Abstract

Pulse propagation of beam waves from a source immersed in a slab of scatterers is investigated by means of analytically solving the diffusion equation. The on- and off-axis pulse intensities are calculated for collimated beam waves transmitted in the two typical directions of the slab. To a good approximation, the effect of finite beamwidth on the pulse intensity appears only through the diffusion factor in the radial direction of the beam, resulting in a faster decrease of the intensity in the tail part compared with that of the corresponding plane wave pulse. Also, the pulse shape does not appreciably change when the beamwidth is changed within the range of practical use. An analytical expression is obtained for the pulse width broadening and is applied to several typical cases leading to some simple expressions. The influence of the scatterers existing behind the source is discussed in some detail particularly in connection with pulse shape and width broadening, showing that, in many situations, the pulse width is determined mostly by the total optical thickness of the slab independently of the relative position of the wave source within the slab.

© 1981 Optical Society of America

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  1. L. M. Erukhimov, I. G. Zarnitsyna, P. I. Kirsh, Radiophys. Quantum Electron. 16, 436 (1973).
  2. V. I. Shishov, Sov. Astron. 17, 598 (1974).
  3. L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).
  4. I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1974).
  5. C. H. Liu, K. C. Yeh, J. Opt. Soc. Am. 67, 1261 (1977).
  6. L. B. Stotts, Appl. Opt. 17, 504 (1978).
  7. C. H. Liu, K. C. Yeh, IEEE Trans. Antennas Propag. AP-26, 561 (1978).
  8. K. Furutsu, J. Math. Phys. 20, 617 (1979).
  9. I. Sreenivasiah, A. Ishimaru, Appl. Opt. 18, 1613 (1979).
  10. S. Ito, Radio Sci. 15, 893 (1980).
  11. E. A. Bucher, Appl. Opt. 12, 2391 (1973).
  12. K. Furutsu, J. Math. Phys. 21, 765 (1980).
  13. K. Furutsu, J. Opt. Soc. Am. 70, 360 (1980).
  14. E. A. Bucher, R. M. Lerner, Appl. Opt. 12, 2401 (1973).
  15. G. C. Mooradian, M. Geller, L. B. Stotts, D. H. Stephens, R. A. Krautwald, Appl. Opt. 18, 429 (1979).
  16. S. Ito, K. Furutsu, J. Opt. Soc. Am. 70, 366 (1980).
  17. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 228.

1980 (4)

S. Ito, Radio Sci. 15, 893 (1980).

K. Furutsu, J. Math. Phys. 21, 765 (1980).

K. Furutsu, J. Opt. Soc. Am. 70, 360 (1980).

S. Ito, K. Furutsu, J. Opt. Soc. Am. 70, 366 (1980).

1979 (3)

1978 (2)

C. H. Liu, K. C. Yeh, IEEE Trans. Antennas Propag. AP-26, 561 (1978).

L. B. Stotts, Appl. Opt. 17, 504 (1978).

1977 (1)

1975 (1)

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).

1974 (2)

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1974).

V. I. Shishov, Sov. Astron. 17, 598 (1974).

1973 (3)

L. M. Erukhimov, I. G. Zarnitsyna, P. I. Kirsh, Radiophys. Quantum Electron. 16, 436 (1973).

E. A. Bucher, Appl. Opt. 12, 2391 (1973).

E. A. Bucher, R. M. Lerner, Appl. Opt. 12, 2401 (1973).

Bucher, E. A.

Erukhimov, L. M.

L. M. Erukhimov, I. G. Zarnitsyna, P. I. Kirsh, Radiophys. Quantum Electron. 16, 436 (1973).

Furutsu, K.

K. Furutsu, J. Opt. Soc. Am. 70, 360 (1980).

S. Ito, K. Furutsu, J. Opt. Soc. Am. 70, 366 (1980).

K. Furutsu, J. Math. Phys. 21, 765 (1980).

K. Furutsu, J. Math. Phys. 20, 617 (1979).

Geller, M.

Hong, S. T.

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1974).

Ishimaru, A.

I. Sreenivasiah, A. Ishimaru, Appl. Opt. 18, 1613 (1979).

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1974).

Ito, S.

