550 Edward V. Lewis and John P. Breslin 
Froude numbers, the surface acts as a rigid ceiling, which causes a suction or attractive 
force and virtually zero moment. This attractive force is considered negative because the 
positive direction is downward. Because this attractive force becomes less negative as Z 
increases, Z, is positive at low Froude numbers. At high Froude numbers, the reverse is 
true, that is, the force is one of repulsion and Z, is negative. The force rate changes sign 
at a Froude number of F = U\/gl = 0.55. Again, with increasing Froude number the moment 
changes from a small bow-up moment (M, < 0) at low Froude numbers, to a bow-down moment 
(M, > 0) for all Froude numbers greater than about 0.33. 
Physically, the vertical attractive force at low Froude numbers tends to be destabilizing 
because any excursions toward the free surface will cause this force to increase. At high 
Froude numbers, the repelling effect is stabilizing. If we consider the effect of M, in the 
absence of the force, then the moment is destabilizing at low Froude numbers and stabilizing 
at high Froude numbers in regard to simple exponential divergence. However, large values 
of M, can lead to oscillatory divergence or instability. 
The magnitudes of Z, and M, at 30 knots (U//gl = 0.62) are significantly different from 
their values at 40 and 60 knots so that we may expect, and do find, a considerable difference 
in the stability or the fin area required for stability over this speed range. Although, other 
derivatives can be expected to be Froude-number dependent, such as M,,, Mz, M;, Z,» and 
Z.;, they have very weak variations with Froude number at the submergence ratio and over 
the Froude numbers considered. 
Exponential Instability 
To establish stability criteria, consideration must be given to the requirement that each 
of the coefficients a, b, c, d, and e of the quartic equation in o be positive. Equation (12) 
shows that a and b are positive; Z,, and M, are both negative. Figure 12 is a graph of b as 
a function of tail area; note that b is large. 
The coefficient c is composed of three terms; the last term is found to be small and, be- 
cause the two remaining terms are composed of those derivatives which are considered to be 
independent of Froude number, the relation for stability arising from c > 0, that is, 
MZ, - M,(m+Z,) > 0, (13) 
is exactly that required for the body when operating deeply submerged. Figure 13 is a graph 
of c as a function of fin area; a fin area of 220 ft? is the minimum required to prevent ex- 
ponential divergence. In computing the changes in the various coefficients as a function of 
fin or tail area, a conservative lift rate of 2.0 was chosen to take into account the attrition 
expected from partial cavitation. 
An inspection of the equation 
i 1 
d = i, (1,2. ) H,2, (14) 
y 
shows it to be positive for high Froude numbers but negative for those ranges of low Froude 
numbers where M, is negative. Thus, this term can cause exponential divergence at low 
