Abstract

We apply the theory of plane-wave scattering by spheres to calculate the variance, the space–time covariance, and the temporal-frequency spectrum of the amplitude fluctuations of a light wave passing through rainfall. The development includes the effects of various sizes of raindrops and their associated terminal velocities. We show that the time-lagged covariance function of two vertically spaced sensors can provide a practical remote measurement of path-averaged rain parameters, such as drop-size distribution and rain rate, without any assumptions about the form of the drop-size distribution.

© 1975 Optical Society of America

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References

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  1. T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 45, 301 (1966)
    [Crossref]
  2. T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 47, 723 (1968).
    [Crossref]
  3. D. Atlas, J. Meteorol. 10, 486 (1953).
    [Crossref]
  4. R. W. Wilson and A. A. Penzias, Nature 211, 1081 (1966).
    [Crossref]
  5. D. Atlas and C. W. Ulbrich, J. Rech. Atmos. VIII, 275 (1974).
  6. S. W. Kurnick, R. N. Zitter, and D. B. Williams, J. Opt. Soc. Am. 50, 578 (1960).
    [Crossref]
  7. A. Arnulf and J. Bricard, J. Opt. Soc. Am. 47, 491 (1957).
    [Crossref]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 4319 (National Technical Information Service, Springfield, Va., 1971).
  10. R. Gunn and G. D. Kinzer, J. Meteorol. 6, 243 (1949).
    [Crossref]
  11. T. S. Marshall and W. McK. Palmer, J. Meteorol. 4, 186 (1948).
    [Crossref]
  12. S. F. Clifford, J. Opt. Soc. Am. 61, 1285 (1971).
    [Crossref]
  13. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, New York, 1965), p. 707.

1974 (1)

D. Atlas and C. W. Ulbrich, J. Rech. Atmos. VIII, 275 (1974).

1971 (1)

1968 (1)

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 47, 723 (1968).
[Crossref]

1966 (2)

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 45, 301 (1966)
[Crossref]

R. W. Wilson and A. A. Penzias, Nature 211, 1081 (1966).
[Crossref]

1960 (1)

1957 (1)

1953 (1)

D. Atlas, J. Meteorol. 10, 486 (1953).
[Crossref]

1949 (1)

R. Gunn and G. D. Kinzer, J. Meteorol. 6, 243 (1949).
[Crossref]

1948 (1)

T. S. Marshall and W. McK. Palmer, J. Meteorol. 4, 186 (1948).
[Crossref]

Arnulf, A.

Atlas, D.

D. Atlas and C. W. Ulbrich, J. Rech. Atmos. VIII, 275 (1974).

D. Atlas, J. Meteorol. 10, 486 (1953).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.

Bricard, J.

Chu, T. S.

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 47, 723 (1968).
[Crossref]

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 45, 301 (1966)
[Crossref]

Clifford, S. F.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, New York, 1965), p. 707.

Gunn, R.

R. Gunn and G. D. Kinzer, J. Meteorol. 6, 243 (1949).
[Crossref]

Hogg, D. C.

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 47, 723 (1968).
[Crossref]

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 45, 301 (1966)
[Crossref]

Kinzer, G. D.

R. Gunn and G. D. Kinzer, J. Meteorol. 6, 243 (1949).
[Crossref]

Kurnick, S. W.

Marshall, T. S.

T. S. Marshall and W. McK. Palmer, J. Meteorol. 4, 186 (1948).
[Crossref]

Palmer, W. McK.

T. S. Marshall and W. McK. Palmer, J. Meteorol. 4, 186 (1948).
[Crossref]

Penzias, A. A.

R. W. Wilson and A. A. Penzias, Nature 211, 1081 (1966).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, New York, 1965), p. 707.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 4319 (National Technical Information Service, Springfield, Va., 1971).

Ulbrich, C. W.

D. Atlas and C. W. Ulbrich, J. Rech. Atmos. VIII, 275 (1974).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Williams, D. B.

