduces to

fn(ρd) ' Qn

(2

π√−Knkwρd

)1/2

cos(√

−Knkwρd − π/4)

. (25)

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102 J Oz and E Heyman

On the other hand, forρd not too close to the complex turning pointρtn we can use thelarge-order approximation for the Airy function in (21). Noting from figure 2(c) that in thisrange the argument of the Airy function is essentially negative, we obtain

f WKBn (ρd) = Qn

(2

π√−Knkwρd

)1/2

cos[ψn(0) − ψn(ρd) − π/4

]' Qn

(2

π√−Knkwρd

)1/2

cos[√

−Knkwρd + π/4 − ψn(0)]

(26)

where the second expression applies for smallρd, and we inserted from (23)ψn(ρd) '√−Knkwρd. The solutions in (25) and (26) can only be matched if

ψn(0) =∫ ρtn

0

√αn(ρ) dρ =

(n + 1

2

)π (27)

thus givingQn = (−1)nQn. Using (8), this condition may be brought to the form∫ ε1/νn

0

√εn − yν dy =

(n + 1

2

)π (28)

which can be evaluated analytically. The standard integral forn = 0 yields the result forε0

ε0 =[√

πν0(1/ν + 3/2)

0(1/ν)

]2ν/(2+ν)

(29)

while for highern we obtain the recursive relation

εn = ε0(2n + 1)2ν/(2+ν). (30)

This yields a good approximation even for the lower-ordern. The recursive relation forn > 1 may, in fact, be used with a better approximation forε0, say the variational one in(16), giving an excellent agreement with direct calculations. Note that (16), (29) and (30)are exact forν = 2.

3.4. Numerical calculation of the first few eigenvalues and eigenfunctions

In order to establish the validity of the above analytic parameterization, we compare thefirst few eigenvalues and eigenfunctions with direct numerical results obtained by applyingthe finite-difference method. In table 1 the calculated values forε0, ε1 andε2 are tabulatedfor ν = 1.5 to ν = 2 in steps of 0.1. The Kolmogorov case is given asν = 1.6667. Theaccuracy can be estimated by examining the result forν = 2 for which the exact values areknown. In table 2 we compare the values forε0, ε1 andε2 as obtained by the WKB methodwith those obtained numerically. As can be verified from the table, the relation given in(30) is indeed in excellent agreement for the higher eigenvalues. Moreover the value ofε0

calculated by the variational method agrees extremely well with the numerical value.

4. Transversal parameterization for power-law media

(i) Transversal correlation width. Within the uniform WKB model, the width of thentheigenfunction is essentially determined by the turning point (see (3)), yielding

1

2k2

wkdCρνd = Im (−Knkw). (31)

Since from (30)Kn+1/Kn = ((2n + 3)/(2n + 1))2ν/(2+ν), the width given in (31) increaseswith increasingn (see e.g. figure 1). Recalling that the longitudinal part of the mode

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Modal theory for the two-frequency mutual coherence function: II 103

Table 1.

ν ε0 ε1 ε2

1.5 1.844 4.8845 7.58131.6 1.9094 5.115 8.06491.6667 1.9254 5.266 8.3861.7 1.9333 5.3419 8.54961.8 1.9562 5.5651 9.03421.9 1.9784 5.7844 9.51762 1.9999 5.9996 9.9991

Table 2.

ν εnum0 εvar

0 εWKB0 εnum

1 εWKB1 εnum

2 εWKB2

1.5 1.844 1.8874 1.9081 4.8845 4.8929 7.5813 7.58091.6 1.9094 1.9113 1.9293 5.115 5.1228 8.0649 8.06691.6667 1.9254 1.9268 1.9425 5.266 5.2738 8.386 8.39011.7 1.9333 1.9344 1.9489 5.3419 5.3485 8.5496 8.55241.8 1.9562 1.9568 1.9672 5.5651 5.5699 9.0342 9.03691.9 1.9784 1.9787 1.9841 5.7844 5.7871 9.5176 9.51962 1.9999 2 2 5.9996 6 9.9991 10

amplitude exp( 12iKnz) decays withn we define the transversal correlation widthρ0(z) as

the width of the highest relevant mode, i.e. the mode for which| exp( 12iKnz)| 6 1. This

mode is therefore identified by

Im (Kn) ' 2/z. (32)

Substituting Im(−Knkw) ' −2kw/z into (31) and solving forρd, we obtain an expressionfor the correlation width

ρ0(z) =(

4

Ck1k2z

)1/ν

. (33)

This expression coincides with the expression given in [4] in the narrow-band approximationwherek1k2 ' k2. Equation (33) holds only as long asz 6 2/ Im (K0) for the relevant mode.Beyond this range it may be used as an upper bound.

