Tuak and Taylor 



esting mathematical feature of this small width limit is that the singu- 

 larity at F = 1 becomes stronger as w/L — ^ 0, changing from in- 

 verse square root (e.g. (3. i 2)) to inverse first power (e.g. (2.15)). 



Another conclusion in the 1967 paper was that the ratio between 

 the sinkage at width w and that at infinite width was almost independ- 

 ent of the shape, s ize or speed of the ship, depending only on the 

 parameter (w/L)V1 - F^. Thus, starting with any estimate (even 

 (3,21) !) of the infinite width sinkage, we may further estimate the 

 effect of finite width by use of the universal curve given in the 1967 

 paper. For example, at low values of F, a channel width of two 

 ship lengths increases the sinkage by 10%, one ship length by 33%, 

 over the infinite width values. For channel widths less than one ship 

 length a one-dinnensional theory as in Section 1 is sufficiently accurate 

 and probably to be preferred. 



IV. THREE-DIMENSIONAL THEORY OF SQUAT IN INFINITE 

 WIDTH, FINITE DEPTH 



We begin the present section by presenting the solution! 

 Suppose S(x) is the cross-section area curve of a slender ship 

 moving at velocity U in water of finite constant depth h, and let 



;*(k) = r 



•-' -J 



Then consider 



.__ poo _, pw 



= -^\^ dkkS''(k)e-"*'*^ d\ 



-00 -00 



S(x) e**** dx. (4.1) 



00 -i\y 



e 



■00 



tqz e" cosh qz k cosh q(z +h) "j ,. ^n 



s inh qh sinh qh(k'^ - /cq tanh qh) J 



where K = g/U and q = (k + \ )^ . 



Although the expression (4.2) is extremely connplicated, it 

 has the following properties, easily checked: 



1 

 (i) V%=0, -h<z<0, r = (y^ +z2)2 ^ 0, (4.3) 



(Li) d<^/dz = on z = - h, (4.4) 



(iii) K{d<^/dz) + 0%/9x^) = on z = 0,y#0, (4.5) 



640 