Wilson, R. W.

R. W. Wilson and A. A. Penzias, Nature 211, 1081 (1966).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.

Zitter, R. N.

Bell Syst. Tech. J. (2)

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 45, 301 (1966)
[Crossref]

T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 47, 723 (1968).
[Crossref]

J. Meteorol. (3)

D. Atlas, J. Meteorol. 10, 486 (1953).
[Crossref]

R. Gunn and G. D. Kinzer, J. Meteorol. 6, 243 (1949).
[Crossref]

T. S. Marshall and W. McK. Palmer, J. Meteorol. 4, 186 (1948).
[Crossref]

J. Opt. Soc. Am. (3)

J. Rech. Atmos. (1)

D. Atlas and C. W. Ulbrich, J. Rech. Atmos. VIII, 275 (1974).

Nature (1)

R. W. Wilson and A. A. Penzias, Nature 211, 1081 (1966).
[Crossref]

Other (4)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 4319 (National Technical Information Service, Springfield, Va., 1971).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, New York, 1965), p. 707.

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Figures (12)

FIG. 1
FIG. 1

Geometry of an incident plane wave scattered by randomly displaced monodisperse raindrops of radius a.

FIG. 2
FIG. 2

Temporal power spectrum of the amplitude scintillations induced by a sheet of rainfall with a rain rate h (mm/h) located x0 = 1 km from the receivers. The wave number of the incident plane wave is k = 107 m−1. A monodisperse size distribution (a=1 mm) of raindrops is assumed. The horizontal axis is the normalized frequency Ω = f/f0; f0 = v/(2πa) and v is the terminal velocity of raindrops of radius a.

FIG. 3
FIG. 3

Temporal power spectrum of the rain-induced amplitude scintillations of a monodisperse (a = 1 mm) rainfall normalized to the variance of the amplitude scintillations vs the normalized frequency Ω.

FIG. 4
FIG. 4

Temporal power spectra of the rain-induced amplitude scintillations of a Marshall–Palmer size distribution of raindrops, normalized to the variance of the amplitude scintillations. The horizontal axis is the frequency in Hz.

FIG. 5
FIG. 5

Frequency distribution of energy of the temporal power spectra shown in Fig. 4.

FIG. 6
FIG. 6

The combination of the rain-induced and turbulence-induced temporal power spectra vs frequency in Hz. Here L = 500 m, va = 2 m/s, λ = 0.628 μm, σ x a 2 = 0.25, h = 7 mm/h.

FIG. 7
FIG. 7

Experimentally measured temporal power spectrum of amplitude scintillation during a rainfall along a 500 m path.

FIG. 8
FIG. 8

The normalized covariance functions of amplitude fluctuations C χ ( z 0 , t ) ¯ / σ χ 2 ¯ for two detectors with a vertical separation z0 = 10 cm vs the time lag t.

FIG. 9
FIG. 9

The normalized covariance functions of amplitude fluctuations C χ ( z 0 , t ) ¯ / σ χ 2 ¯ for a fixed time lag t = 31.6 ms vs the vertical separation z0.

FIG. 10
FIG. 10

The quantity C χ ( z 0 , t ) ¯ / ( t 2 h ¯ ) for a fixed vertical separation z0 = 10 cm vs the time lag t.

FIG. 11
FIG. 11

The quantity C χ ( z 0 , t ) ¯ z 0 / h ¯ for a fixed time lag t = 31.6 ms vs the vertical separation z0.

FIG. 12
FIG. 12

The quantity C χ ( z 0 , t ) ¯ t 7 for fixed z0 = 10 cm vs 1/t2 and the quantity C χ ( z 0 , t ) ¯ z 0 - 6 for fixed t = 31.6 ms vs z 0 2. The corresponding raindrop diameter is also shown on the horizontal axis.

