Morris Kline. # Wave propagation in variable media. Final report under Contract AF 19(604)-3495 online

. **(page 1 of 4)**

Online Library → Morris Kline → Wave propagation in variable media. Final report under Contract AF 19(604)-3495 → online text (page 1 of 4)

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NEW YORK UNIVERSITY

JNSTITUTE OF MATHEMATICAL SCIENCES

L1BRA. or some similar

unproved assumption. The reason for making such assijmptlons is to obtain

determinate equations for . Other examples of dishonest methods will be

described later. In all cases probability is introduced before u is deter-

mined and an \mproved assumption is made about some statistical property of

the random wave motion. The assimiption usually simplifies the problem so

that It becomes solvable.

Dishonest methods have the advantage over honest ones that they simplify

the problem to be solved. As a consequence, a problem vhich can be solved

honestly only by a perturbation method might be solved dishonestly without

the use of a perturbation expansion. Then the dishonest solution would be

applicable for all values of the relevant parameter while the honest solution

would be valid only for small values of it. For this and similar reasons,

many of the significant and non-trivial results in the theory of wave propa-

gation in random media have been obtained by dishonest methods. Many of

these results have compared well with experiment. Thus one of the important

mathematical problems in this field is to justify the dishonest methods by

showing that their results, in some sense, are approximations to honest solu-

tions. A clear vmderstanding of the circumstances in which this is the case

would permit the introduction of many more useful dishonest procedures.

Most of the work on this subject has been done since 19^5^ having been

stimulated by practical problems of radio wave propagation through the atmos-

phere and ionosphere, sound wave propagation in the ocean and the atmosphere,

light transmission through the atmosphere, etc. The recent book by L.A.

Chernov' - ' contains a rathe'r complete bibliography and an understandable

account of the present state of this subject. Additional material is con-

[21

tained in the book by V.I. Tatarski â€¢- -^ . The related subject of random wave

motion in a non-random medium is surveyed by M. Bom and E. Wolf '- â– * -^ .

Some other work is contained in the symposium volumes edited by W.C. Hoffman' - '

and Z. KopalJ- J . Because of the complete bibliographies in these books, we

shall give relatively few references.

Physicists have studied extensively the propagation of waves through

random collections of discrete scatterers. A clear formulation of a prob-

lem of this type, together vlth a new dishonest method for treating it, was

given by L.L. Foldy^ -1 . This method wr.s generalized to other problems by

M. Lax'- -' , who has also reviewed much of the previous work in this field.

Many problems have been treated by this method and others by V. Twersky'- -' .

[91

Recently, J. Bazer'- -' proved that for a one-dimensional scattering problem,

this method does yield the correct result. I. Kay and R.A. Silverman'- -'

investigated the accuracy of another method, the single scattering approxi-

mation, by determining the extent to which randomness reduces the importance

of multiple scattering.

The next section describes how wave propagation in a random medium

applies to the problem of the twixikling of a star. In Section J we consl-.

der light rays in a slightly inhomogeneous medium, and in Section ^ we apply

these results to a random medium obtaining some new results. We then compare

these honest results with corresponding dishonest ones in Section 5, where

we treat a light ray in a random medium as a Mai^off process. In Sections

6 and 7 we use the results of Sections 3 sixd k to determine the phase and

amplitude fluctuations of a wave in a random medium. Our results are exten-

sions of previously known ones. This completes our discussion of geometri-

cal optics in a random medium . In the final section we present a brief new

treatment of the reduced wave equation in a random medium. This section is

independent of the preceding ones.

2. An application

Before Illustrating the techniques used to analyze wave propagation in

random media, we shall describe a physical phenomenon in which such propaga-

tion plays a role. It is the scintillation or twinkling of a star. The most

- 6

appropriate theory of vave propagation to describe this phenomenon is geo-

metrical optics. According to this theory^ light travels along certain

straight or curved paths called rays. These rays are determined by ordinary

differential equations in which appears the index of refraction n(x, t), a

scalar function vhlch characterizes the transmission medium. This medium is

the earth's atmosphere in the present case. If n=l the rays are straight lines

which emanate from the star in all directions. One of them (really a narrow

beam) enters the eye of an observer who is viewing the star. The direction

from which the ray enters the eye is the apparent direction of the star.

