472 TRANSACTIONS OP SECTION G. 



is not less than, say, 60°, and the maximum righting lever is not unlike some previous 

 ship which ' has been to sea and come home again ' safely, this ship will be safe. 

 This assumption is based on the belief that only what has happened to previous ships 

 will happen to the one in question ; that is, that the contingencies of waves will be 

 the same in all cases. But when we find that occasionally ships are missing we are 

 compelled to ask ourselves the question — is it possible that some occasional con- 

 tingencies of sea or ship, or both, may exist which will produce a dangerous and perhaps 

 fatal roll ? 



Mr. W. Froude's investigations were made for a uniform system of waves. He 

 showed that in unresisted rolling if a ship initially at rest and in the upright position 



is acted upon by a uniform series of waves such that — = ? ; where p and g are the 



T, q 

 smallest whole numbers which express this ratio, then the rolling of the ship will be 

 in cycles, the maximum inclination in each roll gradually increasing, and again 

 gradually diminishing, and so on. The period of occurrence of the maximum of 

 maxima will be 2 q T. The number of times the ship passes through the upright 

 in each complete cycle is 2 p or 2 q, whichever is the smaller. The ship is upright 

 at the middle of the cycle, and on either side of this middle there is an equal maximum 



which is approximately 0, — ?- , and never exceeds this value (where 0, is maximum 



q-p 



wave-slope). If T is much larger than T,, and therefore p is much larger than q, 



then the value of 0, q ^ approaches 0, q and is less than the wave- slope. If T 

 q-p p 



is much smaller than T„ then the value of 0, — 9 - approaches 0,. If T is nearly 



q-p 



equal to T,,then0, —2— approaches a high value. 

 q-p 



From this it is seen that our investigations in unresisted rolling may be over a 

 very wide field, but would produce no definite result in the matter of finding cases of 

 large angles of roll in practice. We can only obtain valuable results when resistance 

 is included. 



Mr. R. E. Froude in 1896 was led to deal with the subject of non-uniform rolling of 

 ships in an assumed uniform system of waves which did not synchronise with the ship, 

 such as is dealt with above for unresisted rolling, and he dealt with the effect of 

 resistance in such a case. He pointed out that there is a particular phase- relation 

 between the ship and the wave which will produce uniform rolling, just as there is in 

 the case of a synchronous system of waves. If at any stage for any reason the roll- 

 ing is of the cyclic character considered in non-resisted rolling, then the resistance 

 must gradually introduce uniformity, because the rolling is made up of two sets of 

 oscillations — 



(1) That due to the rolling relatively to the water surface, such as would occur 

 in undisturbed water. 



(2) That due to the oscillation of the water-surface itself, caused by the passage 

 of the wave. 



We have already seen that the resisted oscillation in undisturbed water gradually 

 decreases when the vessel is left free to oscillate, but takes place in practically uniform 

 time T. The oscillation of the water- surface is forced on the ship and causes a simple 

 harmonic oscillation of the ship in time T, in algebraic addition to that due to the 

 free resisted oscillation. When the maximum angle of a roll due to the free oscilla- 

 tion coincides with the maximum angle due to the forced oscillation of the wave, we 

 shall have a maximum extreme inclination which is the sum of that due to the free 

 and the forced. When they are in opposition we shall have a minimum extreme 

 oscillation which is the difference of these two. At stages between coincidence and 

 opposition we shall have extreme angles varying between maximum and minimum. 

 As time goes on the extreme angle due to the free oscillation gradually decreases under 

 resistance, and the sum and the difference referred to above approximate to each 

 other, and the rolling becomes that due to the wave alone. We have seen that in 

 the case of unresisted rolling where the wave and the ship synchronise there is an 

 addition to the angle of inclination for each passage of the wave, and were it not for 

 resistance these accumulated increases would cause the vessel to upset. But in the 