Equations (78)

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E s E 0 = exp ( - i k r + i k x ) i k r [ η 2 J 1 ( η θ ) η θ + O ( 1 ξ ) ] ,
E s E 0 exp ( - i k ρ 2 / 2 x ) i k x η J 1 ( η θ ) θ .
χ = ( E 0 + E s ) / E 0 - 1 = { [ 1 + Re ( E s / E 0 ) ] 2 + Im 2 ( E s / E 0 ) } 1 / 2 - 1 ,
χ ( x , y , z ) = Re [ E s ( x , y , z ) / E 0 ( x , y , z ) ] = - sin [ k ( y 2 + z 2 ) / 2 x ] ( η / k ) × J 1 [ η ( y 2 + z 2 ) 1 / 2 / x ] / ( y 2 + z 2 ) 1 / 2 .
C χ ( t ) χ ( x , y , z - v t ) χ ( x , y , z ) .
d C χ ( t ) = a 2 N ( x ) d x - d y - d z sin [ k ( y 2 + z 2 ) / 2 x ] × sin [ k { y 2 + ( z - v t ) 2 } / 2 x ] J 1 [ η { y 2 + z 2 } 1 / 2 / x ] { y 2 + z 2 } 1 / 2 × J 1 [ η { y 2 + ( z - v t ) 2 } 1 / 2 / x ] { y 2 + ( z - v t ) 2 } 1 / 2 .
N ( x ) = h ( x ) 4.8 π × 10 6 a 3 v .
v 2 ( d ) = 4 3 g d ( ρ w - ρ a ) / ( ρ a C ) ,
v ( a ) = 45.4 ( 2 g a ) 1 / 2 = 200 a 1 / 2 .
N ( x ) = 3.32 × 10 - 10 a - 7 / 2 h ( x ) .
C χ ( z n ( t ) ) 0 L d C χ ( z n ) = 1 2 a 2 0 L N ( x ) [ I 1 ( x , z n ) + I 2 ( x , z n ) ] d x ,
I 1 ( x , z n ( t ) ) = - - d y - d z cos [ α ( y 2 + z 2 - z z n + z n 2 / 2 ) ] × J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ { y 2 + ( z - z n ) 2 } 1 / 2 ] { y 2 + ( z - z n ) 2 } 1 / 2 ,
I 2 ( x , z n ( t ) ) = - d y - d z cos [ α ( z z n - z n 2 / 2 ) ] × J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ { y 2 + ( z - z n ) 2 } 1 / 2 ] { y 2 + ( z - z n ) 2 } 1 / 2 ,
α k x / η 2 = λ x / ( 2 π a 2 ) ,             z n = ( η v / x ) t ,
σ χ 2 = a 2 0 L N ( x ) d x - d y - d z sin 2 [ 1 2 α ( y 2 + z 2 ) ] × J 1 2 [ ( y 2 + z 2 ) 1 / 2 ] / ( y 2 + z 2 ) = a 2 0 L d x N ( x ) F 5 ( α ) .
σ χ 2 π 2 a 2 0 L N ( x ) d x .
ω χ ( f ) 2 - exp ( - i ω t ) C χ ( v t ) d t .
ω χ ( f ) = a 2 - d t exp ( - i ω t ) 0 L d x N ( x ) [ I 1 ( x , z n ( t ) ) + I 2 ( x , z n ( t ) ) ] = a 2 0 L d x N ( x ) ( F 1 + F 2 ) .
ω χ ( Ω ) = 3.32 π × 10 - 12 a - 1 Ω - 2 0 L d x h ( x ) × [ H 1 ( 2 Ω ) + η J 1 2 ( Ω ) ( π / k x ) 1 / 2 sin ( 1 4 π - k x Ω 2 / η 2 ) ] ,
