Ogilvie 



in the equations of motion, (13), require care. Rather than attempt to transform 

 these terms as they appear in Eqs. (13), I find it desirable to retrace steps 

 somewhat. Equation (Al) of Appendix A states that the j -th component of the 

 motion induced force is: 



^jo - E ^jk ^k(t) - Zi b-k a^Ct) - f; c.^ a^(t) 



6 t ^ r' 



(Bl) 



The last two sums are compressed into the single sum of convolution integrals 

 in (13). However, it is now more perspicuous to consider the two separate kinds 

 of integrals. 



In order to study the behavior of the kernels in these integrals, let us as- 

 sume that there is a velocity impulse in the k-th mode, so that 



ak(t) = ^ko S(t) 



Then, for t > , 



X. = X.„ + a 



ko 



t 

 ^jk + L.^Ct) +1 M.,CT)d. 



The term L.^(t) represents the actual unsteady force due to the velocity impulse 

 itself. Physically, this must approach zero for large t, since the impulse mo- 

 tion generates only a finite-energy wave system, and these waves rapidly radiate 

 away. The terms 



jk 



t 



I M.k(T)dT 



represent the force which results from the steady deflection of the translating 

 ship after t = 0. The integral term of this expression must approach zero 

 eventually, because c^i^, by definition, is the constant of proportionality be- 

 tween steady perturbation force and displacement. Obviously, if the integral 

 of Mjj^ approaches zero as t becomes infinite, then M^^ itself approaches zero. 



We can now find the Fourier transforms of the convolutions. For the first, 

 involving Ljj^ and the disturbance velocity, we have from the convolution the- 

 orem: 



I aj^C-^) L.j^(t - r)dr r = a^( oj) L ■ ^^( oj) 



76 



