coherence function for cylindrically spreading waves in a random medium

5
Coherence function for cylindrically spreading waves in a random medium M. M. Fitelson General Electric Company, Advance Development Engineering, Heavy Military Electronic Systems, Syracuse, New York 13201 (Received 21 February 1974) The coherence function for a cylindrically spreading wave in a random medium is obtained as the exact solution of an equation derived from the wave equation when forward scatteringis dominant. The region of validity for this equationis obtainedunder the assumption that the index of refraction is a Gaussianfield. The coherence function for cylindrically spreading waves is comparedwith the corresponding solution for the plane-wavecase. Both the cylindrical and plane-wavesolutionsare shown to have essentially the same range of validity. Subject Classification:30.20, 30.40; 45.10; 20.15. INTRODUCTION In a previous paper,t the author presented a prelimi- nary analysis of the subject of the title. This paper pre- sents a more extensive treatment of the subject, includ- ing a derivation of the region of validity of the expression obtained for the coherence function. The random medi- um of interest here is weakly inhomogeneous, so that random effects will manifest themselves only after prop- agation over distances of many wavelengths. This work is largely motivated by a long-range goal to understand propagation of sound over great distances in the ocean, where turbulence and thermal gradients pro- duce random fluctuations of the index of refraction. In many such cases, there will be a strong path that begins as a spherical wave, but which after many wavelengths of propagation evolves into an approximately parallel ray bundle in the vertical plane with cylindrical spreading in a horizontal plane. To capture the essence of this type of propagation with- in the framework of a tractable model, refraction is ig- nored and a cylindrically spreading wave in a statistical- ly homogeneous and isotropic medium is considered. This model is compared with the corresponding solu- tion for plane-wave propagation. The plane-wave and cylindrically spreading-wave solutions are shown to have the same region of validity. The approach taken for obtaining the region of validity for the cylindrical wave solution is an outgrowth of Tatarskii's approach to the plane-wave case.•' The coherence function is derived in the next section. I. THE COHERENCE FUNCTION Since this report is concerned with situations for which random effects do not manifest themselves until propa- gation overy many wavelengths takes place, let it be as- sumed that the cylindrical wave is generated by a cylin- drical, monochromatic source whose radius • satisfies k( >>1 . (1) The source is radiating into a statistically homogeneous and isotropic medium. The wave equation in cylindrical coordinates is ap•'p• p•' +'az _ (z + •)• a•'• (2) The index of refraction fluctuation g is much smaller than unity, so that it satisfies and is assumed to have zero mean. For simplicity, as- sume that •[•(p, z, ½)•(p', z, = g•exp [-(9 • +9 '•- 299'cos(½- ½')+(z- The parameter a is the correlation distance of the index of refraction. Note that this form of the index of refrac- tion correlation function is chosen for simplicity only, since the mathematical techniques developed below can be applied to any homogeneous isotropic correlation function. In view of the foregoing assumptions, ½ is of the form ½ =½o•)e *•tb(p, ½,z) • ½0(P) ei•t b(p, s), (5) where s represents the pair (½,z). The term is the solution to the wave equation when g = 0, so that ½0(0) satisfies d•½ø(P) + + •½0(P) = 0 (6) do do ' and b(9, s) satisfies the boundary condition 5(•, s)= •. (•) Substituting Eqs. 5 and 6 into Eq. 2, yields +2 0 •+•0•+•0•+2•F•0a=0. (8) Since ok>> 1, by h•othesis Substituting Eq. 0 into Eq. 8 gives 53 'J.Acoust. Soc. Am., Vol. 56, No. 1, July1974 Copyright ¸ 1974 bytheAcoustical Society of America 53 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.91.169.193 On: Sat, 22 Nov 2014 08:22:32

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Page 1: Coherence function for cylindrically spreading waves in a random medium

Coherence function for cylindrically spreading waves in a random medium

M. M. Fitelson

General Electric Company, Advance Development Engineering, Heavy Military Electronic Systems, Syracuse, New York 13201

(Received 21 February 1974)

The coherence function for a cylindrically spreading wave in a random medium is obtained as the exact solution of an equation derived from the wave equation when forward scattering is dominant. The region of validity for this equation is obtained under the assumption that the index of refraction is a Gaussian field. The coherence function for cylindrically spreading waves is compared with the corresponding solution for the plane-wave case. Both the cylindrical and plane-wave solutions are shown to have essentially the same range of validity.

Subject Classification: 30.20, 30.40; 45.10; 20.15.