S. Ito, Radio Sci. 15, 893 (1980).

S. Ito, K. Furutsu, J. Opt. Soc. Am. 70, 366 (1980).

Jokipii, J. R.

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).

Kirsh, P. I.

L. M. Erukhimov, I. G. Zarnitsyna, P. I. Kirsh, Radiophys. Quantum Electron. 16, 436 (1973).

Krautwald, R. A.

Lee, L. C.

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).

Lerner, R. M.

Liu, C. H.

C. H. Liu, K. C. Yeh, IEEE Trans. Antennas Propag. AP-26, 561 (1978).

C. H. Liu, K. C. Yeh, J. Opt. Soc. Am. 67, 1261 (1977).

Mooradian, G. C.

Shishov, V. I.

V. I. Shishov, Sov. Astron. 17, 598 (1974).

Sreenivasiah, I.

I. Sreenivasiah, A. Ishimaru, Appl. Opt. 18, 1613 (1979).

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1974).

Stephens, D. H.

Stotts, L. B.

Yeh, K. C.

C. H. Liu, K. C. Yeh, IEEE Trans. Antennas Propag. AP-26, 561 (1978).

C. H. Liu, K. C. Yeh, J. Opt. Soc. Am. 67, 1261 (1977).

Zarnitsyna, I. G.

L. M. Erukhimov, I. G. Zarnitsyna, P. I. Kirsh, Radiophys. Quantum Electron. 16, 436 (1973).

Appl. Opt. (5)

Astrophys. J. (1)

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).

IEEE Trans. Antennas Propag. (1)

C. H. Liu, K. C. Yeh, IEEE Trans. Antennas Propag. AP-26, 561 (1978).

J. Math. Phys. (2)

K. Furutsu, J. Math. Phys. 20, 617 (1979).

K. Furutsu, J. Math. Phys. 21, 765 (1980).

J. Opt. Soc. Am. (3)

Radio Sci. (2)

S. Ito, Radio Sci. 15, 893 (1980).

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1974).

Radiophys. Quantum Electron. (1)

L. M. Erukhimov, I. G. Zarnitsyna, P. I. Kirsh, Radiophys. Quantum Electron. 16, 436 (1973).

Sov. Astron. (1)

V. I. Shishov, Sov. Astron. 17, 598 (1974).

Other (1)

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 228.

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Figures (10)

Fig. 1
Fig. 1

Coordinate system and geometry of a slab of scatterers confined within the region −ρb < ρ3 < ρf. Transmitter and receiver are denoted by T and R; A and B indicate the case of the beam wave pulses transmitted in the directions parallel with the transverse to the ρ3 axis, respectively.

Fig. 2
Fig. 2

Pulse intensity of a beam wave normally incident on a slab of scatterers within the region 0 < ρ3 < ρf = d (ρb = 0). The normalized on-axis intensity IA/[2cγI0 exp(−γact)] is shown as a function of γct for several values of the optical thickness γρf. W0 = 0.1 m and γ = 0.1 m−1.

Fig. 3
Fig. 3

Normalized on-axis pulse intensity IA/[2cγI0 exp(−γact)] vs γct in case A; the parameter γρb is the optical thickness between the source and the backside boundary of slab. W0 = 0.1m and γ = 0.1 m−1.

Fig. 4
Fig. 4

Normalized pulse intensity IA/[2cγI0 exp(−γact)] vs γct for several values of ρTf. The broken and solid curves show the intensities for ρb = 0 and ρb = ρf, respectively. W0 = 0.1m, γ = 0.1 m−1, and ρT = |ρT|.

Fig. 5
Fig. 5

Normalized pulse intensity IB/[2cγI0 exp(−γact)] vs γct in case B where the beam is transmitted in the direction transverse to the ρ3 axis. Intensities on the ρ1 and ρ2 axes are shown by the solid and broken curves, respectively. W0 = 0.1 m, γ = 0.1 m−1, ρf = 500 m, and ρT = |ρT|.

Fig. 6
Fig. 6

Normalized pulse width 〈Δt21/2/tf vs βf for various values of γa/ηγ, where tf = ρf/c. The solid curves are for the beam wave pulse incident on a slab of scatterers confined within the region 0 < ρ3<d(ρf = d, ρb = 0), while the dot–dash curves are for the corresponding plane wave pulse incidence.

