Abstract

Specular-point flow on a stationary wave-like random surface with respect to a reference frame moving with a source–observer is described. The time rate of change of specular point densities and the differential changes caused by creations and annihilations are used to calculate the divergence of the mean flow field. The divergence has been integrated, and both mean flow and mean velocity fields have been calculated. The flow pattern is particularly complex when the source–observer motion includes a vertical component. In such a case, there is a small but nonzero vertical component of flow, and specular points at extremal surface heights may flow in opposite directions to the major flow which occurs near the mean surface plane.

© 1973 Optical Society of America

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References

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  1. M. S. Longuet-Higgins, Philos. Trans. R. Soc. Lond. A 249, 321 (1957).
    [Crossref]
  2. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960).
    [Crossref]
  3. R. D. Kodis, IEEE Trans. Antennas Propag. 14, 77 (1966).
    [Crossref]
  4. D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
    [Crossref]
  5. E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 88.
  6. M. Loeve, Probability Theory (Van Nostrand, Princeton, N.J., 1963), p. 469.

1968 (1)

D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
[Crossref]

1966 (1)

R. D. Kodis, IEEE Trans. Antennas Propag. 14, 77 (1966).
[Crossref]

1960 (1)

1957 (1)

M. S. Longuet-Higgins, Philos. Trans. R. Soc. Lond. A 249, 321 (1957).
[Crossref]

Barrick, D. E.

D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
[Crossref]

Kodis, R. D.

R. D. Kodis, IEEE Trans. Antennas Propag. 14, 77 (1966).
[Crossref]

Loeve, M.

M. Loeve, Probability Theory (Van Nostrand, Princeton, N.J., 1963), p. 469.

Longuet-Higgins, M. S.

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

M. S. Longuet-Higgins, Philos. Trans. R. Soc. Lond. A 249, 321 (1957).
[Crossref]

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 88.

IEEE Trans. Antennas Propag. (2)

R. D. Kodis, IEEE Trans. Antennas Propag. 14, 77 (1966).
[Crossref]

D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
[Crossref]

J. Opt. Soc. Am. (1)

Philos. Trans. R. Soc. Lond. A (1)

M. S. Longuet-Higgins, Philos. Trans. R. Soc. Lond. A 249, 321 (1957).
[Crossref]

Other (2)

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 88.

M. Loeve, Probability Theory (Van Nostrand, Princeton, N.J., 1963), p. 469.

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Figures (8)

Fig. 1
Fig. 1

Reference frame moving with source–observer so that the x axis lies in the mean surface plane and the origin falls immediately below the source–observer. Figure depicts a specular point on the surface with normal n at coordinates (x, z) and at an angle of incidence γ with respect to the source–observer.

Fig. 2
Fig. 2

Average number of specular points per unit length per unit height for various observer altitudes (rms slope = tan10°, incidence angle = 0°).

Fig. 3
Fig. 3

Average number of specular-point creations and annihilations per unit length, per unit height, per second for a random corrugated surface (observer altitude = 500 m, rms slope = tan10°, υx = 1000 m/s, υz = 0, z = 0).

Fig. 4
Fig. 4

Average number of specular-point creations and annihilations per unit length, per unit height, per second for a random corrugated surface (observer altitude = 500 m, rms slope = tan10°, υx = 0, υz = −1000 m/s, z = 0).

Fig. 5
Fig. 5

Mean-flow field for specular points on a random corrugated surface (observer altitude = 200 m, rms surface slope = tan10°, υx = 1000 m/s, υz = 0, maximum vector length depicted represents flow of 7.34 specular points per meter, per second).

Fig. 6
Fig. 6

Mean-flow field for specular points on a random corrugated surface (observer altitude = 200 m, rms surface slope = tan10°, υx = 0, υz = 1000 m/s, maximum vector length depicted represents flow of 0.234 specular points per meter per second).

Fig. 7
Fig. 7

Mean-velocity field for specular points on a random corrugated surface (observer altitude = 200 m, rms surface slope = tan10°, υz = 1000 m/s, υz = 0, maximum vector length depicted represents velocity of 1385 m/s).

Fig. 8
Fig. 8

Mean-velocity field for specular points on a random corrugated surface (observer altitude = 200 m, rms surface slope = tan10°, υx = 0, υz = 1000 m/s, maximum vector length depicted represents velocity of 218 m/s).

