Appendix A 
HEAVE RESPONSE OF AN ELEVATED PLATFORM 
by H. S. Zwibel 
In order to assist in the initial elimination of unsuitable MOBS 
candidates, computer program HEAVE was written to determine the heave 
response for a raised deck platform. In this appendix, the mathematics 
for the computer program is discussed. 
A general motion of a floating object requires the specification 
of six degrees of freedom. If the floating object is sufficiently 
symmetric and the waves are coming at it from a symmetry direction, 
then these motions become uncoupled. The MOBS elevated platform is 
sufficiently symmetric so that with head-on waves, the heave motion is 
decoupled from the other five degrees of freedom. 
Even with the restriction to head-on seas and a consideration of 
only the heave motion, the exact solution of the problem is extremely 
difficult. As a first step in simplification, the motion is considered 
to be small amplitude and the water inviscid. Fortunately these approx- 
imations do not invalidate the results that will be obtained. The 
general philosophy is that if the candidate is unsuitable for small 
waves it will undoubtably be unsuitable for large waves. Even with the 
small amplitude approximation, the problem is still difficult. Consider 
for a moment the sequence of events for waves of definite frequency 
incident head-on on our floating structure. These incident waves are 
diffracted by the structure giving rise, therefore, to two sets of waves, 
the incident wave and the diffracted wave. The combination of these 
two waves produce forces that set the structure into motion. The motion 
of the floating structures, in turn, generates its own wave trains which 
carry energy away and out to infinity. The general problem, therefore, 
consists of solving for the reflected wave, the radiated waves, and the 
motion of the object. The incident and the reflected waves do not de- 
pend on the motion of the platform; the transmitted waves, however, are 
directly dependent on the amplitude of motion for the platform. These 
radiated waves give rise to vertical forces on the structure that can 
be considered to be broken up into two parts. One part is in phase with 
the acceleration and the other part is in phase with the velocity. The 
former is usually considered to be an added mass effect since it is in 
phase with the object's acceleration. The latter is a damping force 
since damping forces are proportional to the velocity. A rigorous 
solution to the problem, therefore, would be to calculate for given 
incident waves, the reflected waves and for a given vertical motion of 
the object, the radiated waves. Once these radiated waves are determined 
the added mass force and damping force could be calculated. In terms 
of these parameters and assuming steady-state harmonic motion, the 
following equation is satisfied by the heave motion Z(t) for the platform. 
