Abstract

The number of specular points reflected in a random Gaussian surface is determined theoretically under the following alternative conditions: (1) when the surface is perfectly long crested (two-dimensional); (2) when the surface is three-dimensional but isotropic; (3) for quite general surfaces, provided that the observer and the source of radiation are both at a great distance from the surface.

The results can be applied to the similar problem when the surface forms the boundary of two refracting media.

© 1960 Optical Society of America

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References

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  1. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960).
    [Crossref]
  2. A. H. Schooley, Trans. Am. Geophys. Union 36, 273 (1955).
    [Crossref]
  3. M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A249, 321 (1957).
  4. S. O. Rice, Bell System Tech. J. 23, 282 (1944); Bell System Tech. J. 24, 46 (1945).
    [Crossref]
  5. J. L. Doob, Stochastic Processes (John Wiley & Sons, Inc., New York, 1953).
  6. M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. (to be published).

1960 (1)

1957 (1)

M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A249, 321 (1957).

1955 (1)

A. H. Schooley, Trans. Am. Geophys. Union 36, 273 (1955).
[Crossref]

1944 (1)

S. O. Rice, Bell System Tech. J. 23, 282 (1944); Bell System Tech. J. 24, 46 (1945).
[Crossref]

Doob, J. L.

J. L. Doob, Stochastic Processes (John Wiley & Sons, Inc., New York, 1953).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960).
[Crossref]

M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A249, 321 (1957).

M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. (to be published).

Rice, S. O.

S. O. Rice, Bell System Tech. J. 23, 282 (1944); Bell System Tech. J. 24, 46 (1945).
[Crossref]

Schooley, A. H.

A. H. Schooley, Trans. Am. Geophys. Union 36, 273 (1955).
[Crossref]

Bell System Tech. J. (1)

S. O. Rice, Bell System Tech. J. 23, 282 (1944); Bell System Tech. J. 24, 46 (1945).
[Crossref]

J. Opt. Soc. Am. (1)

Phil. Trans. Roy. Soc. London (1)

M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A249, 321 (1957).

Trans. Am. Geophys. Union (1)

A. H. Schooley, Trans. Am. Geophys. Union 36, 273 (1955).
[Crossref]

Other (2)

J. L. Doob, Stochastic Processes (John Wiley & Sons, Inc., New York, 1953).

M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. (to be published).

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Figures (2)

Fig. 1
Fig. 1

The mean number of images N, as a function of the parameter α, defined by Eq. (2.15).

Fig. 2
Fig. 2

The mean number of images as a function of A, defined by Eq. (4.6): (a) an isotropic surface, (b) a surface consisting of two long-crested systems at right angles.

Equations (82)