ω χ ( x 0 , Ω ) = 3.32 π × 10 - 12 a - 1 Ω - 2 × h [ H 1 ( 2 Ω ) + η J 1 2 ( Ω ) ( π / k x 0 ) 1 / 2 sin ( 1 4 π - k x 0 Ω 2 / η 2 ) ] .
ω χ ( Ω ) = 3.32 π × 10 - 12 a - 1 Ω - 2 H 1 ( 2 Ω ) L h ¯ ,
( π / 2 ) f 0 ω χ ( Ω ) / σ χ 2 = H 1 ( 2 Ω ) / Ω 2 .
F ¯ 0 d a p ( a , x ) F ( a ) .
σ ¯ χ 2 = 5.22 × 10 - 10 0 L d x h ( x ) 0 d a a - 3 / 2 p ( a ) .
ω χ ( f ) ¯ = 1.06 × 10 - 8 f - 2 0 L d x h ( x ) 0 d a p ( a , x ) a - 2 H 1 ( α 1 f a ) ,
p ( a , x ) = N ( a , x ) 4 3 π a 3 0 N ( a , x ) 4 3 π a 3 d a .
N ( a , x ) = N 0 e - 2 Λ ( x ) a ,
N 0 = 8 × 10 6 m - 4 ,
Λ ( x ) = 4100 h ( x ) - 0.21 m - 1 ,
p ( a , x ) = 16 6 Λ 4 ( x ) a 3 e - 2 Λ ( x ) a .
σ χ 2 ¯ = 8.57 × 10 - 5 0 L d x h ( x ) 0.685
ω χ ( f ) ¯ = 0.119 f 2 0 L d x h ( x ) 0.58 0 d δ δ e - δ H 1 ( α 2 f δ ) ,
ω χ ( f ) ¯ σ χ 2 ¯ = 1390 f 2 0 L d x h ( x ) 0.58 0 d δ δ e - δ H 1 ( α 2 f δ ) 0 L d x h ( x ) 0.685 .
d 2 ω χ ( f ) ¯ d f 2 | f = 0 ~ 0 L d x h ( x ) 0 d a p ( a , x ) = h ¯ L .
d 2 d f 2 [ H 1 ( α 1 f a ) f 2 ] | f = 0 a 2 .
C χ ( z 0 , t ) ¯ = 0 L d x 0 d a 1 2 a 2 p ( a , x ) N ( a , x ) [ I 1 ( x , z 0 - v t ) + I 2 ( x , z 0 - v t ) ] = 0 L d x ( F 3 + F 4 ) ,
C χ ( z 0 , t ) ¯ = 8.85 × 10 - 12 t - 1 0 L d x h ( x ) p ( a , x ) × [ 1 + 9.24 × 10 - 6 ( k / x ) 1 / 2 z 0 2 / t 2 ] ,
a = z 0 2 / ( 4 × 10 4 t 2 ) .
C χ ( z 0 , t ) ¯ = 8.85 × 10 - 12 t - 1 0 L d x h ( x ) p ( a , x ) ~ p ( a ) ¯ / t ,
0 d a t C χ ( z 0 , t ) ¯ = 8.85 × 10 - 12 0 d a 0 L d x h ( x ) p ( a , x ) = 8.85 × 10 - 12 h ¯ L .
d a = - 5 × 10 - 5 z 0 2 t - 3 d t .
h ¯ = 5.65 × 10 6 z 0 2 0 d t C χ ( z 0 , t ) ¯ t - 2 / L ,             for fixed z 0 .
d a = 5 × 10 - 5 z 0 t - 2 d z 0 .
h ¯ = 5.65 × 10 6 t - 1 0 d z 0 z 0 C χ ( z 0 , t ) ¯ / L ,             for fixed t .
N ( a ) ¯ ~ z 0 - 6 t 7 C χ ( z 0 , t ) ¯ ;             a = z 0 2 / ( 4 × 10 4 t 2 ) .
( π a 2 ) ( L / d ) d 2 .
d = N - 1 / 3 .
L ( π a 2 N ) - 1 = K m - 1 ,
K m ( a ) = 1.04 × 10 - 9 a - 3 / 2 h ( a ) ,
h ( a ) d a = h ¯ p ( a ) d a .