Equations (58)

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Γ ( x ) = h 2 exp ( x 2 / l 2 ) .
f ξ 0 , , ξ n ( x 0 , , x n ) = 1 ( 2 π ) n / 2 1 | K | 1 2 exp { 1 2 j , k = 0 n x j K j k x k } ,
K j k = ( 1 ) k d j + k d x j + k Γ ( x ) | x = 0 j , k = 0 , 1 , .
[ K j k / h 2 ] = ( 1 0 2 / l 2 0 0 2 / l 2 0 12 / l 4 2 / l 2 0 12 / l 4 0 0 12 / l 4 0 120 / l 6 ) .
z = ζ ( x , t )
ζ x = x / ( H 0 z ) x / H 0 ;
f ( x , t ) = ζ ( x , t ) x 2 / 2 H 0 .
P 1 ( x , z ) = d x d z × 1 / H 0 ( 1 H 0 z x x ) f ζ , ζ x , ζ x x ( z , x / H 0 , z x x ) d z x x .
P 2 ( x , z ) = d x d z × 1 / H 0 ( 1 H 0 z x x ) f ζ , ζ x , ζ x x ( z , x / H 0 , z x x ) d z x x ,
f ζ , ζ x , ζ x x ( z , x / H 0 , z x x ) = f ζ x ( x / H 0 ) · f ζ , ζ x x ( z , z x x ) = l 3 ( 4 π ) 3 2 2 h 3 exp ( l 2 x 2 4 h 2 H 0 2 ) · exp ( 3 z 2 4 h 2 l 2 z z x x 4 h 2 l 4 z x x 2 16 h 2 ) .
ν i ( x , z ) = P i ( x z ) / d x d z , i = 1 , 2
ν ( x , z ) = i = 1 2 ν i ( x , z ) .
ν 1 ( α , β ) = exp ( 6 β 2 z 2 / 2 h 2 ) 2 π 3 2 h t [ e α 2 + π α erfc ( α ) ] ,
ν 2 ( α , β ) = ν 1 ( α , β ) ,
ν ( α , β ) = 2 exp ( 6 β 2 z 2 / 2 h 2 ) π 3 2 h l [ e α 2 + π α erfc ] ,
α = l 2 4 h H 0 + z 2 h ,
β = l x 2 6 h H 0 ;
N L ( β , β 0 ) = ν [ α ( z ) , β ] d z = 6 π l exp ( 6 β 2 ) [ e β 0 2 + π β 0 erf β 0 ] ,
β 0 = l 2 / 2 6 h H 0 .
N T ( β 0 ) = N L [ β ( x ) , β 0 ] d x = 2 ( 6 π ) 1 2 h H 0 l 2 [ e β 0 2 + π β 0 erf β 0 ] .
z = ζ ( x + υ x t ) .
H 0 ( t ) = H 0 ( 0 ) + υ z t .
f x = 0 , f x x = 0
P t w = d x d t d z [ | ( f x , f x x ) ( x , t ) | × f ζ , ζ x , ζ x x , ζ x t , ζ x x x ( z , z x , z x x , z x t , z x x x ) ] d ( f x t ) d ( f x x x ) , f x = f x x = 0 .
z x = x / H 0 , z x x = 1 / H 0 ,
f x x x = z x x x , f x t = 1 H 0 ( υ x + υ z x / H 0 ) ,
P t w = d x d t d z H 0 | υ x + υ z x / H 0 | | z x x x | × f ζ , ζ x , ζ x x , ζ x x x ( z , x / H 0 , 1 / H 0 , z x x x ) d z x x x .
f x t · f x x x < 0 ,
f x t · f x x x > 0 ,
Z + = 2 ( υ x + υ z x / H 0 ) H 0 × 0 z x x x f ζ , ζ x , ζ x x , ζ x x x ( z , x / H 0 , 1 / H 0 , z x x x ) d z x x x ,
Z = 2 ( υ x + υ z x / H 0 ) H 0 × 0 z x x x f ζ , ζ x , ζ x x , ζ x x x ( z , x / H 0 , 1 / H 0 , z x x x ) d z x x x .
