Coupled Ship Motions 



where 



x/^i + y/^2 ■^ ^^3 



(2.5) 



Note that as indicated, s. is independent of x,y,z for i = 1,2,3. Note also 

 that Eq. (2.4) is valid only for small x. (i = 4, 5, 6). 



Now let 0-(x, y, z, t) be the disturbance potential associated with the small 

 displacement x.(t) in the ith mode only, where x.(t) - for t <0. Then in 

 addition to being a solution of Laplace's equation, 0. also satisfies the following 

 conditions: 



3z g 3t^ 



g Bx3t g 3x2 



(z = 0, t > 0) 



(2.6) 



3n 





+ Vx (a.xV^) 



(x,y, z) on S„ , t > 0. 



(2.7) 



In Eq. (2.7) [l], n is the unit normal to S^ pointing into the fluid, d/3n is 

 differentiation in the direction of n, and v^(x,y, z) is the velocity at (x,y,z) of 

 the steady flow generated by the body at rest in the uniform stream. 



The two initial conditions on 0. , applied at t = 0+ on the undisturbed free 

 surface, are 



'/'i(x,y,0,0+) = 



(2.8) 



— 0i(x,y,O,t),.O+ = 



(2.9) 



In case x.(0+) = , Eq. (2.8) follows from the fact that 0- vanishes not 

 only on z = at t = 0+, but throughout the fluid. Equation (2.9) is then a con- 

 sequence of Eq. (2.8) combined with the fact that the free surface elevation due 

 to the body motion is zero at t = 0+ . 



In case the body suddenly acquires a finite velocity at t = 0+, i.e., 

 x.(0+) ^ , then 0j vanishes on z := at t = 0+ , though not in general vanish- 

 ing throughout the fluid. This follows from the equations of impulsively gener- 

 ated motion [2] combined with the fact that the pressure is zero on the free sur- 

 face. Equation (2.9) then follows as before. 



Now by modifying a procedure used by Cummins [3] we can write the follow- 

 ing representation for the potential ^.^(x,y,z,t): 



t 



0.(x,y,z, t) = i.(t) 0.(x,y,z) + r i. (r) i/^j .(x, y, z, t -T)dT . 



(2.10) 



409 