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ζ / x = - κ x ,
κ = 1 2 [ ( 1 / h 1 ) + ( 1 / h 2 ) ] ,
κ = ( μ 1 h 1 + μ 2 h 2 ) / ( μ 2 - μ 1 ) h 1 h 2 .
( 1 / x ) ( ζ / x ) = ξ 1 ,             2 ζ / x 2 = ξ 2 ,
d ξ 1 = ξ 1 / x d x = ( 1 / x ) ξ 1 - ξ 2 d x
N x d x = - p ( ξ 1 , ξ 2 ) 1 x ξ 1 - ξ 2 d x d ξ 2 ,
N = - N x d x .
ζ ( x ) = n = 1 n 0 c n cos ( k n x + n ) ,
1 2 c n 2 = E ( k ) d k ,
( ξ i ξ j av ) = ( m 2 / x 2 0 0 m 4 ) ,
m r = 0 E ( k ) k r d r
p ( ξ 1 , ξ 2 ) = x 2 π ( m 2 m 4 ) 1 2 exp [ - 1 2 ( ξ 1 2 x 2 / m 2 + ξ 2 2 / m 4 ) ] .
N = 1 2 π ( m 2 m 4 ) 1 2 - - ξ 2 + κ × exp [ - 1 2 ( ξ 1 2 x 2 / m 2 + ξ 2 2 / m 4 ) ] d x d ξ 2 .
N = ( 2 π ) 1 2 [ α exp ( - 1 2 α - 2 ) + 0 1 / α exp ( - 1 2 z 2 ) d z ] ,
α = m 4 1 2 / κ .
N = 1 ,
N ~ ( 2 / π ) 1 2 α ,
ζ / x = - κ x ,             ζ / y = - κ y ,
( 1 / x ) ( ζ / x ) ,             ( 1 / y ) ( ζ / y ) = ξ 1 , ξ 2 , 2 ζ / x 2 ,             2 ζ / x y ,             2 ζ / y 2 = ξ 3 , ξ 4 , ξ 5 ,
N x y d x d y = - - - p ( ξ 1 , , ξ 5 ) | ( ξ 1 , ξ 2 ) ( x , y ) | × d x d y d ξ 3 d ξ 4 d ξ 5 ,
( ξ 1 , ξ 2 ) ( x , y ) = | ( 1 / x ) ( ξ 3 - ξ 1 ) ( 1 / x ) ξ 4 ( 1 / y ) ξ 4 ( 1 / y ) ( ξ 5 - ξ 2 ) | = 1 x y [ ( ξ 3 + κ ) ( ξ 5 + κ ) - ξ 4 2 ] .
N = - - N x y d x d y .
ζ ( x , y ) = n = 1 n 0 c n cos ( u n x + v n y + n ) ,
1 2 c n 2 = E ( u , v ) d u d v .
( ξ i ξ j av ) = ( m 20 / x 2 m 11 / x y 0 0 0 m 11 / x y m 02 / y 2 0 0 0 0 0 m 40 m 31 m 22 0 0 m 31 m 22 m 13 0 0 m 22 m 13 m 04 ) ,
m p q = 0 0 E ( u , v ) u p v q d u d v .
p ( ξ 1 , , ξ 5 ) = p ( ξ 1 , ξ 2 ) p ( ξ 3 , ξ 4 , ξ 5 ) ,
p ( ξ 1 , ξ 2 ) = x y 2 π ( m 20 m 02 - m 11 2 ) 1 2 × exp [ - m 02 x 2 ξ 1 2 - 2 m 11 x y ξ 1 ξ 2 + m 20 ξ 2 2 2 ( m 20 m 02 - m 11 2 ) 1 2 ] , p ( ξ 3 , ξ 4 , ξ 5 ) = M i j 1 2 ( 2 π ) 3 2 exp [ - 1 2 i , j = 3 , 4 , 5 M i j ξ i ξ j ] ,
( Ξ i j ) = ( m 40 m 31 m 22 m 31 m 22 m 13 m 22 m 13 m 04 ) = ( M i j ) - 1 .
N = M i j 1 2 ( 2 π ) 3 2 κ 2 - - - ( ξ 3 + κ ) ( ξ 5 + κ ) - ξ 4 2 × exp [ - 1 2 i , j = 3 , 4 , 5 M i j ξ i ξ j ] d ξ 3 d ξ 4 d ξ 5 .
ξ i = j = 1 3 a i j η j ,             i = 3 , 4 , 5 ,
M i j ξ i ξ j = η 1 2 + η 2 2 + η 3 2 , ξ 3 ξ 5 - ξ 4 2 = l 1 η 1 2 + l 2 η 2 2 + l 3 η 3 2 .
4 l 3 - 3 H l - Δ = 0 ,
3 H = m 40 m 04 - 4 m 31 m 13 + 3 m 22 2 , Δ = ( Ξ i j ) = ( M i j ) - 1 .
l 1 + l 2 + l 3 = 0 , l 2 l 3 + l 3 l 1 + l 1 l 2 = - 3 4 H , l 1 l 2 l 3 = 1 4 Δ .
( a 31 + a 51 ) 2 + ( a 32 + a 52 ) 2 + ( a 33 + a 53 ) 2 = D , ( a 31 + a 51 ) 2 / l 1 + ( a 32 + a 52 ) 2 / l 2 + ( a 33 + a 53 ) 2 / l 3 = 4 ,
D = m 40 + 2 m 22 + m 04 ,
( ξ 3 ξ 5 - ξ 4 2 ) + κ ( ξ 3 + ξ 5 ) + κ 2 = ( l 1 η 1 2 + l 2 η 2 2 + l 3 η 3 2 ) + κ j = 1 3 ( a 3 j + a 5 j ) η j + κ 2 = j = 1 3 l j { η j + κ ( a 3 j + a 5 j ) / 2 l j } 2 ,
( ξ 3 , ξ 4 , ξ 5 ) / ( η 1 , η 4 , η 5 ) = ( a i j ) = ( M i j ) 1 2 ,
N = 1 ( 2 π ) 3 2 κ 2 - - - j = 1 3 l j ( η j + y j ) 2 × exp [ - 1 2 j = 1 3 η j 2 ] d η 1 d η 2 d η 3