Fig. 7
Fig. 7

Normalized pulse width 〈Δt21/2/td vs ρf/d for several values of the optical thickness γd, where td = d/c.

Fig. 8
Fig. 8

Normalized pulse width 〈Δt21/2/td vs γd for several values of βf/d. td = d/c.

Fig. 9
Fig. 9

Dependence of the normalized pulse width 〈Δt21/2/tf on ρbf for several values of the optical thickness γρf. tf = ρf/c.

Fig. 10
Fig. 10

Off-axis pulse width 〈Δt21/2/tf vs ρTf for several values of βf. ρb =0, tf = ρf/c, and βf = ηγρf.

Equations (96)

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[ γ a + c 1 / / t ( η γ ) 1 ( / ρ ) 2 ] I ( ρ ¯ ) = J ( ρ ¯ ) ,
η = 3 ( 1 a 1 ) a 1 = d Ω ( Ω · Ω ) σ ( Ω · Ω ) / d Ω σ ( Ω · Ω ) , γ = d Ω σ ( Ω · Ω ) , J ( ρ ¯ ) = γ I c ( ρ ¯ ) ,
I ( ρ ¯ ) = ( η γ ) 1 ( / ρ ) I ( ρ ¯ ) ,
I ( ρ ¯ ) = 2 ( η γ ) 1 ( n · / ρ ) I ( ρ ¯ ) = 0 ,
n · I ( ρ ¯ ) = ½ I ( ρ ¯ ) ,
I ( λ , ρ 3 , ν ) = d ρ T dt I ( ρ T , ρ 3 , t ) exp [ I ( λ , ρ T ν t ) ] ,
[ γ a + ( η γ ) 1 λ 2 + i ν / c ( η γ ) 1 ( / ρ 3 ) 2 ] I ( λ , ρ 3 , ν ) = γ I c ( λ , ρ 3 , ν ) ,
I ( λ , ρ f , ν ) = γ d ρ 3 G λ ν ( ρ f ρ 3 ) I c ( λ , ρ 3 , ν ) ,
G λ ν ( ρ f ρ 3 ) = ( 2 k λ ) 1 η γ [ 1 A 2 exp ( 2 k λ d ) ] 1 ( 1 + A ) × exp ( k λ ρ f ) × { exp ( k λ ρ 3 ) + A exp [ k λ ( ρ 3 + 2 ρ b ) ] } ,
k λ = [ ( γ a + i ν / c ) η γ + λ 2 ] 1 / 2 , Re ( k λ ) > 0 , A = [ ( η γ ) 1 k λ ½ ] / [ ( η γ ) 1 k λ + ½ ] .
I c ( λ , ρ 3 , ν ) = { I 0 exp [ ( W 0 f 3 ) 2 λ 2 / 4 ( γ t + i ν / c ) ρ 3 ] , ρ 3 0 , 0 , ρ 3 < 0 ,
f 3 2 = 1 + ρ 3 2 / ( k 0 W 0 2 ) 2 , k 0 = ω 0 / c ,
( γ t 1 / k 0 W 0 2 ) 2 1 ,
I c ( λ , ρ 3 , ν ) I 0 exp [ W 0 2 λ 2 / 4 ( γ t + i ν / c ) ρ 3 ] , ρ 3 0 .
I A ( λ , ρ f , ν ) = γ 0 ρ f d ρ 3 G λ ν ( ρ f ρ 3 ) I c ( λ , ρ 3 , ν ) = γ I 0 exp ( W 0 2 λ 2 / 4 ) [ g A 1 ( λ , ν ) + g A 2 ( λ , ν ) / f ( λ , ν ) ,
g A 1 ( λ , ν ) = [ G ( λ , ν ) G + ( λ , ν ) ] ,
g A 2 ( λ , ν ) = [ G + ( λ , ν ) exp ( k λ ρ f ) G ( λ , ν ) exp ( k λ ρ f ) ] × exp [ ( i ν / c + γ t ) ρ f ] ,
f ( λ , ν ) = { [ ( η γ ) 1 k λ + ½ ] 2 exp ( k λ d ) [ ( η γ ) 1 k λ ½ ] 2 exp ( k λ d ) } / ( η γ ) 1 k λ ,
