470 TRANSACTIONS OF SECTION G. 



relation between the period of the ship and that of the wave normal is a dominating 

 feature in the resulting movement of the ship. Mr. W. Froude's mathematical 

 solution of this relation is the basis of our belief that we understand the behaviour 

 of a ship in the uniform system of waves when the vessel is placed broadside on to 

 the waves. To obtain this solution he assumed that within the limits considered, the 

 moment of stability varied as the angle of inclination. In the curve of righting levers 

 of a ship, usually known as a curve of stability, this condition holds generally for angles 

 up to about 10°. The curve usually reaches a maximum value at about 30° to 40° 

 and vanishes at 60° to 80°, so that for large angles of roll the assumption does not 

 hold. On this assumption, however, he showed that the motion of a ship amongst 

 such a system of waves is the same as for still water plus a motion composed of two 

 si.ie terms. The amplitude of this latter motion depends upon the maximum slope 



T 



of the waves and the ratio (the period of the ship in undisturbed water to the period 



of the wave). If the ship starts from rest in the upright, is the maximum angle of 

 inclination of the ship and fl, the maximum wave-slope ; then 



1 



0=0, - 



1- 



T,- 



He considered several solutions of the equation of motion : — 



(1) T=T, ; this is synchronism, and the angle of inclination gradually increases. 

 Each wave- impulse adds something to the ship's inclination, and without any resist- 

 ance to rollinj the vessel would capsize. 



T 



(2) . = ; this is the case of the ship's period being very small compared with that 



-m 

 of the wave. will then be positive and equal to 0,. In other words, the ship will 

 place herself normally to the wave- slope. The maximum amplitude will only be the 

 maximum wave- slope. 



T 



(3) <fl- In this case the wave- period is greater than that of the ship and is 



!•] 



always positive and greater than r The vessel always inclines away from the wave- 

 slope. If 



T =i, fl=- 16 0,. If T =i,0= 4 0,. If Z = b e = !f fl i 

 T, * 15 ' T, * 3 ' T, 4 7 



T 

 Tha nearer . is to unity the larger is the maximum amplitude. 

 Ti 



(<>!• 



In this case the wave- period is less than that of the ship, and is 



-M 



always negative. The vessel inclines towards the wave-slope. 



If ^ = 1-1, then = - 4-76 0, ; 



-M 



£ =1-26, then 9= -*,; 



1 = 2-0, then = l 0, ; 



^= 2-235, then = £0,. 



This shows the advantage of having T greater than T,. 



The ship goes through a cycle of changes of extreme angle of roll. Mr. W. Froude 



T T 5 



considered the effect of variations of upon these cycles. — —'- is better than 



I, 1, 4 



T 4 T 



- = , so that it is better to lengthen T than to shorten it. Similar results for — = 2 

 1 j 5 J-i 



and £ respectively gave better results by lengthening than shortening T. In each of 