Actually n(x^t) differs from unity by a small amount. As a consequence, the

rays deviate slightly from straight lines and enter the observer's eye from

slightly different directions at successive Instants of time. Therefore the

star appears to be moving about its mean position. Its apparent Intensity

also fluctuates.

We would like to calculate the apparent direction and intensity as func-

tions of time. To do so it would be necessary to know the index n(x,t). This

is practically impossible becatise the variation of n(x_,t) with time and posi-

tion results from the trubulent motion of the atmosphere. Consequently analy-

sis would appear to be impossible. In the face of this difficulty we treat

the atmosphere as a random medium. The random medivmi must be so chosen that

its Important statistical properties correspond to measurable properties of

the atmosphere. In tMs way we give up the possibility of calculating the

apparent direction and intensity at any particular time. Instead we can calcu-

late statistical properties of the apparent direction and intensity which may

be related to the actual temporal distribution of apparent directions and in-

tensities. We shall consider this example further in the next section. In

doing so we shall make use of the fact that a ray traverses the atmosphere so

7 -

qiilckly that the index does not change significantly during the traversal.

Consequently we may assvune that the index is independent of time in deter-

mining the rays.

5. Light rays in a slightly inhomogeneous medium

Let n(x, e) denote the index of refraction of a random medium, vhlch may

be written as

n(x,e) = 1 + e |a(x) . (l)

The quantity â‚¬ measures the deviation of the index from unity. The index also

depends upon a parameter co which we shall not write explicitly. We wish to

determine the ray x(s,â‚¬) which starts from the origin in the direction of the

unit vector u. Here s denotes arclength along the ray. The ray also depends

upon w. The equations which x satisfies are

{n^')' = Vn (2)

x(0) = (5)

x'(0) = u , (u^ = 1) . (4)

These equations have a unique solution which depends continuously upon n and u

in an appropriate norm, if n is continuously differentiable. Thus the problem

(2) - (k) is well posed.

To find X we shall determine its derivatives with respect to â‚¬ at e =

and then express x by means of its Taylor series in e. Thus we shall employ

an honest method, in the first phase of which probability plays no role. Let

us first set â‚¬ = in (2) - (^4-) and denote x(s,0) by x (s). Then we obtain

x" = (5)

o

x^(o) = (6)

x;(o) = u . (7)

The solution of (5) - (7) is

X (s) = us .

(8)

Nov ve differentiate (2) - (k) vith respect to e, set e = 0, and denote

X (s^O) by X (s)_, obtaining

x" = V ^(x )

â‚¬ ' O'

x' â€¢ V ^(x )

^ O

(9)

X (0) = x'(0) =

(10)

The right side of (9) is just the component of Vp. vhich is normal to x' = u.

Let us call it the transverse gradient and denote it by V p . Then the solu-

tion of (9) and (10) is

s

x^(s) = J (s-t) V^ n(ut)dt. (11)

o

Differentiating (2) - {k) tvlce with respect to e at e = yields

x-(s) = 2(x^ . V^)V^ ^^(x^)-V^ H^(x^)-2^;x^ . V^ H(x^)-2^^x; â€¢ V n(xj (12)

X (0) = x' (0) =

The solution of (l2) and (15) is

(15)

x^^(s) = 2 / (s-t)

""e ' ^T ^ â– I "^T ^^-^^^T ^^^-^^^' '^t^)^T ^

dt. (l4)

In the integrand, the argument of |i is ut. Thus to the second order in e "we

have

^ 2 ^

x(s,e) = x^(s) + e x^(s) + ^ \^i^) + 0(â‚¬^) . (15)

- 9 -

k. Light rays In a random medium

Let us now take account of the randomness and compute some statistical

properties of x(s,â‚¬). Let us begin with the mean value which is

just the s\am of the mean values of the terms on the right side of (15) â€¢

Since x (s) = us is independent of n and therefore of w, = us. From

(11)^ by interchanging the order of taking the mean value with integration

and differentiation, which we assume to be permissible, we obtain

Ik -

Let us finally calculate the mean square value of x'(s^e) - u. This is

the mean square value of 2 - 2 cos a(s^e) where a is the angle between x' and

u. From (15) we have

(x' -Zf = e^{x^f + 0{e^) . (58)

From (58) and (ll) we obtain

= 2M - j = he^

For s large compared to a this becomes

N(s)-l-s / r'hi dr

+ 0(e

(^0)

3n

NEW YORK UNIVERSITY

JNSTITUTE OF MATHEMATICAL SCIENCES

L1BRA. or some similar

unproved assumption. The reason for making such assijmptlons is to obtain

determinate equations for . Other examples of dishonest methods will be

described later. In all cases probability is introduced before u is deter-

mined and an \mproved assumption is made about some statistical property of

the random wave motion. The assimiption usually simplifies the problem so

that It becomes solvable.