G ± ( λ , ν ) = exp ( ± k λ ρ b ) [ ( η γ ) 1 k λ ± ½ ] × ( k λ i ν / c γ t ) 1 / ( η γ ) 1 k λ .
I c ( λ , ρ 3 , ν ) = I 0 π W 0 ( γ t + i ν / c i λ 1 ) 1 × exp ( ρ 3 2 W 0 2 W 0 2 λ 2 2 4 ) , λ = ( λ 1 , λ 2 ) ,
I B ( λ , ρ f , ν ) = γ ρ b ρ f d ρ 3 G λ ν ( ρ f ρ 3 ) I c ( λ , ρ 3 , ν ) γ I 0 exp ( W 0 2 λ 2 2 / 4 ) g B ( λ , ν ) / f ( λ , ν ) .
g B ( λ , ν ) = 2 ( γ t + i ν / c i λ 1 ) 1 [ ( η γ ) 1 k λ cosh ( k λ ρ b ) + ½ sinh ( k λ ρ b ) × exp ( W 0 2 k λ 2 / 4 ) / ( η γ ) 1 k λ ,
I A , B ( λ , ρ f , t ) = ( 2 π ) 1 d ν I A , B ( λ , ρ f , ν ) exp ( i ν t ) .
is = k λ d = ( η γ ) 1 k λ β s , β s = η γ d , i ν / c = η γ β s 2 s 2 γ a ( η γ ) 1 λ 2 ,
f ( s ) f ( λ , ν ) = 2 [ ( β s 1 s 2 + β s / 4 ) s 1 sin ( s ) + cos ( s ) ] = 0
s n n π [ 1 ( 2 / n π ) tan 1 ( 2 n π / β s ) ] , n = 1 , 2 , 3 , ,
i ν n / c = η γ β s 2 s n 2 γ a ( η γ ) λ 2 , f ( s ) s s = s n = 2 ( 1 ) n [ β s 2 / 4 + β s + s n ] / β s s n .
I A ( λ , ρ f , t ) = 2 c γ I 0 exp [ γ a ct ( η γ ) 1 λ 2 ct W 0 2 λ 2 / 4 ] × n = 11 exp ( η γ β s 2 s n 2 ct ) { ( 1 ) n 1 a on ( λ ) cos ( s n ρ b / d ) b on ( λ ) exp [ β s 1 s n 2 ρ f / d γ ρ f + ( η γ ) 1 λ 2 ρ f ] + ( 1 ) n 1 c on ( λ ) sin ( s n ρ b / d ) } ,
I B ( λ , ρ f , t ) = 2 c γ I 0 exp [ γ a ct ( η γ ) 1 λ 2 ct W 0 2 λ 2 2 / 4 ] × n = 11 ( 1 ) n 1 d on exp ( η γ β s 2 s n 2 ct ) / [ e ( λ ) β s 2 s n 2 + i γ I 0 exp [ γ t ct + i λ 1 ct W 0 2 λ 2 2 / 4 ] × Res [ g B ( λ , ν p ) ] / f ( λ , ν p ) ,
d on = 0 for even n ,
I A , B ( ρ T , ρ f , t ) = ( 2 π ) 2 d λ I A , B ( λ , ρ f , t ) exp ( i λ · ρ T ) ,
I A ( ρ T , ρ f , t ) = 2 c γ I 0 exp ( γ a ct ) D A ( ρ T , t ) × n = 1 ( 1 ) n 1 exp ( η γ β s 2 s n 2 ct ) { a on ( λ A ) cos ( s n ρ b / d ) + c on ( λ A ) sin ( s n ρ b / d ) } , t > ρ f / c ,
D A ( ρ T , t ) = [ 4 π ϕ ( t ) ct ] 1 η γ exp { [ 4 ϕ ( t ) ct ] 1 η γ ρ T 2 } , λ A = i [ 2 ϕ ( t ) ct ] 1 η γ ρ T ,
ϕ ( t ) = 1 + η γ W 0 2 / 4 ct ,
| ρ T / ϕ ( t ) ct | 2 4 η 1
a on ( λ A ) a on a on ( 0 ) , c on ( λ A ) c on c on ( 0 ) ,