INTRODUCTION

In a previous paper, t the author presented a prelimi- nary analysis of the subject of the title. This paper pre- sents a more extensive treatment of the subject, includ- ing a derivation of the region of validity of the expression obtained for the coherence function. The random medi-

um of interest here is weakly inhomogeneous, so that random effects will manifest themselves only after prop- agation over distances of many wavelengths.

This work is largely motivated by a long-range goal to understand propagation of sound over great distances in the ocean, where turbulence and thermal gradients pro- duce random fluctuations of the index of refraction. In

many such cases, there will be a strong path that begins as a spherical wave, but which after many wavelengths of propagation evolves into an approximately parallel ray bundle in the vertical plane with cylindrical spreading in a horizontal plane.

To capture the essence of this type of propagation with- in the framework of a tractable model, refraction is ig- nored and a cylindrically spreading wave in a statistical- ly homogeneous and isotropic medium is considered.

This model is compared with the corresponding solu- tion for plane-wave propagation. The plane-wave and cylindrically spreading-wave solutions are shown to have the same region of validity.

The approach taken for obtaining the region of validity for the cylindrical wave solution is an outgrowth of Tatarskii's approach to the plane-wave case. •'

The coherence function is derived in the next section.

I. THE COHERENCE FUNCTION

Since this report is concerned with situations for which random effects do not manifest themselves until propa- gation overy many wavelengths takes place, let it be as- sumed that the cylindrical wave is generated by a cylin- drical, monochromatic source whose radius • satisfies

k( >> 1 . (1)

The source is radiating into a statistically homogeneous

and isotropic medium. The wave equation in cylindrical coordinates is

ap•' p • p•' • +'az •

_ (z + •)• a•'• (2)

The index of refraction fluctuation g is much smaller than unity, so that it satisfies

and is assumed to have zero mean. For simplicity, as- sume that

•[•(p, z, ½)•(p', z,

= g• exp [-(9 • +9 '•- 299'cos(½- ½')+(z-

The parameter a is the correlation distance of the index of refraction. Note that this form of the index of refrac-

tion correlation function is chosen for simplicity only, since the mathematical techniques developed below can be applied to any homogeneous isotropic correlation function.

In view of the foregoing assumptions, ½ is of the form

½ = ½o•)e *•t b(p, ½, z) • ½0(P) ei•t b(p, s), (5)

where s represents the pair (½, z). The term is the solution to the wave equation when g = 0, so that ½0(0) satisfies

d•½ø(P) + + •½0(P) = 0 (6) do • do '

and b(9, s) satisfies the boundary condition

5(•, s)= •. (•)

Substituting Eqs. 5 and 6 into Eq. 2, yields

+2 0 •+•0•+•0•+2•F•0a=0. (8) Since ok>> 1, by h•othesis

Substituting Eq. 0 into Eq. 8 gives

53 'J. Acoust. Soc. Am., Vol. 56, No. 1, July 1974 Copyright ¸ 1974 by the Acoustical Society of America 53 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.91.169.193 On: Sat, 22 Nov 2014 08:22:32

Page 2: Coherence function for cylindrically spreading waves in a random medium

54 M.M. Fitelson' Cylindrically spreading waves 54

a•'b(p, s t) 2ik ab(p, s•) + +At(p)b(p st) 8p 8p •' '

+2/z(p, sz)k•'b(p, sz)=0,

and the conjugate equation

•"•*(p, s•) •,.(p)•*(o, - 2ik Ob*(p, s•) + + 8p 8p •'

(•o)

where

+2/z(p, sz)k•'b*(p, s•.):0, (11)

i 0 2 •)2

",(p)--• • +•,•.' •: •, •, (12)

b(p, s•) -= b(p, •, z•).

The terms OZb/Op z and OZb*/Op z contribute backscattering to the solution of Eq. 10 and Eq. 11. They will • disre- garded here, and their effect will be evaluated in the ne• section.

The coherence function is defined by

r(p, s•, s•.):E[5(O, s•)5*(p, s•.)]. (13)

Before proceeding to calculate F, it will be convenient to explore certain pertinent properties of b(p, s) and •(p,s).