(ii) Inner and outer scale effectsThe conditions under which the inner and outer scaleeffects may be neglected have already been discussed in section 2.2. For the present case ofa power-law medium withD(ρd) = Cρν

d, the criterion of equation (5) in section 2 impliesthat the outer scale effect may be neglected if the width of the highest relevant mode (i.e.the mode whose longitudinal part is not exponentially negligible) is smaller thanLm (seefigure 1). This leads to the requirement that the correlation widthρ0(z) should be muchsmaller thanLm, giving

4

Ck1k2zL−ν

m � 1. (34)

This equation may be considered as a lower-bound condition onz. (The z-dependence in(34) follows from the fact that the number of relevant modes depends onz.)

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104 J Oz and E Heyman

The inner scale effect may be neglected if the width of the zeroth-order mode is greaterthan lm (see equation (4) and figure 1). This condition yields

Im (K0) � 1

2Ck1k2l

νm. (35)

Using the dimensional expression forK0 extracted from (7), equation (35) now becomes

ε0

(1

2Ckdk

2w

)−ν/(2+ν)

k2w sin(π/(2 + ν)) � lνm. (36)

In this expression the positive parameterε0 can be estimated by the variational value in(16).

Alternatively, (35) may be viewed as a condition on the frequency separation andmust be weighted against the requirement for the frequencies to lie within the coherencebandwidth of the pulse, i.e. if Im(K0) 6 2/z. If we add this additional restriction to thecondition in (35) we obtain the upper-bound condition onz:

4

Ck1k2zl−νm � 1. (37)

5. Explicit results

5.1. Quadratic structure function

The quadratic structure function is given byD(ρd) = Cρ2d with C being a dimensional

constant.

(i) Transversal correlation width. From (33) we obtain

ρ0(z) = 2√Ck1k2z

. (38)

(ii) Inner and outer scale effects. Next we assume that the structure function is known tobe quadratic only in the rangeρd < Lm. From (34) we see that the finite outer scale effectis negligible if

z � 4

Ck1k2L2m

. (39)

(iii) Asymptotic behaviour. At large propagation distances only the first mode remains.From (16) and (17) the behaviour of01 then becomes

01 ' 2 exp

(i

2K0z − 1

2K0kwρ2

d

)K0 =

√2ikdC. (40)

5.2. Kolmogorov structure function

Here, the structure function is given byD(ρd) = Cρ5/3d whereC is a dimensional constant.

(i) Transversal correlation width. Using (33) we obtain

ρc(z) =(

4

Ck1k2z

)3/5

(41)

(ii) Inner and outer scale effects. Next we assume that the structure function is known tohave the Kolmogorov formD(ρd) = Cρ

5/3d only in the rangelm < ρd < Lm.

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Modal theory for the two-frequency mutual coherence function: II 105

From (34) it follows that the outer scale effect can be ignored for

z � 4

Ck1k2L5/3m

(42)

and the inner scale effect can be ignored if

1.9268

(1

2Ckdk

2w

)−5/11

k2w sin(3π/11) � l5/3

m (43)

where here we used the variational approximation forε0 from (16).The upper-bound condition onz becomes

z � 4

Ck1k2l5/3m

. (44)

In [4] the conditions for the structure function to be approximated asCρ5/3d are given

in the narrow-band approximation. By comparison we see that if we setk1k2 ' k2, (42)is identical to the lower-bound condition onz and (44) to the upper-bound condition. Forpropagation distances not in this range, finite outer and/or inner scales must be taken. Modeswith continuous eigenvalues can no longer be neglected in the expansion and have to betaken into account. However, since they are not confined transversally, they violate theparabolic assumption, so that the parabolic equations cannot be useda priori.

