low and Stern Waves and Small-Wave -Ship Singularities 



where 



Z 



(ikpk sec 0)^"-i -ikoksece 

 (2n-j)! "■ 



iknk sec 6 



L 



= 2 n + 1 



(ik„k sec 0)^ 



-ik,, k sec 



•0' 



ik„k sec e y-i ( ik^k sec d)^ 



j = 2n + 1 



j! 



(30) 



and where the absolute value of this becomes smaller when n becomes larger, 

 since the series 



L 



j =0 



(ikgk sec 8)^ 



is a converging series mainly due to the factor 1 j ! . Therefore, i^ has no 

 singularity except k = 1 + (i/u/kg) cos by which the regular wave and the local 

 disturbance can be separated. Due to the factor k^^", for small Froude num- 

 bers (= 1 '\¥^), iCnl becomes smaller when n becomes larger. In general, for 

 a usual smooth ship, the influence of the entrance angle or the term of a^ is 

 dominant when the Froude number is less than 0.3. Therefore the bulb for the 

 term a^, in this case, will be adequate to cancel almost all the regular waves 

 (7). Then the wave height will be mainly the local disturbance proportional to 

 0^ due to the ship double model. Even cutting off the bulb at a certain distance 

 will not affect the disturbance too much, for the same reason that truncated 

 bulbs work effectively to reduce the regular wave height. 



The surface ship whose source distribution is distributed both in the x and 

 the z direction on the xz plane, can be treated in a similar manner as above, 

 since only superposition in the draftwise direction (2) is involved. 



LOCAL DISTURBANCE AND IMAGE SYSTEM OF A 

 NO-WAVE SINGULARITY 



It has been known that a combination of two higher order singularities, a 

 doublet in the z direction and a quadrupole in the x direction produce no regu- 

 lar waves (2). The local disturbance due to this no-wave singularity can easily 

 be analyzed as in the previous sections. 



The wave height due to the doublet in the z direction located at (0,0. - zj is 



^-^-1 I^ 



i2k^ sec 9 e 



k(i^-z,) 



- i/Li sec t? 



dkdei , 



(31) 



691 