First, consider the covariance function for /z in Eq. 4, which may be written as

c(o- p', s- s'): •[•(p, s)•(o, s')]

= •" exp [- (p• - p•.cos•)•' ]. , 4'--(•- •., (14)

,E•Zl- Z2 ß

In an expression which involves an integral with the right-hand side of Eq. 14, the following simplification can be made. Since this paper is concerned with long- range propagation, p• and/or p•. will extend to values much larger than a, and exp[- (Pz - P2COSd?)2/a2] will be- have like a sharply peaked function compared to the scale of pz and/or p•., so that one can make the approxi- mation

exp [- (p• - p•.cos • y'] •-• j = V•- a6 (p z - p•.cos qb ) . (15) Then,

C(p - p', s- s') -•4• 1• •' aa(p• - p•.cos•b)

(16)

Furthermore, in long-range propagation the angular separation •b of the points being correlated may be small, while the horizontal separation p l •bl is much larger than a. Since the coherence function will drop to some as- ymptotic value for separations much larger than a, it is su/ficient to restrict oneself to small angular separa- tions. Then Eq. 16 becomes

C(p p' ' •2 P2)exp(_ P••') ( z•') - ,s-s)-•• a6(p•- -- .exp -• . (17)

The approximations in Eqs. 16 and 17 are similar to one used by Tatarskii for the plane-wave case.

In Appendix A, it is shown that b(p, s) is a functional ß of /z built up from values of /z(p t s) with ½ <pt < P. Fur-

thermore, from the form of Eq. A3, it is seen that since /z(p, s) is statistically homogeneous and isotropic, b(p,s) is statistically invariant with respect to translation in qb and z. Thus,

r(p, s•, s•.) = r(p, •bz - •b•., z• - z•.) . (18)

Now, return to the calculation of I'o Noting that •'b/ 8p•' and •'b*/•p •' have been disregarded, multiplying Eq. 10 by b*(p, s•.), Eqo 11 by b*(p, s•), and subtracting Eqo 11 from Eq. 10, yield

2ik(O/Op)[b(o, sz)b*(o, sz)] + [A,(p)- Az(p)][b(p,s,)b*(p, sz) ]

+2k•'[•(o, s•)- •(p, s•.)]a(o, sOb*(O, s•.) :0. (•9)

Equation 19 may be written in the form

exp [ik•(p; sz, s2)]' 2ik(O/Op){exp(- ik•[p; sz, s2)]

x b(p, st)b*(p, s2)} = [Az(p) -- At(p)]b(p , st)b*(p, sz), (20) where

p ß (p; s•, s•.)-= [•(p', s•)- •(p', s•.)]do'=,•- %, (2•)

4•' ' ' 2 •-- /z(p, s•)dp , i =1, .

Multiplying both sides of Eq. 20 by exp(-ik•), integrat- ing over p, and multiplying both sides of the resulting equations by exp(ik•)yield

2ikb(p, sz)b*(p, s•.)

= 2ik exp[ik•(p; sz, s•.)]

+ f,• dptexp{ikf•, • [/z(p tt, s,)- lz(p tt, sz)]dp t•} x [Az(p t) - Az(pt)]b(p t, sz)b*(p t, s2) ß (22)

The boundary condition

b(½, sz)= b*(½, s•.)= 1 (23)

was used in obtaining Eqo 22ø

Before proceeding further, it will be assumed that since p>> a, f• •(p, s)dp t is Gaussian, either because /•(p, s) satisfies an appropriate central limit theorem or because /.L(p, S) iS a Gaussian field. Then, since /.L(p, S) has a sharply peaked correlation function and b(p t, s) is built up from values of /•(ptt s) only for ½ <ptt < pt • [fL(p" Sz)-- •(p" s•.)]dp" is statistically independent of b(p', s•) and b*(p', s•.).

In view of the foregoing, taking expectation values of both sides of Eq. 22 yields

2ikF(p, s•, s z)

= 2ikE(exp[ik$(p, s•, s•.)]

+ f•o dp'E(exp{ikf•, • [tz(p", s•)- g(p", s•)]dp"}) x{[•.(p') - •,(p')]r(p, s,, s•.)}. (24)

Since I' is a function only of p, qbz- qb•. and z•- z•., one has

J. Acoust. Soc. Am., Vol. 56, No. 1, July 1974 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.91.169.193 On: Sat, 22 Nov 2014 08:22:32

Page 3: Coherence function for cylindrically spreading waves in a random medium

55 M.M. Fitelson: Cylindrically spreading waves 55

[A,.(p') - At(p')]I'(p' , st, s,.) = O, (25) so that

r(p, st, s,.)=E{exp[ik•(p; st, s•)]} . (26)

Taking note of Eq. 21 and the normality of •t and •, Eq. 26 becomes

r(p, s,, sz)=e• •(E• +E•- 2E•,•z . (27) Assuming small angular separations and usi• Eq. 17 yields

E$• = E• • • a fø f' 6(pt - Oz)dO,dOz =• a(p- ()•• ap, p>> (, p>> a, (28)

and

x exp (- P•)dp,dpz . exp( - zZ/a z) • [erf(p½/a)] exp(- z•/a•),

O >>(, O >>a, I½l<<z, where

(30)

Therefore, the coherence function is given by

P(P, s,, s=) = exp (- • k =a

x z-• (o• j'exp(-zVa •) , o>>a, o>>•, [½l<<z- (3z)

In the ne• section the region of validity of the fo•ard- sca•ering approximation is c•culated.

