function K(t) does not vanish sufficiently rapidly as t -* 

 °°. Precise information about the asymptotic behavior of 

 the function K(t) as t — <» is required in this respect. 

 Figure 2 depicts the real and imaginary parts of the 

 functions (1 + t^)K(t) and t(I +t^)K(t) appearing in the 

 integrands of the integrals (5a) and (5b), respectively, for 

 < t < 16 and for a simple ship-bow shape which is 

 considered further on in this study. Differentiation under 

 the integral sign in expression (3a) is clearly not justified 

 in the case corresponding to figure 2. 



6'(v,a) and fl"(t;o) are depicted in figures 3a, b, c, 

 respectively. 



0.4S 



)1+f . i ii. i 







z,^ '~\ j'^.x^.fVH''i| H 





iw — ' • 1' 'f 'p' l;p' 





 -0.2S 



iy'^\j\ / "^""v r-5*i.ft^i^^,^'Hn-^vit^l*%i^^ 



Fig. 2 — Real and Imaginary Parts of the Functions 



(1 -l-t2)K(t) and t(l +t2)K(t) for a Simple Ship Bow Shape 



in the Zeroth-Order Slender-Ship Approximation 



ASYMPTOTIC EVALUATION OF THE 

 KELVIN WAKE 



For large values of |x| and z = 0, analytical 

 approximations to the integrals (1), (3a, b) and (5a, b) can 

 be obtained by taking advantage of the rapid oscillations 

 of the exponential functions E ^. defined by equation (7). 

 These functions may be expressed in the form 

 E = exp[ixe(t;o)], (9) 



where 0(t;o) is defined as 



e(t;o) = (l-fft)(l-ft2)'''2. (10) 



The functions E^(t;x,a) and E (t;x,o), where a > 0, 

 correspond to the function E(t;x,o) with a > and a < 0, 

 respectively. In the limiting case o = 0, we have E^ = 

 exp[ix(l -i-t^)'^^] = E_ . The derivatives of the phase- 

 function 0(t;x,o) with respect to t are given by 

 e'(t;o) = -(o-t + 2otV(l+t^)'''^, (11a) 



e"(t;o) = (l-3at-2ot^)/(l+t^)^^^, (lib) 



9"'(t;o) = -3{o + l)/{l+t^)^^^. (lie) 



The phase 9(t;a) and its first and second derivatives 



-1 -%• -^4 -.2 -.1 a > .OS 



J^ 



\\ \i/ai«\.3 \ 



v,VV,\\ \ \, 



~\' 



■ \\\ 



1 



2 3 4 



S 



6 



Fig. 3a — The Phase Function e(V,a) for < t < 6 and 

 Several Values of a 



Fig. 3b — The Function e'(t;o) for < t < 6 and 

 Several Values of a 



fg 







o = -.8 



~— _ 





-.6 



— - .___ 





-.4 



^~~^ — — -^ZIZ 





-.2 



-.1 



V"::^N> 



--.-.~~~~— - --_iiim 







^~...^]^~~ — 





.2 



C""~- - — — ^_^_ 





.3 



"-^^-^-^iiizii: 





.4 





.S 



0:^-^^111; 





.8 





.8 





a = 1 







1 2 





3 



Fig. 3c — The Function e"(t;o) for « t < 3 

 Several Values of a 



and 