K m ( a ) = 2.77 × 10 - 9 Λ 4 h ¯ a 3 / 2 e - 2 Λ a ,
K m = 1.7 × 10 - 4 h ¯ 0.685 .
L m = 5.9 × 10 3 ( h ¯ ) - 0.685 .
F ( u ) = a b f ( t ) exp [ i u g ( t ) ] d t ;
F ( u ) [ 2 π u g ( t 0 ) ] 1 / 2 exp [ i { u g ( t 0 ) 1 4 π } ] [ f ( t 0 + 0 ) + f ( t 0 - 0 ) ] / 2 ,
( a ) F 1 - exp ( - i ω t ) I 1 ( x , v t ) d t = - - d y - d z - d t exp ( - i ω t ) × cos [ α { ( y 2 + z 2 ) - z v t + ( v t ) 2 / 2 } ] × J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ { y 2 + ( z - v t ) 2 } 1 / 2 ] { y 2 + ( z - v t ) 2 } 1 / 2 ,
F 1 = - 2 v - d y - d z exp ( - i ω z ) × 0 d z 1 cos ( ω z 1 ) cos [ α y 2 + α z 2 / 2 + α z 1 2 / 2 ] × J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ ( y 2 + z 1 2 ) 1 / 2 ] ( y 2 + z 1 2 ) 1 / 2 ,
cos ( ω z 1 ) cos [ α { y 2 + z 2 / 2 + z 1 2 / 2 } ] = 1 2 { cos [ α y 2 + α z 2 / 2 + ω z 1 + α z 1 2 / 2 ] + cos [ α y 2 + α z 2 / 2 - ω z 1 + α z 1 2 / 2 ] } .
F 1 = - ( 1 v ) ( 2 π x k ) 1 / 2 - d y - d z cos [ π 4 + α y 2 + α z 2 2 - ω 2 2 α ] × exp ( - i ω z ) J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ ( y 2 + ω 2 / α 2 ) 1 / 2 ] ( y 2 + ω 2 / α 2 ) 1 / 2 .
F 1 = 2 2 π η k v 0 d z sin [ α z 2 2 - ω 2 2 α ] cos ( ω z ) J 1 ( z ) z J 1 ( ω / α ) ω / α .
F 1 = 2 π 3 / 2 η 2 k 3 / 2 v J 1 2 ( Ω ) Ω 2 sin ( π 4 - k x Ω 2 η 2 ) / x ,
( b ) F 2 - exp ( - i ω t ) I 2 ( x , v t ) d t = - d y - d z - d t cos [ α ( z v t - ( v t ) 2 / 2 ) ] × exp ( - i ω t ) J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ { y 2 + ( z - v t ) 2 } 1 / 2 ] { y 2 + ( z - v t ) 2 } 1 / 2 ,
F 2 = 4 π η k v 0 d y J 1 2 [ { y 2 + Ω 2 } 1 / 2 ] ( y 2 + Ω 2 ) .
F 2 = 2 π η k v H 1 ( 2 Ω ) Ω 2 ,
( c ) F 3 = 0 d a p ( a , x ) I 1 ( z 0 - v t ) ( 1 2 a 2 ) N ( a , x ) = - 1 2 0 d a 0 L d x - d y - d z p ( a , x ) a 2 N ( a , x ) × cos [ α { ( y 2 + z 2 ) - z ( z 0 - v t η / x ) + ( z 0 - v t η / x ) 2 / 2 } ] × J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ { y 2 + ( z - z 0 + v t η / x ) 2 } 1 / 2 ] { y 2 + ( z - z 0 + v t η / x ) 2 } 1 / 2 ,
