Ogilvie 



i|j2(x,y ,z ,t). Thus, we must retain two terms in the time-dependent 

 part of the potential function. (The problem is still linear, however, 

 in terms of 5.) It is not convenient to be repeatedly attaching sub- 

 scripts to the symbols, and so I shall simply write out equations 

 and conditions which are asymptotically valid to the order of mag- 

 nitude appropriate to keeping e terms in the expansion of 

 i|j(x,y,z,t). 



In the far field, the effect of the oscillating ship can be repre- 

 sented in terms of line distributions of singularities. Again, we 

 try to get along with just a distribution of sources, and we are 

 successful if we allow for the existence of both steady and pulsating 

 sources. The steady-source distribution is exactly the same as in 

 the steady-motion problem. Let the density of the unsteady sources 

 be given by o-(x)e"^*; define (r(x) = for the values of x Ijeyond 

 the bow or stern. The corresponding potential function must satisfy 

 the Laplace equation in three dimensions, a radiation condition, and 

 the usual linearized free-surface condition: 



(ioo)^4; + ZiooUqj^ + U^^jj^x + g^z = ^ on z = 0. (3-37) 



Then it can be shown that: 



ioit poo 



i|j(x,y,z,t)~ p- \ dk e 



r°°.v Jkx *., > r di exp[ii|y| +zVk^+i^] 



*/i \ r di ex 



"^-^ -'-cx) ^C Vk +i - v(l+Uk/co) 



(3-38) 



where (r (k) is the Fourier transform of (r(x) , and the contour C 

 is taken as in Fig. (3-4), where k, and kg are the real roots 

 (k| < kg) of the equation: 



There are two real roots if t = ooU/g > l/4; the other two roots 

 are a cornplex pair. Since we assume that co = 0(i/ve), then also 

 T = 0(l/Ve), and we are assured that t » l/4. However, if 

 T \ 1/4, the complex pair come together, and our estimates are 

 all very bad. Of course, it is well known that the ship-motion 

 problem is singular at T = 1/4. For still smialler values of T, 

 there are four real roots of the above equation, and the solution 

 can again be interpreted physically and mathematically. From 

 experimental evidence, it appears that our final formulas can be 

 applied for any forward speed, at least in head seas, but the 

 presence of a singularity at t = 1/4 shows that this is accidental. 

 Our theory is a high-frequency, finite-speed theory, and it really 

 should not be possible to let U vary continuously down to zero. 



760 