| Z + | = n CR if υ x + υ z x / H 0 < 0 , n AN if υ x + υ z x / H 0 > 0 ,
| Z | = n CR if υ x + υ z x / H 0 > 0 , n AN if υ x + υ z x / H 0 < 0 .
δ = Z + Z ,
f , ζ x , ζ x x , ζ x x x ( z , x / H 0 , 1 / H 0 , z x x x ) = l 6 64 3 π 2 h 4 exp ( 5 l 2 x 2 8 h 2 H 0 2 + l 4 x z x x x 8 h 2 H 0 + l 6 z x x x 2 96 h 2 + 3 z 2 4 h 2 + l 2 z 4 h 2 H 0 + l 4 16 h 2 H 0 2 ) .
Z + = 3 2 π 2 h 2 H 0 ( υ x + υ z x / H 0 ) exp ( 15 β 2 + α 2 + z 2 / 2 h 2 ) × [ 1 3 π β e 9 β 2 erfc ( 3 β ) ] ,
Z = 3 2 π 2 h 2 H 0 ( υ x + υ z x / H 0 ) exp ( 15 β 2 + α 2 + z 2 / 2 h 2 ) × [ 1 + 3 π β e 9 β 2 erfc ( 3 β ) ] ,
δ = 3 2 l x 4 π 3 2 h 3 H 0 ( υ x + υ z x / H 0 ) exp ( 6 β 2 + α 2 + z 2 / 2 h 2 ) .
Δ ( x ) = δ ( x , z ) d z = ( υ x + υ z x / H 0 ) 6 l x 2 π h 2 H 0 2 exp ( 6 β 2 + β 0 2 ) .
Δ T = Δ ( x ) d x = 2 6 h υ z π l 2 e β 0 2 .
· q + ν t δ = 0 ,
ν t = l 2 π 3 2 h 2 H 0 2 [ π 2 erf α x 2 h H 0 ( e α 2 + π α erf α ) ] × exp ( 6 β 2 + z 2 / 2 h 2 ) .
· q = x l 2 2 π 3 2 h 3 H 0 2 exp ( 6 β 2 + α 2 + z 2 / 2 h 2 ) ( 3 υ x + x υ z / H 0 ) + l υ z 2 π h 3 H 0 3 erf α exp ( 6 β 2 + z 2 / 2 h 2 ) × ( h H 0 / 2 α x 2 ) .
d U d x + N L t Δ ( x ) = 0 ,
U ( x ) = u ( x , z ) d z
d U d x = 6 l x υ x 2 π h 2 H 0 2 exp ( 6 β 2 + β 0 2 ) l υ z π H 0 2 h erf α exp ( 6 β 2 ) ( 1 12 β 2 ) .
d W d z = υ z e z 2 / 2 h 3 2 π l 2 ( e α 2 + π z h erf α ) ,
W ( z ) = w ( x , z ) d x ,
· q d x = u ( x , z ) + w z d x + C ,
· q d x = [ x υ z 2 π 3 2 h H 0 l exp ( 6 β 2 + z 2 / 2 h 2 ) ( 2 π α erf α 1 ) 3 υ x 2 π 3 2 h l exp ( 6 β 2 + α 2 + z 2 / 2 h 2 ) ] + [ υ z 2 π l 2 exp ( α 2 + z 2 / 2 h 2 ) · erf ( 6 β ) ( 1 3 π h e α 2 erf α ) ] .
u ( x , z ) = x υ z 2 π 3 2 h H 0 l exp ( 6 β 2 + z 2 / 2 h 2 ) ( 2 π α erf α 1 ) 3 υ x 2 π 3 2 h l exp ( 6 β 2 + α 2 + z 2 / 2 h 2 ) + u ( z ) ,
w ( x , z ) = υ z 2 π l H 0 erf α exp ( 6 β 2 + z 2 / 2 h 2 ) + w ( z ) .
U ( x ) = x l υ z π H 0 2 h e 6 β 2 erf β 0 6 υ x π l exp ( 6 β 2 + β 0 2 ) + U 0 ,
W ( z ) = h υ z ( 2 π ) 1 2 l 2 e z 3 / 2 h 3 erf α + W 0 ,
x ˙ p = x υ z 2 H 0 ( 2 π α erf α 1 π α erf α e α 2 ) 3 2 υ x ( e α 2 e α 2 + π α erf α ) ,
z ˙ p = π h υ z 2 H 0 ( erf α e α 2 + π α erf α ) ,
V s p = x ˙ p e x + z ˙ p e z .
lim H 0 X ˙ p = υ x ,