[ 4 ϕ ( t ) ct ] 1 η γ ρ T 2 γ ct ϕ ( t ) , γ ct 1 ,
I A ( ρ T , ρ f , t ) = D A ( ρ T , t ) I p ( ρ f , t ) ,
I p ( ρ f , t ) = 2 c γ I 0 exp ( γ a ct ) n = 1 ( 1 ) n 1 exp ( η γ β s 2 s n 2 ct ) × { a on cos ( s n ρ b / d ) + c on sin ( s n ρ b / d ) } ,
I B ( ρ T , ρ f , t ) = 2 c η γ I 0 exp ( γ a ct ) D B ( ρ T , t ) × n = 1 ( 1 ) n 1 d on exp ( η γ β s 2 s n 2 ct ) ,
D B ( ρ T , t ) = [ 4 π ϕ ( t ) ct ] 1 η γ exp { ( η γ / 4 ct ) [ ρ 1 2 + ρ 2 2 / ϕ ( t ) ] } ,
λ B = I ( η γ / 2 ct ) ( ρ 1 , ρ 2 / ϕ ( t ) ) .
η γ W 0 2 / 4 ct < η γ W 0 2 / 4 ρ f 1 , ϕ ( t ) 1 ,
D A ( ρ T , t ) ( η γ / 4 π ct ) exp [ η γ ρ T 2 / 4 ct ] .
t 1 exp [ γ a ct ( s 1 / β s ) 2 η γ ct ( η γ / 4 ct ) ρ T 2 ] ,
t m = dt t m I A ( ρ T , ρ f , t ) / dt I A ( ρ T , ρ f , t ) ,
t m = ( i / ν ) m I A ( ρ T , ρ f , ν ) | ν = 0 / I A ( ρ T , ρ f , ν ) | ν = 0 .
I A ( ρ T , ρ f , ν ) = ( 2 π ) 2 d λ I A ( λ , ρ f , ν ) exp ( i λ · ρ T ) ,
I A ( λ , ρ f , ν ) = I p ( ρ f , ν i c λ 2 / η γ ) exp ( W 0 2 λ 2 / 4 )
t m = t m 1 p I p ( ρ f , ν ) ν = 0 / 0 dX I p ( ρ f , ν Xi ) ν = 0 , ρ T = 0 ,
Δ t 2 = t 2 t 2 = T A [ t p T A ] ,
t p = ( i / ν ) log I p ( ρ f , ν ) ν = 0 , T A = I p ( ρ f , ν ) ν = 0 / 0 dX I p ( ρ f , ν Xi ) ν = 0 .
Δ t 2 1 / 2 / t f = β f 1 [ g ( y a ) T D ( ρ f ) f ( y a ) ] 1 / 2 × [ g ( y a ) g ( y a ) + f ( y a ) f ( y a ) g ( y a ) T D ( ρ f ) f ( y a ) ] 1 / 2 ,
t f = ρ f / c , β f = η γ ρ f , y a = γ a / η γ ,
y a = γ a / η γ 1 ,
Δ t 2 1 / 2 / t f β s 1 [ η ( 1 + η / 2 ) / T D ( ρ f ) ] 1 / 2 [ ( ¼ ) sinh ( β s y a ) / y a + cosh ( β s y a ) ] 1 × { ( β s / 2 + 1 ) sinh ( β s y a ) / y a + ( ¼ ) ( sinh ( β s y a ) y a ) η ( 1 + η / 2 ) / T D ( ρ f ) } 1 / 2 .
Δ t 2 1 / 2 / t f β f 1 [ η ( 1 + η / 2 + β b / 2 ) T D ( ρ f ) ( β s / 4 + 1 ) ] 1 / 2 × { η + η 2 [ 3 + ( η 1 + ½ ) β b ] β b + β b 3 / 12 ( 1 + η / 2 + β b / 2 ) + β s ( 1 + β s / 2 + β s 2 / 24 ) β s / 4 + 1 η ( 1 + η / 2 + β b / 2 ) T D ( ρ f ) ( β s / 4 + 1 ) } 1 / 2 .
Δ t 2 1 / 2 / t f β s 1 ( β s / 4 + 1 ) 1 [ η ( 1 + η / 2 ) / T D ( ρ f ) ] 1 / 2 × { β s ( 1 + β s / 2 + β s 2 / 24 ) η ( 1 + η / 2 ) / T D ( ρ f ) } 1 / 2 ,
