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1877 л Ё 1826, Daniel Colladon, Charles Sturm, , 1687, 1490, : л... , , ...Ё


r r p r v r r r r + g + v + (v, )v = - t r + div( v ) = 0 t
r v


r r + + div v 3


r r v p r + g =- t / r + div v = 0 : rrr r r t v = v 0 + v v 0 = 0 v - r p = p 0 + p p << p
p
0 0

r =g

0 0

= 0 +

<<


r v

t r v

t r r p 0 p r =- - +g t 0 0

=-

r p

r +g

r v

=-

r (p 0 + p) 0 +

r +g

r -g

r v t

=-

r p
0




t (0 + ) t t

r + div v = 0 r + div (0 + ) v = 0

r r + div (0 v + v) = 0
t r + 0 div v = 0


r r r v p (F) =- = t t 0 r 1 p + + 0 div v = 0 2 c t t

-

r p
0

r 0 div F = 0


p = c 2 p = s


r r v = F


r (F) t

=-

r p
0



F t

=-

p
0

r 1 p + 0 div F = 0 2 c t F
F
2



r r v = F
p = - F
0

t

2

- c F = 0
2

t


c

(1)
2

c = 1.26 340 430 / c
R= -1



p = 0
6 2

c = 1000 1500 1.5 10 / c
R=1



w = 0
6 2

c = 3000 4000 12 10 / c




(2)

c = 1.26 340 430 / c
2



R= -1

p = 0
6 2

c = 1000 1500 1.5 10 / c
R<1


2

w 1 = w 2 , p1 = p
6 2

c = 3000 4000 12 10 / c




(3)

c = 1.26 340 430 / c
2





c = 1000 1500 1.5 10 / c
6 2





c = 3000 4000 12 10 / c
6 2




T, S z()

1480 / c T >0

<

c

< 1545 / c p >0









()
Deep Sound Channel SOFAR Channel (Sound Fixing and Ranging)
~1000


()


William Maurice Ewing 1906 н 1974
american geophysicist and oceanographer


1917-2005
, ,


2-










p R T p 340 / c = CV І s
q T=const q ()










z
u ~ lnz








()
q , q q q


q q q


q , q q q



()

q ,


,


c 0 f ( )



c = 0 / n = f ( )
1.00029 1.33




(z)

n (z) sin ( z) = const

n(z)
z




n -1 1
2 2

n +2 n = 1 + n, n << 1 n ( z) = 0 e = const л Ё

=r





-z / H


n (z) = 1 + (n 0 - 1)e

-z / H



z z



n 0 sin 0 = 1 sin
1



0

n0

n









0.5










-

, , ( ) , н ,

*** , *** ,






Etruscan Vase Sunset
The mirage is produced by light refracted by a low lying warm air layer. by Michael Myers at Cape Hatteras, North Carolina.


,


(1729 .)

dI = - I dx

I( x ) = I 0 e



- x

Pierre Bouguer 1698-1758
French mathematician, geophysicist, geodesist, and astronomer



= ( )




3

: q " ": 0.3-1.3 ( ) q 1.5-1.8 ( ) q 2.0-2.6 ( ) q 7.0-15.0 ( ) q 0.5 10 (. )




I( x ) = I 0 e = ( )

- x


()






=0.27 -1 =0.023 -1 =0.005 -1
0

100


0

1000


:
1. ( ) 2. (лЁ)

()

D>>

()

D<<









I()


1 2 I ~ 4 1 + cos

[

]






I~

1
4


I( x ) = I 0 e


- x

=+



(=0.55 )


,


-1

,

-1

, -1

0.07 0

0.16 2§10-4

0.23 2§10-4





0.023 < < 3.3

-1


,
:
1. 2.


л-Ё

R f ( )
q нл Ё q


,






R=1


sin =z

sin =z/n

z





z

R>>


+ (-2 )+(- )= =3 - = -(- ) = 4 - 2

-

z



-






z



= -6+2


=-6 +2

= 4 -2








,








21.7

o


(Sundogs)

diamond dust





( )


Atmospheric Optics
http://www.atoptics.co.uk/