Dishonest methods have the advantage over honest ones that they simplify

the problem to be solved. As a consequence, a problem vhich can be solved

honestly only by a perturbation method might be solved dishonestly without

the use of a perturbation expansion. Then the dishonest solution would be

applicable for all values of the relevant parameter while the honest solution

would be valid only for small values of it. For this and similar reasons,

many of the significant and non-trivial results in the theory of wave propa-

gation in random media have been obtained by dishonest methods. Many of

these results have compared well with experiment. Thus one of the important

mathematical problems in this field is to justify the dishonest methods by

showing that their results, in some sense, are approximations to honest solu-

tions. A clear vmderstanding of the circumstances in which this is the case

would permit the introduction of many more useful dishonest procedures.

Most of the work on this subject has been done since 19^5^ having been

stimulated by practical problems of radio wave propagation through the atmos-

phere and ionosphere, sound wave propagation in the ocean and the atmosphere,

light transmission through the atmosphere, etc. The recent book by L.A.

Chernov' - ' contains a rathe'r complete bibliography and an understandable

account of the present state of this subject. Additional material is con-

[21

tained in the book by V.I. Tatarski â€¢- -^ . The related subject of random wave

motion in a non-random medium is surveyed by M. Bom and E. Wolf '- â– * -^ .

Some other work is contained in the symposium volumes edited by W.C. Hoffman' - '

and Z. KopalJ- J . Because of the complete bibliographies in these books, we

shall give relatively few references.

Physicists have studied extensively the propagation of waves through

random collections of discrete scatterers. A clear formulation of a prob-

lem of this type, together vlth a new dishonest method for treating it, was

given by L.L. Foldy^ -1 . This method wr.s generalized to other problems by

M. Lax'- -' , who has also reviewed much of the previous work in this field.

Many problems have been treated by this method and others by V. Twersky'- -' .

[91

Recently, J. Bazer'- -' proved that for a one-dimensional scattering problem,

this method does yield the correct result. I. Kay and R.A. Silverman'- -'

investigated the accuracy of another method, the single scattering approxi-

mation, by determining the extent to which randomness reduces the importance

of multiple scattering.

The next section describes how wave propagation in a random medium

applies to the problem of the twixikling of a star. In Section J we consl-.

der light rays in a slightly inhomogeneous medium, and in Section ^ we apply

these results to a random medium obtaining some new results. We then compare

these honest results with corresponding dishonest ones in Section 5, where

we treat a light ray in a random medium as a Mai^off process. In Sections

6 and 7 we use the results of Sections 3 sixd k to determine the phase and

amplitude fluctuations of a wave in a random medium. Our results are exten-

sions of previously known ones. This completes our discussion of geometri-

cal optics in a random medium . In the final section we present a brief new

treatment of the reduced wave equation in a random medium. This section is

independent of the preceding ones.

2. An application

Before Illustrating the techniques used to analyze wave propagation in

random media, we shall describe a physical phenomenon in which such propaga-

tion plays a role. It is the scintillation or twinkling of a star. The most

- 6

appropriate theory of vave propagation to describe this phenomenon is geo-

metrical optics. According to this theory^ light travels along certain

straight or curved paths called rays. These rays are determined by ordinary

differential equations in which appears the index of refraction n(x, t), a

scalar function vhlch characterizes the transmission medium. This medium is

the earth's atmosphere in the present case. If n=l the rays are straight lines

which emanate from the star in all directions. One of them (really a narrow

beam) enters the eye of an observer who is viewing the star. The direction

from which the ray enters the eye is the apparent direction of the star.

Actually n(x^t) differs from unity by a small amount. As a consequence, the

rays deviate slightly from straight lines and enter the observer's eye from

slightly different directions at successive Instants of time. Therefore the

star appears to be moving about its mean position. Its apparent Intensity

also fluctuates.

