Developments in Theory of Bulbous Ships 



m (a) 



m(x) 



(11) 



According to the theory developed by Havelock, ^^g and ^^j. are understood as 

 bow waves and stern waves respectively. 



The regular wave heights (5), (9) and (10) all have a form 



i = 3(6) sin [kj sec^^ {(x-a) cos 61 + y sin 0}] dd 



n/ 2 



C(e) cos [kj sec^f^ {(x-a) cos 6* + y sin 6}] dd 



(12) 



Havelock (1934a) showed that the integrands in (12) indicate one-dimensional 

 waves propagating from the point (a,o,0) with the speed Vcos d in the direction 6. 



Indeed it can be easily understood if we recognize: 



(x - a) cos 6 + y sin 6 = t 



(13) 



is the equation of the straight line 1(t,0) on the plane z = o with the distance, r , 

 from the point (a, 0,0) to the line I, and the angle between the normal to the 

 line l and the x axis, 0; the wave speed c = Vcos in the deep sea satisfies 



C^ = -^ = V^ cos' 



277 



(14) 



where \ is the wavelength. Hence the one-dimensional wave in the direction 

 angle 6^ is 



^^ = A sin -^ (r-Ct) 



= A sin — sec^(9 {(x-a) cos + y sin - Vt cos 



If we replace x - vt by x and nondimensionalize by l 



^jj = A sin Ik J sec^6^ {(x-a) cos + y sin 0} j 



(15) 



Therefore these Kelvin regular surface waves are a superposition of the one- 

 dimensional sine and cosine waves with the respective amplitude S(l-) and C(0) 

 in the direction -77/2 < < tt/2. He named these one-dimensional waves "ele- 

 mentary waves" and S(0) and C(t^) , amplitude functions. We may omit the word 

 "elementary" in this report except to avoid ambiguities. 



1071 



