III. _ SQUARE FLUCTUATION OF THE WAVE FRONT 
DIRECTION (BEARING FLUCTUATION). 
if we consider the phase at two points 
(L,r/2,0) and (L,-r/2,0) where r<<L then the 
angle @ made by the normal to the intersection 
of the wave front at (L,0,0) and the plane 
z = 0 and the x axis is given approximately by 
eos 
$(L,r/2,0) - S)(L,-¥/2,0) 
<r rk 
(11) 
Similarly, the angle made in the plane y = 0 
is given approximately by 
S,(L,0,r/2) - S,(L,0,-r/2) 
rk 
(12) 
~ 
a 
If the angles ® and ® are small, then the 
angle @ between the “normal to the wave front 
at(L,0,0) and the x axis is given by the 
relationship 
2 
2 2 
 ~ a (13) 
Squaring (11) and (12), substituting in 
(13), averaging the sum and noting that 
[s, (t,+/2,0)]” = [s, (L,-r/2,0)] ° = 
[s,(1,0,2/2)]” = [s,(1,0,-r/2)] ° 
and 
S, (L,r/2,0) S¢(L,-2/2,0) = 
S, (L,0,r/2)S¢(L,0,-r/2) 
(13) becomes 
ro 33 [s, (L,r/2,0)] ° = 
Dt 
S, (L,r/2,0) S¢(L,-r/2,0) - (14) 
If r is very small, then 
[s,(1,+/2,0)] % [5,(1,0,0)]® - 2,(0) 
whence (14) becomes 
[zs(0) ~ ag(+)] 6 
As r->0 (15) becomes an equality or 
[asco = a.(=)| 0 
Using equation (6), we may write (16) as 
“2 4 
Cx ss (15) 
r-k- 
“2 lim 4 
(16) 
r>0 rk? 
33 
#255 wetfc nf a] 
2 
x cone [step] dx. (17) 
Carrying out the limiting operation we get 
oe. 47 2 Jace nn fe cos °[ xe) ax. 
(18) 
Evaluating the inner integral and making the 
substitution n = K/K_ where K_ is defined 
above, (18) becomes P 
uli J { 4 
+ Looe firrn®]c(n)+ein[ 3770 *Js(0] 
ae, [eca[s77a]s(n)-nin [srrn] ein] 
X n° gy(n) m do (19) 
where C(n) and S(n) are the Fresnel integrals. 
IV. EFFECTS OF THE COMPOSITION OF THE INDEX 
OF REFRACTION SPECTRUM (THERMAL SPECTRUM) . 
From the, theory of isotropic-turbulent 
scalar fields the thermal spectrum, or the 
index of refraction spectrum, can be repre= 
sented approximately by a Von Karman type 
interpolation formula, or 
stays 5[7(5/6) 
ai enc 2) P(1/3)K° o[t+(aK, > /x? ay D) 
(20) 
where n, is the wave number of the smallest size 
component of the refractive index fluctuations 
in terms of the process wave number and <cis the 
rms fluctuation of the index of refraction 
variations. Fig.1 shows the plot of the spectrum. 
Although the spectrum of the thermal 
variations in the sea may not correspond exactly 
to that of an isotropic-turbulent sealar field, 
the use of the above approximation should result 
in reasonably good qualitative relationships 
between the thermal spectrum and the spectra 
of the phase and amplitude fluctuations, especi- 
ally over the small size components. In any 
event, should the spectrum be considerably 