y j = κ ( a 3 j + a 5 j ) / ( 2 l i ) .
1 ( 2 π ) 3 2 κ 2 j = 1 3 l j ( 1 + y j 2 ) ( 2 π ) 3 2 = 1
N + 1 = 2 ( 2 π ) 3 2 κ 2 [ j = 1 3 l j ( η j + y j ) 2 ] × exp [ - 1 2 j = 1 3 η j 2 ] d η 1 d η 2 d η 3 ,
l 1 1 2 ( η 1 + y 1 ) = r , ( - l 2 ) 1 2 ( η 2 + y 2 ) = r sin θ cos ϕ , ( - l 3 ) 1 2 ( η 3 + y 3 ) = r sin θ sin ϕ .
- < r < ,             0 θ π / 2 ,             0 ϕ < 2 π ,
( η 1 , η 2 , η 3 ) / ( r , θ , ϕ ) = ( l 1 l 2 l 3 ) - 1 2 r 2 cos θ sin θ .
N + 1 = 2 1 2 π 3 2 κ 2 Δ 1 2 - d r 0 π / 2 d θ 0 2 π d ϕ r 4 cos 3 θ sin θ × exp [ - 1 2 ( P r 2 + 2 Q r + R ) ] ,
P = l 1 - 1 - l 2 - 1 sin 2 θ cos 2 ϕ - l 3 - 1 sin 2 θ sin 2 ϕ , Q = y 1 l 1 - 1 2 + y 2 ( - l 2 ) - 1 2 sin θ cos ϕ + y 3 ( - l 3 ) - 1 2 sin θ sin ϕ , R = y 1 2 + y 2 2 + y 3 2 .
N + 1 = 2 e - 1 2 R π κ 2 Δ 1 2 0 π / 2 d θ 0 2 π d ϕ cos 3 θ sin θ × ( 3 P 2 + 6 Q 2 P + Q 4 ) P - 9 / 2 exp ( Q 2 / 2 P ) .
H = ( 1 / 16 ) D 2 ,             Δ = ( 1 / 64 ) D 3 .
l 1 , l 2 , l 3 = 1 4 D ,     - 1 8 D ,     - 1 8 D .
( a 31 + a 51 ) 2 = D , ( a 32 + a 52 ) 2 + ( a 33 + a 53 ) 2 = 0 ,
y 1 , y 2 , y 3 = κ l 1 - 1 2 , 0 , 0 ,
P = 4 D - 1 ( 1 + 2 sin 2 θ ) , Q = 4 D - 1 κ , R = 4 D - 1 κ 2 .
D / ( 4 κ 2 ) = A ,
N + 1 = 4 A 0 π / 2 d θ · cos 3 θ sin θ exp [ - sin 2 θ A ( 1 + 2 sin 2 θ ) ] × [ 3 A 2 ( 1 + 2 sin 2 θ ) 2 + 6 A ( 1 + 2 sin 2 θ ) + 1 ] × ( 1 + 2 sin 2 θ ) - 9 / 2 .
1 + 2 sin 2 θ = s - 2
N + 1 = 1 A 1 / 3 1 ( 3 s 2 - 1 ) ( 3 A 2 + 6 A s 2 + s 4 ) × exp [ ( s 2 - 1 ) / 2 A ] d s ,
N + 1 = 2 + ( 2 A / 3 ) e - ( 1 3 A - 1 ) ,
N = 1 + ( 2 A / 3 ) e - ( 1 3 A - 1 ) .
( 2 ζ x 2 + 2 ζ y 2 ) 2 av = ( ξ 3 + ξ 5 ) 2 av = ξ 3 2 av + 2 ξ 3 ξ 5 av + ξ 5 2 av = m 40 + 2 m 22 + m 04
N ~ 1
N ~ 2 A / 3 ,
ζ ( x , y ) = ζ 1 ( x ) + ζ 2 ( y )
ζ 1 / x = - κ x ,             ζ 2 / y = - κ y .
N = N ( 1 ) N ( 2 ) ,
N ( i ) = ( 2 π ) 1 2 [ α i exp [ - ( 1 2 α i - 2 ) ] + 0 1 / α i exp ( - 1 2 z 2 ) d z ]
α 1 = m 40 1 2 / κ ,             α 2 = m 04 1 2 / κ .
α 1 = α 2 = ( 1 2 D ) 1 2 / κ = ( 2 A ) 1 2 ,
N = 2 π [ ( 2 A ) 1 2 e - ( 1 4 A - 1 ) + 0 1 / ( 2 A ) 1 2 exp ( - 1 2 z 2 ) d z ] .
N ~ 4 A / π .
N ~ 2 ( 2 π ) 3 2 κ 2 [ j = 1 l j η j 2 ] × exp [ - 1 2 j = 1 3 η j 2 ] d η 1 d η 2 d η 3 .
N ~ 4 π κ 2 ( l 2 l 3 ) 1 2 [ ( l 2 - l 1 l 2 ) 1 2 E ( k ) - ( l 2 l 2 - l 1 ) 1 2 F ( k ) ] ,
E ( k ) = 0 π / 2 ( 1 - k 2 sin 2 ϕ ) 1 2 d ϕ F ( k ) = 0 π / 2 ( 1 - k 2 sin 2 ϕ ) - 1 2 d ϕ
k 2 = l 1 ( l 3 - l 2 ) / l 3 ( l 1 - l 2 ) .
N ~ ( A / π ) ( l 1 / D ) Φ ( - l 2 / l 1 ) ,
Φ ( ρ ) { ρ ( 1 - ρ ) } 1 2 [ ( 1 + ρ ρ ) 1 2 E ( k ) - ( ρ 1 + ρ ) 1 2 F ( k ) ] , ρ = - l 2 / l 1 ,             k 2 = ( 1 - 2 ρ ) / ( 1 - ρ 2 ) .
H = 1 1 2 D 2 sin 2 θ 0 ,             Δ = 0 ,
l 1 , l 2 , l 3 = 1 4 D sin θ 0 , 0 ,     - 1 4 sin θ 0 .
N ~ ( 4 A / π ) sin θ 0 .
N ~ D / 2 3 κ 2 = 2 A / 3
sin - 1 ( π / 2 3 ) = 66 ° 3 0 .