126 M. P. Tulin 
that such turbulence is generally much more intense near the ground than at high altitudes. 
I do not think, however, that ever before has a quantitative comparison of gust intensities 
just beneath the surface of the sea, and in the atmosphere been made. In Fig. 3, the hori- 
zontal axis represents the root-mean-square of the vertical velocities (in feet per 
second) either in air or water, and the vertical axis the probability of exceeding a given 
value of root mean square velocity. Shown plotted are curves for the North Atlantic at a 5- 
foot depth averaged the year around and curves for averages for the atmosphere at low alti- 
tudes (0 to 10,000 feet) and moderate altitudes (30 to 50,000 feet). The values for air have 
been taken from Ref. 5. The values for the North Atlantic at 5-foot depth were theoretically 
derived from experimental all-year observations of significant wave heights in the North 
Atlantic made by the U.S. Weather Bureau, and have been taken from Ref. 6. It will be 
observed that a root-mean-square value of 1 foot per second is exceeded only 17 percent of 
the time at moderate altitudes in the atmosphere, 53 percent of the time at low altitudes, and 
89 percent of the time at a depth of 5 feet in the North Atlantic. For an rms value of 3 feet 
per second, the respective percentages are 3, 16, and 28 percent. Only for high-intensity 
gusts (which occur infrequently) is the atmosphere at low altitudes statistically gustier 
than the ocean. The intensity of motions in the sea of course varies with depth of submer- 
gence, and in Fig. 4 this dependence is shown for a sea generated by a 20-foot-per-second 
wind, which about corresponds to a state 3 sea. It is to be noted that the rms vertical 
velocity is about twice as great at the surface as it is at 5-foot submergence and is further 
reduced about 50 percent as the depth increases to 10 feet. 
The motion of the environment induces loads on the vehicle structure and subsequent 
rigid body motions and structural flexing. The seacraft, operating as it does in such close 
proximity to the sea surface must maintain its altitude relative to the instantaneous sea 
surface within very strict limits lest the hull impact or the foils and propeller broach. For 
that reason it is generally desirable for the craft to “follow” or respond to waves which are 
as long or longer than the craft itself. Both surface-piercing and variable-incidence foils 
are designed with such response in mind. Of course, the actual response of the vehicle to 
a certain sea will particularly depend on the frequency with which it encounters the waves 
that compose that sea. It so happens that the spectrum of a real, fully-risen, wind-gen- 
erated sea seems to be sharply peaked, most of the energy residing in waves traveling with 
a celerity close to the wind speed and with corresponding lengths [7]. The consequence of 
this fact is that a boat traversing such a sea will sense a dominant frequency of encounter, 
which will depend on the wind speed that generated the sea, the boat’s speed, and its direc- 
tion relative to the wind. In Fig. 5 is plotted this frequency of encounter versus boat speed 
from motion into (to the left) and with (to the right) the wind, for wind speeds from 7 to 23.5 
knots, corresponding to sea states from 2 through 5. These curves are based on studies 
made by F’. Turpin and M. Martin [8]. It is to be noted that a response to frequencies, w, 
in excess of 2.5 radians per second or about0.4 cps (averaging the “with” wind and “into” 
wind encounters) is required for a 60-knot boat operating in state 5 sea, and even more 
rapid responses for faster craft. The rms accelerations experienced by a boat responding 
to such a sea are 0.707 w24%p, where ‘Ip is the semiamplitude of the vehicle’s heaving 
motion relative to fixed axes. 
Such accelerations must themselves be limited especially on account of human comfort 
limits. It turns out, in fact, that our tolerance to vertical accelerations is quite small. 
Shown in Fig. 6 are curves based on recent studies of factors influencing hydrofoil boat 
handling qualities by P. Eisenberg [9] which delimit acceleration and frequency ranges for 
which vertical motions are perceptible, uncomfortable, and intolerable. If an rms acceleration 
of 0.15 g is taken as an upper design limit, then in a state 5 sea, Ho could not exceed 
about 6 inches. However, the rms wave semiamplitude in the same sea is about 3 feet and 
the 10 percent highest waves have an rms semiamplitude of 5.4 feet. It would seem that 