(iv) Asymptotic behaviour. Based on (16) and (17) we obtain the behaviour at largez

01 ' 2.13 exp

[1

2(C/2)6/11k

5/11d (k1k2)

1/11ei3π/11z − 3

11

√10

3

√iC

kdk1k2ρ

11/6d

]. (45)

6. Summary and conclusions

In this paper we have completed the discussion of the new modal solution for the plane-wave two-frequency coherence function which we began in a previous paper [1]. While[1] is mainly concerned with the formal theory, as well as its immediate implication forparameterization of the longitudinal phenomena and observables, the emphasis in this secondpart has been placed on the transversal structure of the solution.

The new modal solution is applicable toany transversally homogeneous and isotropicstructure function. It reduces the problem of solving the coherence function equation toa transversal eigenvalue problem. Once this solution is found forz = 0, it provides thesolution for all z. This greatly reduces the computational difficulties in calculating thecoherence function, in particular for largez.

The modal solution also provides a new parameterization of many phenomena andobservables. In particular, the large-z structure of the coherence function (i.e. the linearphase accumulation and exponential decay) is directly described by the dominant eigenvaluesK0 and eigenfunctions. Such behaviour has been reported previously for the Kolmogorovcase, using a direct numerical marching solution. Here, it has been established that thisbehaviour characterizesany medium. Furthermore, considering the error accumulationthat identifies any numerical marching scheme, the present solution provides a theoreticalverification of the numerical result.

The correlation width, the conditions under which finite scale effects are negligible andthe series convergence are all features which originate from the transversal behaviour of01.They were introduced in section 2.

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106 J Oz and E Heyman

A major part of this paper has been devoted to exploring several analytical techniquesfor obtaining approximate expressions for the eigenfunctions and eigenvalues in a mediumcharacterized by a general power-law structure function.

In section 3.1 it was shown that the eigenfunctions all depend on the dimensionlesscoordinatey = λρ

(2+ν)/2d . Consequently, certain statistical parameters in a general medium,

and in particular in a Kolmogorov medium, have been approximately determined by a properdimensionalization of the solution for a quadratic medium (see e.g. (4)–(5) and equation (43)of [5]). Next, in section 3.2 we used the variational method to determine the eigenfunctionand the eigenvalue of the first mode in a general power-law medium. This mode is themost important one since it dominates the behaviour of01 everywhere and, in particular,in the asymptotic regime where01 is described practically exclusively by this mode (see(39) and (45)). Then, in section 3.3 we found the WKB expression for the eigenfunctionsof the higher-order modes and a recursive approximation for the eigenvalues. Finally, insection 3.4 we presented a numerical accuracy check the alternative asymptotic solutions(see tables 1 and 2).

In section 4 the transversal parameterization, which was introduced in section 2 ina general manner, was analysed for power-law media based on the analytical results insection 3. The two-frequency correlation width was derived. In the narrow-band limit itreduces to the expression in equations (20)–(97) of [4]. The conditions under which finitescale effects could be neglected were stated. It has been shown that they are compatiblewith constraints on the Kolmogorov medium to be represented via a power-law structurefunction, as given in section 20-7 of [4].

Acknowledgments

This work is supported by the US–Israel Binational Science Foundation, Jerusalem, Israel,under grant no 92-00273. EH would also like to acknowledge partial support by the US AirForce Office for Scientific Research under grant no F49620-93-1-0093. The authors wouldlike to thank Dr Boris Gisin for performing the numerical calculations for the eigenvalues.

References

[1] Oz J and Heyman E 1997 Modal theory for the two-frequency mutual coherence function in random media:general theory and plane wave solution: IWaves Random Media7 79–93

[2] Oz J 1997 Modal theory for the two-frequency mutual coherence function in random media: point sourceWaves Random Media7 107–17

[3] Oz J and Heyman E 1997 Modal theory for the two-frequency mutual coherence function in random media:beam waveWaves Random Mediasubmitted

[4] Ishimaru A 1978Wave Propagation and Scattering in Random Mediavols 1–2 (New York: Academic)[5] Kamke E 1959Differentialgleichungenpart 1 (New York: Chelsea)[6] Morse P M and Feshbach H 1953Methods of Theoretical PhysicsPart II (New York: McGraw-Hill) ch 9.4[7] Olver F W J 1974Asymptotics and Special Functions(New York: Academic) ch 11[8] Felsen L B and Marcuvitz N 1973Radiation and Scattering of Waves(Englewood Cliffs, NJ: Prentice-Hall)

section 3.5c

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