It is known that the transverse metacentric height of a ship, when trimmed or on the wave crest, is reduced, sometimes even becoming negative. Therefore when she is pitching on the wave, she will become unstable instantaneously if the angle of pitch is large. In this paper, the dynamical properties of this transve se instability are treated According to the calculation, the reduction of a metacentric height can only in very small measure be attributed to the reduction of the trimmed water plane area contrary to the general belief, and also the effect of the wave form is not so large for an ordinarily formed ship, assuming the standard wave adopted for strength calculation, but the reduction should be considered to be mainly owing to the rise of the centre of gravity of the ship due to trim. Therefore the reduction is large as the longitudinal metacentric height is large and the angle of trim increases. The rolling motion of such a ship is called the “quasi-harmonic oscillation”, and has the critical ranges of periods for which the instability of motion occurs, compared with the only one critical value of period for the simple harmonic oscillation. The problem for such a motion can be generally solved by the Hill's method.
In consequence of the executed calculation, the dynamical transverse instability occurs not only when the angle of pitch is so large as the instantaneous negative GM appears, but even when the pitching angle is small, although in less magnitude.
The critical range of instability increases as the angle of pitch increases, and also as the longitudinal metacentric height gets more. This tendency is not much altered, if the effect of the change of water plane area due to the trim and wave form is rejected. The multiplication factor of the amplitude during a single swing is adopted for the difinition of the grade of instability which occurs in the critical period. This grade of instability depends mainly on the ratio of the longitudinal metacentric height to the transverse one. For the ordinary value of this ratio, approximately 200, the grade of instability is nearly two, assuming the max, pitching angle 5° This value may be taken as the standard of the ratio of the two kinds of metacentric height, and if the ratio is less than this standard, the ship will be more stable; if greater, more unstable and the manoeuvering will be more difficult. When a ship rolls, lying transversely on the wave, the motion is less as the metacentric height is less, while, when lying longitudinally on the wave, the motion is more stable as the metacentric height is greater, -this fact may be noteworthy. As the result of the analysis, the lower limit of the transverse metacentric height, from the point of view of the transverse instability due to pitch, is obtained as follows:
M
0_??_C.φ/12C
b•(L/100)
2/d
where 
M0: transverse metacentric height. 
Cb: block coef.
L: length of ship, 
d: draught of ship.
φ: function of water plane coef. 
Cw and φ=C
ω/3-2C
ωC: const.{4 for merchant ships, 3 _??_ warships.
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