Abstract

The radiance N, resulting from an initially narrow collimated beam, propagating in the ocean, is investigated with the aid of the small-angle-scattering theory of electron scattering. The resulting expression for N is calculated approximately and compared both with experiments and numerical calculations. Various properties of the approximate expression for N are explored; an observer, located off the axis of the beam at an axial distance z from the source, who attempts to determine the direction to the source by seeking the maximum of N with respect to ray direction, would sight a point on the beam axis at a distance z/3 from the source.

© 1972 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. G. Jerlov, Optical Oceanography (Elsevier, Amsterdam, 1968).
  2. D. M. Bravo-Zhivotovsky, I. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, Izv. Atmos. Ocean. Phys. 5, 83 (1969).
  3. D. Bauer and A. Morel, Ann. Geophysique 23, 122 (1967).
  4. H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
    [CrossRef]
  5. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).
  6. S. A. Makarevich, A. P. Ivanov, and G. K. Il’ich, Izv. Atmos. Ocean. Phys. 5, 40 (1969).
  7. L. M. Romanova, Izv. Atmos. Ocean. Phys. 4, 679 (1968).
  8. E. Fermi, Nuclear Physics (Univ. Chicago Press, Chicago, 1950).

1969 (2)

D. M. Bravo-Zhivotovsky, I. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, Izv. Atmos. Ocean. Phys. 5, 83 (1969).

S. A. Makarevich, A. P. Ivanov, and G. K. Il’ich, Izv. Atmos. Ocean. Phys. 5, 40 (1969).

1968 (1)

L. M. Romanova, Izv. Atmos. Ocean. Phys. 4, 679 (1968).

1967 (1)

D. Bauer and A. Morel, Ann. Geophysique 23, 122 (1967).

1949 (1)

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[CrossRef]

Bauer, D.

D. Bauer and A. Morel, Ann. Geophysique 23, 122 (1967).

Bravo-Zhivotovsky, D. M.

D. M. Bravo-Zhivotovsky, I. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, Izv. Atmos. Ocean. Phys. 5, 83 (1969).

Dolin, I. S.

D. M. Bravo-Zhivotovsky, I. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, Izv. Atmos. Ocean. Phys. 5, 83 (1969).

Fermi, E.

E. Fermi, Nuclear Physics (Univ. Chicago Press, Chicago, 1950).

Il’ich, G. K.

S. A. Makarevich, A. P. Ivanov, and G. K. Il’ich, Izv. Atmos. Ocean. Phys. 5, 40 (1969).

Ivanov, A. P.

S. A. Makarevich, A. P. Ivanov, and G. K. Il’ich, Izv. Atmos. Ocean. Phys. 5, 40 (1969).

Jerlov, N. G.

N. G. Jerlov, Optical Oceanography (Elsevier, Amsterdam, 1968).

Luchinin, A. G.

D. M. Bravo-Zhivotovsky, I. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, Izv. Atmos. Ocean. Phys. 5, 83 (1969).

Makarevich, S. A.

S. A. Makarevich, A. P. Ivanov, and G. K. Il’ich, Izv. Atmos. Ocean. Phys. 5, 40 (1969).

Morel, A.

D. Bauer and A. Morel, Ann. Geophysique 23, 122 (1967).

Romanova, L. M.

L. M. Romanova, Izv. Atmos. Ocean. Phys. 4, 679 (1968).

Savel’yev, V. A.

D. M. Bravo-Zhivotovsky, I. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, Izv. Atmos. Ocean. Phys. 5, 83 (1969).

Scott, W. T.

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[CrossRef]

Snyder, H. S.