F 3 = 2 π k 0 L d x x 0 d a p ( a , x ) a 4 N ( a , x ) × sin [ α ( z 0 - v t η / x ) 2 4 ] J 1 2 [ ( z 0 - v t η / x ) / 2 ] ( z 0 - v t η / x ) 2 .
F 3 = 1.66 × 10 - 10 π k 0 L h ( x ) d x x 0 d a p ( a , x ) a - 3 / 2 × sin [ k ( z 0 - v t ) 2 4 x ] J 1 2 [ k a ( z 0 - v t ) / ( 2 x ) ] ( z 0 - v t ) 2 / 4 x 2 .
F 3 = 2.09 × 10 - 7 k 0 L h ( x ) d x x 0 d v v 2 p ( v 2 4 × 10 4 , x ) × sin [ k ( z 0 - v t ) 2 4 x ] J 1 2 [ k v 2 ( z 0 - v t ) / ( 8 × 10 4 x ) ] ( z 0 - v t ) 2 / 4 x 2 .
F 3 = 8.18 × 10 - 17 k z 0 2 t 3 0 L d x x - 1 / 2 h ( x ) p ( a , x ) ,
( d ) F 4 = 0 d z p ( a , x ) I 2 ( z 0 - v t ) ( 1 2 a 2 ) N ( x ) = 1 2 0 L d x - d y - d z 0 d a p ( a , x ) a 2 N ( x ) × cos [ α { z ( z 0 - v t η / x ) - ( z 0 - v t η / x ) 2 / 2 } ] J 1 [ ( y 2 + z 2 ) 1 / 2 ] ( y 2 + z 2 ) 1 / 2 J 1 [ { y 2 + ( z - z 0 + v t η / x ) 2 } 1 / 2 ] { y 2 + ( z - z 0 + v t η / x ) 2 } 1 / 2 .
F 4 = 6.65 × 10 - 8 0 L h ( x ) d x - d z - d y 0 d v v 2 p ( v 2 4 × 10 2 , x ) cos [ k { z ( z 0 - v t ) - ( z 0 - v t ) 2 / 2 } / x ] × J 1 [ k v 2 ( y 2 + z 2 ) 1 / 2 / ( 4 × 10 4 x ) ] ( y 2 + z 2 ) 1 / 2 J 1 [ k v 2 { y 2 + ( z - z 0 + v t ) 2 } 1 / 2 / ( 4 × 10 4 x ) ] { y 2 + ( z - z 0 + v t ) 2 } 1 / 2 .
F 4 = 6.64 × 10 - 8 ( 2 π / k t 2 ) 1 / 2 0 L d x x 1 / 2 h ( x ) - d y - d z p [ ( z 0 - z ) 2 / 4 × 10 4 t 2 , x ] cos ( k z 2 / 2 x - 1 4 π ) × J 1 [ k ( z 0 - z ) 2 ( y 2 + z 2 ) 1 / 2 / ( 4 × 10 4 x t 2 ) ] ( y 2 + z 2 ) 1 / 2 J 1 [ k y ( z 0 - z ) 2 / ( 4 × 10 4 x t 2 ) ] y ( z 0 - z ) 2 / t 2 .
F 4 = 6.64 × 10 - 8 2 π t k 0 L d x h ( x ) x - d y p ( a , x ) J 1 2 [ k y a / x ] y 2 z 0 2 .
0 d y J 1 2 ( α y ) y 2 = 4 α 3 π .
F 4 = 8.85 × 10 - 12 t - 1 0 L d x h ( x ) p ( a , x ) ,
( e ) F 5 = - d y - d z sin 2 [ α ( y 2 + z 2 ) / 2 ] J 1 [ ( y 2 + z 2 ) 1 / 2 ] / ( y 2 + z 2 ) = 1 2 - d y - d z J 1 2 [ ( y 2 + z 2 ) 1 / 2 ] / ( y 2 + z 2 ) - 1 2 - d y - d z cos [ α ( y 2 + z 2 ) ] J 1 [ ( y 2 + z 2 ) 1 / 2 ] / ( y 2 + z 2 ) .
F 5 = π 0 d ρ J 1 2 ( ρ ) / ρ .
F 5 = π / 2.