( β s / 8 3 ) [ 1 ( 24 / β s ) 2 ] 1 / 2 / ( 1 12 / β s ) .
Δ t 2 1 / 2 / t f 2 1 β f 1 / 2 y a 1 / 4 ( β f y a 1 / 2 + 1 ) 1 .
η γ ρ T 2 4 ϕ ( t ) ct
Δ t 2 1 / 2 / t f = η γ ρ T 2 { n = 1 F n 2 ( n = 1 F n 1 ) 2 / n = 1 F n 0 } 1 / 2 / ( n = 1 F n 0 ) 1 / 2 ,
F n m = ( 1 ) n 1 p n q n m / 2 K m ( q n 1 / 2 ρ T / ρ f ) , m = 0 , 1 , 2 ,
p n = a on cos ( s n ρ b / d ) + c on sin ( s n ρ b / d ) ,
q n = ( γ a ρ f + β f s n 2 / β s 2 ) β f , ρ T = | ρ T | ,
I A ( λ , ρ f , t ) = 2 c η γ β s 2 exp [ γ a ct ( η γ ) 1 λ 2 ct ] × n = 1 s n exp [ η γ β s 2 s n 2 ct ] Res [ I A ( λ , ρ f , s n ) ] ,
Res [ I A ( λ , ρ f , s n ) ] = γ I 0 exp ( W 0 2 λ 2 / 4 ) × [ g A 1 ( λ , s n ) + g A 2 ( λ , s n ) ] / f ( s ) s s = s n ,
a on ( λ ) b on ( λ ) = { [ ± ½ + η 1 ( η γ ) 2 λ 2 ] β s 2 s n 2 } s n 2 / Q n ( λ , s n ) ,
c on ( λ ) = ½ { [ η 1 ( η γ ) 2 λ 2 ] β s 2 3 s n 2 } β s s n / Q n ( λ , s n ) ,
Q ( λ , s n ) = β s ( β s 2 / 4 + β s + s n 2 ) × { s n 2 + [ β s η 1 β s ( η γ ) 2 λ 2 β s 1 s b 2 ] 2 } .
d on = β s 1 s n ( β s 2 / 4 + β s + s n 2 ) 1 [ s n cos ( s n ρ b / d ) + ( β s / 2 ) sin ( s n ρ b / d ) ] exp ( s n 2 W 0 2 / 4 d 2 ) ,
e ( λ ) = η 1 ( η γ ) 2 λ 2 i ( η γ ) 1 λ 1 .
Res [ g B ( λ , ν p ) ] = 2 i 1 c exp ( s p 2 ( λ ) W 0 2 / 4 d 2 ) { s p ( λ ) × cos [ s p ( λ ) ρ b / d ] + ( β s / 2 ) sin [ s p ( λ ) ρ b / d ] } / s p ( λ ) ,
s p ( λ ) = i [ ( γ a + i ν p / c ) η γ + λ 2 ] 1 / 2 d = β s e ( λ ) ,
1 f ( λ , ν p ) = 1 f ( s p ) = n = 1 2 S n ( s p 2 s n 2 ) f ( s ) / s s = sn = n = 1 ( 1 ) n β s s n 2 ( β s 2 + β s + s n 2 ) [ β s 2 e ( λ ) s n 2 ] ,
c γ I 0 exp [ γ a ct ( η γ ) 1 λ 2 ct W 0 2 λ 2 2 / 4 ] n = 1 ( 1 ) n 1 × exp [ η γ β s 2 s p 2 ( λ ) ct ] { s p ( λ ) cos [ s p ( λ ) ρ b / d ] + ( β s / 2 ) sin [ s p ( λ ) ρ b / d ] } × exp [ s p 2 ( λ ) W 0 2 4 d 2 ] s n 2 / { β s ( β s 2 / 4 + β s + s n 2 ) s p ( λ ) [ e ( λ ) β s 2 s n 2 ] } ,
I A ( 0 , ρ f , ν ) = η γ 4 π c 0 dX I p ( ρ f , ν Xi ) × exp ( η γ W 0 2 X / 4 c ) , X = c λ 2 / η γ ,