II. THE REGION OF VALIDITY OF THE FORWARD- SCATTERING APPROXIMATION

It is desired to estimate the importance of the term 8•'b/Sp •' in Eqo 10o Let the solution to Eq. 10, with 82b/ 8p •' included, be denoted by B(p, s), so that

2ik OB(p, s[) OZB(p, s) +A(p)B(p, s)+2kZg(p, s)B(p, s)=O. Op +' Op •' (32)

Suppose Eqo 32 were to be solved by successive approxi- mation. Then B(p, s) is expanded in a perturbation se- ries,

B(p, s)=b(p, s)+b,(p, s)+... . (33)

Now, since b(p, s) is the solution to Eq. 10 with OZb/ Op •' absent, it represents the zeroth-order solution, while b•(p, s) is the first-order correction, so that

2ikb(P, s•) +A•(p)b(p, s•)+2k•'l•(p, s•)b(p, s•)=0 (34a) •p ,

- 2ik Ob•*(p, %) + Az(p)b•.(p ' sz) + 2kg(p, sz)b•*(p, sz) 8p

=_ ½b*(p, s•) 8p*' '

Note that

r• (p, s•, s,)

=•[s(o, s,)s*(o, s,)]

= r(o, s•, s,) + r•(o, s•, s,) + r•*(o, s,, s•) +--.

= r(o, s•, s,) + •(o, s•, s,) +-o- , where

Let

r(o, s,, s,)--•[v(o, s,)V*(o, s,)],

r,(o, s,, s,)= •[v(o, s,)V•*(o, s,)],

•(o, s•, s,) = r•(o, s•, s,) + rl•(o, s,, s•).

(34b)

(35)

(36)

R=•fr . (37) The region of validity of the forward-scattering approxi- mation is determined by the conditions under which

I nl << x. Multiplying Eq. 34a by b•*(p, s•.), Eq. 34b by b(p, st), and performing manipulations much like those which led to Eq. 22 yields

2ikb(p, s•)b•'(p, sz)

L (,L' ) = dp'exp k , [/.t(p", st)- ß

x [a,(o') a,(p')] b(p, s,)b, (o, s,)

+ b(p' s•) OZb*(P" st) } (39) where the boundary condition

b•(•, s) = o (40) was used in obtaining Eqo 39.

Before proceeding further, it is necessary to assume that g(p, s) is a Gaussian field. Then it can be shown 4 that, if A(g) is a functional of

z[•(o, ½, z)•(•)]

= f f f dp'd½'dz'E[g(O, ½, z)/z(p',

x E{[6A(g)]/[6g(p', ½', z')]}, (41)

where the variational derivative 6A/61• is defined as fol- lOWSo Let 6/•(p, ½, z) be a variation of g which is zero, except for a small neighborhood A(p, ½, z) about the point (p, ½, z). Then,

•,4(•) { [/•(• +•)-A(•)] } 6 g (p, ½, z) = lim . (42) •-o Ia 6Ix(p, ½, z)dpdCdz

For a discussion of the variational derivative, see Refo 5ø

Since b(p, s) is a functional of !.t(p t, s) only for ß _< p• _< p, one has by the definition of the variational deriva- tive

J. Acoust. Soc. Am., Vol. 56, No. 1, JulY, 1974

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.91.169.193 On: Sat, 22 Nov 2014 08:22:32

Page 4: Coherence function for cylindrically spreading waves in a random medium

56 M.M. Fitelson: Cylindrically spreading waves 56

•b(p, s) p, 6p.(p', s) =0' > p' (43) Taking the expectation value of both sides of Eq. 39, us- ing EqSo 41 and 43, and the sharply peaked nature of the covariance function of g, and a great deal of manipula- tion yield

/)H •1 = •Ai(p)• (44) •P • ' where

•(•, •, z)

• • •a • - •. [ (•/a) J' e•(- zVa •) , a• (•/•) ß •, (4•)

Z•Z1 -- Z 2 ß

The interested reader is referred to Rei. 6 for the de-

tails of the derivation of Eq. 44. Reference 6 is avail- able from the author (MMF) upon request.

Fu•her computation a results in the inequality

I•1• I • I/r • • (g•)ak4a + •(g•bP) • ß (46) Thus• if

3 (•p)S•a << 1 (47a) and

•(•=kp) • << 1 , (47b)

the forward-scattering approximation is valid. In the next section, a comparison between the cylindrical and plane-wave cases will be made.