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

Ann. Geophysique (1)

D. Bauer and A. Morel, Ann. Geophysique 23, 122 (1967).

Izv. Atmos. Ocean. Phys. (3)

D. M. Bravo-Zhivotovsky, I. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, Izv. Atmos. Ocean. Phys. 5, 83 (1969).

S. A. Makarevich, A. P. Ivanov, and G. K. Il’ich, Izv. Atmos. Ocean. Phys. 5, 40 (1969).

L. M. Romanova, Izv. Atmos. Ocean. Phys. 4, 679 (1968).

Phys. Rev. (1)

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[CrossRef]

Other (3)

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

N. G. Jerlov, Optical Oceanography (Elsevier, Amsterdam, 1968).

E. Fermi, Nuclear Physics (Univ. Chicago Press, Chicago, 1950).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

F. 1
F. 1

For a source at (0,0) and a beam axis 0-z, an observer at (r,z) sees an apparent source at (0,z0).

F. 2
F. 2

Log10(P/F) approximated here (○-○-○), and calculated numerically (Ref. 2) (——) vs γθ for various σz.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

Σ ( θ ) = γ 2 π θ e γ θ , γ 10
σ z 1 , γ 1 .
1 c d N d t = ( 1 c t + z + u r ) N ( u | r , z , t ) = ( σ + α ) N + σ d 2 u Σ ( | u u | ) N ( u | r , z , t ) + N 0 ( u | r , t ) δ ( z ) ,
N 0 ( u | r , t ) = f 0 ( u | r ) g ( t ) ,
N ( u | r , z , t ) = f ( u | r , z ) g ( t z / c )
( z + u r ) f = ( α + σ ) f + σ d 2 u Σ ( | u u | ) f ( u | r , z ) + f 0 δ ( z ) .
F ( p | k , z ) = d 2 r d 2 u e i ( k r + p u ) f ( u | r , z ) ,
Q ( p ) = d 2 u e i p u Σ ( u ) = 2 π 0 d u u J 0 ( p u ) Σ ( u ) .
( z k p ) F = ( σ Q σ α ) F + F 0 δ ( z ) ,
f ( u | r , z ) = ( 2 π ) 4 e α z d 2 k d 2 p F 0 ( p + z k | k ) G e i ( p u + k r ) , G = exp [ σ z + σ 0 z d ξ Q ( p + ξ k ) ] .
Q ( p ) = γ ( p 2 + γ 2 ) 1 2 .
( p + ξ k ) 2 γ 2 1
f 0 ( u | r ) = ( π u 0 r 0 ) 2 exp ( u 2 / u 0 2 r 2 / r 0 2 ) .
f = ( π U r R 0 ) 2 exp [ α z ( r r m ) 2 / R 0 2 u r 2 / U r 2 u ϕ 2 / U ϕ 2 ]
= ( π U ϕ R 1 ) 2 exp { α z [ ( u r u m ) 2 + u ϕ 2 ] / U ϕ 2 r 2 / R 1 2 } ,
l = ( r 0 / z ) 2 γ 2 / σ z , V = u 0 2 γ 2 / σ z , R 0 2 = σ z 3 γ 2 1 + 2 υ + 6 l + 3 l V 3 ( 2 + V ) , R 1 2 = σ z 3 γ 2 2 + 3 ( l + V ) 3 , U ϕ 2 = σ z γ 2 1 + 2 υ + 6 l + 3 l V 2 + 3 ( l + V ) , U r 2 = σ z γ 2 ( 2 + V ) , u m = 3 ( 1 + V ) 2 + 3 ( l + V ) ( r z ) , r m = 1 + V 2 + V z U r .
z 0 = z r u m = 1 3 l 1 + V ( z 3 )
h ( r , z , t ) 4 π d Ω N ( u | r , z , t ) d 2 u N g e α z ( π R 1 2 ) 1 exp ( r 2 / R 1 2 ) .
F ( z , t ) d 2 r h = g e α z .
r 2 = F 1 d 2 r r 2 h = 2 σ z 3 / 3 γ 2 .
r 2 = σ z 3 λ 2 / 3 a 0 2
γ a 0 / λ .
E ( r , z , t ) d Ω N ( u z ̂ ) h .
P ( a , z , t ) 2 π 0 a d r r E ( r , z , t ) = F ( 1 e a 2 / R 1 2 ) ,
P / F = γ θ 0 [ x + ( 1 + x 2 ) 1 2 ] σ z / x J 0 ( γ θ x ) ,