I p ( ρ f , ν Xi ) γ I 0 g A 1 ( λ = 0 , ν Xi ) / f ( λ = 0 , ν Xi ) ,
( i / ν ) m I p ( ρ f , ν Xi ) = ( / X ) m I p ( ρ f , ν Xi ) ,
( i / ν ) m I A ( 0 , ρ f , ν ) = η γ 4 π c ( i / ν ) m 1 [ I p ( ρ f , ν ) π W 0 2 I A ( 0 , ρ f , ν ) ] η γ 4 π c ( i / ν ) m 1 I p ( ρ f , ν ) , m 1 ,
y = ( η γ ) 1 k λ λ = 0 ,
I p ( ρ f , ν ) = η 1 I 0 g ( y ) / f ( y ) ,
g ( y ) η γ g A 1 ( λ = 0 , ν ) / 2 = [ ( y + η 1 + ½ ) cosh ( β b y ) + ½ ( 3 y + η 1 ) × sinh ( β b y ) / y ] / [ ( y + η 1 ) 2 y ] , f ( y ) f ( λ = 0 , ν ) / 2 = ( y + ¼ ) sinh ( β s y ) / y + cosh ( β s y ) ,
t p = ( c η γ ) 1 { g ( y ) / g ( y ) f ( y ) / f ( y ) } y = y a , = y a γ a / η γ ,
g ( y a ) g ( y a ) = 2 ( y a + η 1 ) 1 ( y a + η 1 ) 2 y a + cosh ( β b y a ) + ½ [ 3 + ( y a + η 1 + ½ ) β b ] sinh ( β b y a ) y a + ½ ( 3 y a + η 1 ) ( sinh ( β b y a ) y a ) ( y a + ½ + η 1 ) cosh ( β b y a ) + ½ ( 3 y a + η 1 ) sinh ( β b y a ) / y a ,
f ( y a ) f ( y a ) = ( β s 2 + 1 ) sinh ( β s y a ) y a + ( y a + ¼ ) ( sinh ( β s y a ) y a ) ( y a + ¼ ) sinh ( β s y a ) / y a + cosh ( β s y a ) ,
T D ( ρ f ) = ( c γ I 0 ) 1 0 dX I p ( ρ f , ν Xi ) ν = 0
T A = t f g ( y a ) β f T D ( ρ f ) f ( y a ) ,
t f = ρ f / c , β f = η γ ρ f , y a = γ a / η γ ,
x = ( η γ ) 1 k λ ν = 0 , X = c λ 2 / η γ ,
T D ( ρ f ) = 2 y a dx x ( x + ½ ) 1 exp ( β s x ) [ ( x ½ x + ½ ) ( x + y t ) 1 exp ( β b x ) ( x y t ) 1 exp ( β b x ) ] / [ 1 ( x ½ x + ½ ) 2 exp ( 2 β s x ) ] , y t = γ t / η α .
[ 1 ( x ½ x + ½ ) 2 exp ( 2 β s x ) ] 1 ,
T D ( ρ f ) = 2 [ A 1 1 ( y a + ½ , β 1 ) + A 2 ( y a + ½ ) 1 2 ( y a + ½ , β 1 ) + B 1 ( y a + y t , β 1 ) C 1 ( y a + ½ , β 2 ) D 1 ( y a y t , β 2 ) ,
1 , 2 ( x 0 + α , β ) exp ( α β ) E 1 , 2 [ ( x 0 + α ) β ] , β 1 = β s + β b , β 2 = β s β b = β f ,
A 1 = 1 6 y t ( 2 y t 1 ) 2 , A 2 = 1 2 y t 1 , B = 2 y t ( 2 y t + 1 ) ( 2 y t 1 ) 2 C = 1 2 y t + 1 , D = 2 y t 2 y t + 1 .
T D ( ρ f ) 32 η ( 1 + / 2 ) ( 1 12 / β s ) β s 3 ,

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