III. COMPARISON OF THE CYLINDRICAL AND

PLANE-WAVE CASES

To compare the cylindrical-wave case with the plane- wave case, assume that the two points being correlated lie in the same horizontal plane (z•- z2 = 0). Let •b be the angle between the points. Then the cylindrical-wave coherence function (see Eq. 27) is given by

- •, (•-755 J '

d = pdp . (48)

For the plane-wave case, one ignores cylindrical spreading and assumes a collimated plane wave is prop- agating through the medium. The coherence œunction for plane waves has been computed by Tatarskii v and Huf- nagel and Stanley. 8 Adapting their result to the index of refraction correlation function assumed above (see Eq. 4), one obtains

r•(p, d)= exp(- • k•'oa[1 - exp(d•'/a•')]}. (49) Let

6 --- • k•'pa, x---d/a . (50)

In Figs. 1 and 2, I' is plotted against x for 6---0.2, 1, and 3. Note that the difference between Fc and I'o be- comes more pronounced as /5 grows.

B2110

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.4 0.8 1.2 1.6 2.0 2 4 2 8 3.2 3.6 4.0 4.4 x

FIG. 1. Cylindrical vs plane-wave coherence function (5= 0.2, 1).

J. Acoust. Soc. Am., Vol. 56, No. 1, July 1974 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.91.169.193 On: Sat, 22 Nov 2014 08:22:32

Page 5: Coherence function for cylindrically spreading waves in a random medium

57 M.M. Fitelson' Cylindrically spreading waves 57

B2111

1.0(

0.9

0.8

0.7

0.6

0.5

0.4

0.3

,

0.2

0.1

CYLINDRICAL WAVE O exp - • (1 - %/7r/2 erf (x)/x) I I '

k PLANE WAVE m exp-• (1-exp(-x2))

5=3

--

,.} • •. n .--, n -• n rl • 0 i i I I I I I I I i I

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4

FIG. 2. Cylindrical vs plane-wave coherence function (6= 3).

Tatarskil •' has obtained an expression for the first- order correction to I',. If one uses the foregoing cor- relation function in his formula, the following conditions for the validity of the plane-wave forward-scattering equation are

3 ' (/•'0)3 k4a << 1 , (51) •(•2 kp)z << 1 . (52)

These conditions are the same as the conditions for the

cylindrical-wave case.

APPENDIX A: THE DEPENDENCE OF b(p,s) ON p.

Upon removing 8•'5/8p •' from Eq. 10,

2ik[Sb(p, s)/Op] + A(p)b(p, s)+ 2kZg(p, s)b(p, s) = 0, (A1)

with the boundary condition

b((, s) = 1 . (A2)

Equation A1 has the formal solution

b(o, s)=exp A(p')dp'+ik g(p', s)dp ß (A3)

The right-hand side of Eq. A3 is to be interpreted as an operator operating to the right on the constant 1. H one writes

= i}f, ø s)ap', then exp(A +B) is evaluated via the expansion

(A4)

exp(A + B)= (,4 + B)- 1 + (A •' + AB + BA + B •') 2!

=B+ ! .1+.-- . (A5)

Note that A and B do not commute, so that Eq. A5 is more difficult to work with than might be apparentø In any case, Eqo A3 indicates that b(p, s) is determined only by values of g(p• s) for ½ < p• , <P.

1M. M. Fitelson, "Cylindrically Spreading Waves in a Random Medium," Gert. Elect. ADE Tech. Memo No. 105 (Oct. 1973).

2V. I. Tatarskii, "The Effects of the Turbulent Atmosphere on Wave Propagation," Israel Program for Scientific Transla- tion (1971). (Available in U.S. through NTIS pp. 405-410).

3Ref. 2, pp. 378, 389--401. 4Ref. 2, pp. 378-379. 5Ref. 2, pp. 441-448. 6M. M. Fitelson, "The Coherence Function for Cylindrically

Spreading Waves in a Random Medium," (Sec. II), Gen. Electr. TIS No. R74EMH1 (Feb. 1974).

•See Ref. 2, pp. 381-382, 414. 8R. E. Hufnagel and N. R. Stanley, "Modulation Transfer Func-

tion Associated with Image Transmission Through Turbulent Media," J. Opt. Soc. Am. 54, 52-61 (1964).

J. Acoust. Soc. Am., Vol. 56, No. 1, July 1974 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.91.169.193 On: Sat, 22 Nov 2014 08:22:32