We would like to calculate the apparent direction and intensity as func-

tions of time. To do so it would be necessary to know the index n(x,t). This

is practically impossible becatise the variation of n(x_,t) with time and posi-

tion results from the trubulent motion of the atmosphere. Consequently analy-

sis would appear to be impossible. In the face of this difficulty we treat

the atmosphere as a random medium. The random medivmi must be so chosen that

its Important statistical properties correspond to measurable properties of

the atmosphere. In tMs way we give up the possibility of calculating the

apparent direction and intensity at any particular time. Instead we can calcu-

late statistical properties of the apparent direction and intensity which may

be related to the actual temporal distribution of apparent directions and in-

tensities. We shall consider this example further in the next section. In

doing so we shall make use of the fact that a ray traverses the atmosphere so

7 -

qiilckly that the index does not change significantly during the traversal.

Consequently we may assvune that the index is independent of time in deter-

mining the rays.

5. Light rays in a slightly inhomogeneous medium

Let n(x, e) denote the index of refraction of a random medium, vhlch may

be written as

n(x,e) = 1 + e |a(x) . (l)

The quantity â‚¬ measures the deviation of the index from unity. The index also

depends upon a parameter co which we shall not write explicitly. We wish to

determine the ray x(s,â‚¬) which starts from the origin in the direction of the

unit vector u. Here s denotes arclength along the ray. The ray also depends

upon w. The equations which x satisfies are

{n^')' = Vn (2)

x(0) = (5)

x'(0) = u , (u^ = 1) . (4)

These equations have a unique solution which depends continuously upon n and u

in an appropriate norm, if n is continuously differentiable. Thus the problem

(2) - (k) is well posed.

To find X we shall determine its derivatives with respect to â‚¬ at e =

and then express x by means of its Taylor series in e. Thus we shall employ

an honest method, in the first phase of which probability plays no role. Let

us first set â‚¬ = in (2) - (^4-) and denote x(s,0) by x (s). Then we obtain

x" = (5)

o

x^(o) = (6)

x;(o) = u . (7)

The solution of (5) - (7) is

X (s) = us .

(8)

Nov ve differentiate (2) - (k) vith respect to e, set e = 0, and denote

X (s^O) by X (s)_, obtaining

x" = V ^(x )

â‚¬ ' O'

x' â€¢ V ^(x )

^ O

(9)

X (0) = x'(0) =

(10)

The right side of (9) is just the component of Vp. vhich is normal to x' = u.

Let us call it the transverse gradient and denote it by V p . Then the solu-

tion of (9) and (10) is

s

x^(s) = J (s-t) V^ n(ut)dt. (11)

o

Differentiating (2) - {k) tvlce with respect to e at e = yields

x-(s) = 2(x^ . V^)V^ ^^(x^)-V^ H^(x^)-2^;x^ . V^ H(x^)-2^^x; â€¢ V n(xj (12)

X (0) = x' (0) =

The solution of (l2) and (15) is

(15)

x^^(s) = 2 / (s-t)

""e ' ^T ^ â– I "^T ^^-^^^T ^^^-^^^' '^t^)^T ^

dt. (l4)

In the integrand, the argument of |i is ut. Thus to the second order in e "we

have

^ 2 ^

x(s,e) = x^(s) + e x^(s) + ^ \^i^) + 0(â‚¬^) . (15)

- 9 -

k. Light rays In a random medium

Let us now take account of the randomness and compute some statistical

properties of x(s,â‚¬). Let us begin with the mean value which is

just the s\am of the mean values of the terms on the right side of (15) â€¢

Since x (s) = us is independent of n and therefore of w, = us. From

(11)^ by interchanging the order of taking the mean value with integration

and differentiation, which we assume to be permissible, we obtain

Ik -

Let us finally calculate the mean square value of x'(s^e) - u. This is

the mean square value of 2 - 2 cos a(s^e) where a is the angle between x' and

u. From (15) we have

(x' -Zf = e^{x^f + 0{e^) . (58)

From (58) and (ll) we obtain

= 2M - j = he^

For s large compared to a this becomes

N(s)-l-s / r'hi dr

+ 0(e

(^0)

3n

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