52 Dr. G. Green on Ship- Waves, and on Waves in 



where the pressure is impulsively applied by <£ (a\ y, z, t} 

 and £q(%, y, z, t) respectively, the corresponding results for 

 the same pressure in motion with velocity v are given by 



and 



<j>{x,y,z,t)=\ dr<f> {(a:-vT),y,z,(t~T)\, . (19) 



t(tf,>,^0=f dTZo{(x-VT),y,z,(t-T)\. . (20) 



By means o£ these equations the solution for any moving- 

 pressure problem may be derived from the solution of the 

 corresponding impulsive pressure problem by a single inte- 

 gration. All that is required is to change x into (x, — vr) and 

 t into (t — r) in the expressions representing the motion due 

 to an impulsive pressure, and then to integrate with respect 

 to r. 



i§ 2. General Treatment of Fluid Motion due to 

 Motion of Submerged Bodies. 



We now proceed to consider the application of the results 

 contained in § 1 to the problems in which the wave-dis- 

 turbance is due to the motion of a submerged solid. The 

 velocity-potential in this case, in addition to satisfying 

 equations (1) and (2), must satisfy 



p — 0, at the free surface, (21) 



and the condition that the fluid in contact with the solid has 

 no velocity normal to the surface of the solid, 



!£ = „cos<>,#), (22> 



where v is the velocity of the solid in the direction of x. 



Let us assume that the solid is moving at a uniform- 

 velocity, its centroid being at a constant depth beneath the 

 free surface, large in comparison with the dimensions of the 

 solid. A velocity-potential satisfying all required condi- 

 tions can then be obtained by the following system of suc- 

 cessive approximations. 



(«) Find first the velocity-potential <pi, corresponding to 

 the motion of the given solid in an infinite mass of liquid- 

 This fulfils required conditions at the boundary of the solid, 

 but involves a certain impulsive pressure at the free surface 

 when the motion of the solid commences and also a certain 

 surface-elevation. Each of these leads to a violation of 

 condition (21). 



