bow region given by x = (1 + l/4bt)/4 for l/4b < t < 

 oo, so that we have 1/2 > x > 1/4 for l/4b < t < » 

 with X -► 1/2 as t -» l/4b and x -♦ 1/4 as t -* <*>. We 

 thus have one point of stationary phase in the stern 

 region for < t < l/4b and two points of stationary 

 phase, one in the stern region and one in the bow region, 

 for l/4b < t < oo. The two points of stationary phase 

 approach the shoulders x = ±1/4, where dy/dx = 0, as 

 t -* oo. 



The asymptotic approximation (50) and equations 

 (51)-(58) and (59)-(67) defining the contributions of the 

 bow and stern and of the stationary-phase point(s) on 

 the waterline, respectively, show that the low-Froude- 

 number behavior of the far-field wave-amplitude 

 function is strongly influenced by the shape of the hull in 

 the vicinity of the waterline. More precisely, for a value 

 of t for which there is one (or more) point of stationary 

 phase on the mean waterHne where the hull has flare, the 

 contribution of this stationary-phase point dominates the 

 contribution of the bow and stern and is of order 1/F, 

 that is we have K(t) = 0(1/F). On the other hand, for a 

 value of t for which either there corresponds no point of 

 stationary phase or the hull has no flare at the point(s) 

 of stationary phase, the dominant contribution stems 

 from the bow and stern, and it is of order 1 , that is we 

 then have K(t) = 0(1). However, if n^ = or n^ = 

 at both the bow and the stern, that is if the bow and the 

 stern are either cusped or round, their contribution is 

 O(F^) and the contribution of the stationary-phase 

 point(s), which is 0(F) if there is no flare (as is assumed 

 here), is dominant; so that we then have K(t) = 0(F). 



For a ship form that is everywhere wall sided, the 

 contribution of the bow and stern is dominant for all 

 values of t, and we have K(t) = 0(1) for < t < °°. On 

 the other hand, for a hull form that has flare over a 

 portion of the waterline and is wall sided elsewhere, the 

 contribution of the bow and stern is dominant, and 0(1), 

 only for those values of t for which the corresponding 

 points of stationary phase fall outside the range of flare; 

 for the range of values of t for which the corresponding 

 points of stationary phase are within the range of flare, 

 the contribution of these stationary-phase points is 

 dominant, of order 1/F. In this instance, the function 

 K(t) is 0(1/F) for a range of values of t (corresponding 

 to the region of flare) and 0(1) for other values to t. If 

 the region of flare is of small extent, and the slope 

 dy/dx of the waterline does not vary widely within that 



region, the range of values of t for which K(t) = 0(1/F) 

 is also small. The far-field wave-amplitude function K(t) 

 can then exhibit a sharp peak for some value of t in the 

 low-Froude-number limit. In fact, several isolated peaks 

 of the function K(t) can exist if the hull form has several 

 distinct regions of flare within which the slope of the 

 waterline varies gradually. 



It should be noted that the result K(t) = 0(1/F) 

 for values of t for which the hull has flare at the 

 corresponding points of stationary phase does not imply 

 that the corresponding free-surface elevation becomes 

 unbounded in the zero-Froude-number limit F = 0. 

 Indeed, the asymptotic approximation (6), where we have 

 (i4) = (X,Z)g/u2, then yields Zg/U^ = 0[l/F(-4)'^2] ^ 

 0[l/(-X/L)'''2] as F - and -X/L - «. The free- 

 surface elevation Z thus is of order (U^/g)/(-X/L)'^^ as 

 F -* and -X/L -* <», and equation (15) shows that 

 the corresponding wave steepness s is 0[l/(-X/L)'^^]. 

 For values of t for which K(t) = 0(1) it is seen that 

 Zg/U^ and s are 0[F/(-X/L)'^2] as F - and -X/L 



It should also be noted that the asymptotic 

 approximation (51)-(55) for the contribution of the bow 

 and stern is not uniformly valid for the values of t for 

 which the bow or the stern is a point of stationary 

 phase, that is for which the waterline slope dy^/dxg at 

 the bow or the stern is equal to - 1/t or 1/t, 

 respectively. Indeed, it may be shown that we have 

 1 - n^^ - p^n^^ = at a point of stationary phase, so 

 that the first-order approximation to the ampHtude 

 functions Ag g given by equation (53) becomes 

 unbounded. The asymptotic approximation (60)-(62) for 

 the contribution of a point of stationary phase likewise is 

 not uniformly valid at a stationary-phase point where th 

 waterline has an inflexion point. Indeed, the radius of 

 curvature r at such an inflexion point is infinite and 

 equation (60) yields an unbounded contribution K^. 

 Asymptotic approximations valid for these special cases 

 may easily be obtained and are given in Noblesse 

 (1986b). It will only be noted here that the far-field 

 wave-amplitude function K(t) for a hull form having 

 flare at a point where the waterline has an inflexion may 

 be expected to exhibit a particularly pronounced peak at 

 the value of t corresponding to the Inflexion point since 

 we have K(t) = 0{\/F*^'^) as F - for this particular 

 value of t. 



