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III. On Ship- Waves, and on Waves in Deep Water due 

 to the Motion of Submerged Bodies. By George Green, 

 D.Sc, Lecturer in Natural Philosophy in the University of 



Glasgow *i 



Bote by Professor Gray. 



The following paper was ready for publication at the be- 

 ginning of 1916, but was put aside on account of war work. 

 It was further deferred by Dr. Green's appointment to the 

 Royal Engineers and his departure to France on military 

 service. Recently, when he was in Glasgow on leave, I 

 advised him to revise the MS. in order that, if possible, it 

 might be published without further delay. The paper may 

 be regarded as a continuation of Lord Kelvin's work on 

 Waves, with which Dr. Green was associated for some time 

 before Lord Kelvin's death in 1907. 



Glasgow, Feb. 16, 1918. A. GRAY. 



THE present paper may be regarded as a continuation 

 of Lord Kelvin's work on Ship- Waves. It deals first 

 with the fundamental problem of Ship-Waves, which is — to 

 determine the wave-motion produced by any arbitrary applied 

 surface-pressure. The method used to obtain the solution of 

 this problem is virtually that used by Lord Kelvin in his last 

 paper on Water- Waves, — but here extended to apply to any 

 arbitrary conditions of applied surface-pressure. The paper 

 then proceeds to indicate how we may use the solution given 

 for the case of an arbitrary surface-pressure to obtain the 

 solution of any problem involving the motion of submerged 

 bodies ; and a complete discussion is given of the wave-dis- 

 turbance due to a cylinder and a sphere moving with con- 

 stant velocity at a considerable depth beneath the surface. 



§ 1. Arbitrary Surface Pressure. 



Taking an origin in the free undisturbed surface of an 

 infinitely extended mass of liquid, with x and y axes in the 

 surface and the z axis drawn downwards, we can express the 

 conditions to be fulfilled by the velocity-potential, <j)(a, y, z, t), 

 corresponding to any possible motion of the fluid by the 

 equations — 



v 2 4> = 0, (1) 



/»/*> = *(«+0-|*. (2> 



* Communicated by Prof. A. Gray, F.R.S. 



