Yim 



Now let us investigate the last (third) term of the above inte- 

 gral I. From the slender body theory 



\ o-(x,y ,z) exp {ti (x, cos 6 + y| - y sin 9) +tz} dc 



d^ 



dx Jc(x) 



,2 



^d_MU) g^ ^^^. ^^g Q ^ sine)} (46) 



dx 



With this approximation, we go back to the wave-height Eq« (42) and 

 consider the values only on y| = 



f%Q pTT ^CB , ti(x|-x)cos5 , ,^ 



^r dxM"(x)r [^^il±_l ^tJQ_ 



•I ^-ir*Jc) it(t-ksec^e - ijisece) 



,,(x„0).-^\ dxM-(x)3 \ -; ^ 21^ (47) 



Changing the contour of integration with respect to t in a complex 

 plane such as Havelock [13] used, we have 



r (^ 0) = - J^ f dx M"(x) r ^ec e de r°°«^^(l^^^ sec 6) ^^ ^^ 

 '3* I ktrU J., ' J-fl. Jo m(I +m) 



+ \ dxM"(x)\ Ztt sec 9 cos (kx, -x sec 9) dG 

 = ml dxM"(x)[j|log|x,-x|-Yo(kix,-x|)} 



X sign (x,-x) + Ho(k^r;^)] 



"^ml dxM"(x)Yo(k|x,-x|) (48) 



where Y is the Neumann function, H is the Struve function, and 

 sign (X|-x) is +1 for x,-x>0 and -1 for X|-x<0. 



When only the integral of the first termi of I in Eq. (45) is 

 taken, we may approximate the wave height as follows, for X| < 0: 



584 



