Bow and Stern Waves and Small-Wave-Ship Singularities 



^ = ^ . (3) 



The wave height due to a simple point source of strength m at (xj,o,-Zj), 

 with Zj > 0, is given by 



^ = -5^"^ 



77 00 



2k sec 6 e 



k(ia;-Zi) 



kp sec 6 - i/j. sec 



dkde , 



(4) 



where m is the fictitious frictional force and 



0) - (x- Xj) cos 6 + y sin 



(5) 



or, letting C - ^r+ ^/' where i^ refers to an integral representing the regular 

 wave pattern and ii refers to a double integral representing the local disturb- 

 ance, 



I- 4^k 



- (rr/ 2) + i 



(77/ 



/ 



•'-(77/2) + 



exp(-kQ z, sec^i9) sec^ (9 cos (kg oj sec^^)d6' 



(7T/2) + 



2m r I exp(-ta>)t sec 



77V 



(77/2) + 8 ^(,2 sec'* 5+ t2 



kg sects' sin(tZj)-t cos (tZj) 



where 



S = arctan 



dtd(9 , (6) 



(7) 



The wave height due to a point doublet of strength -/Xj located at (0,0, -zJ is 



y ^1 r, f r :<Jk' e ' dkcieT' /o\ 



""i-l^^'i J ^ ^^^ 



2k 2 ^'^(^---'i) ^^g 

 k - kg sec 2^ - i/^i sec 



or 



C<, = -4^K.^J 



exp(-kQZj sects') sec'' (9 sin (kgOj sects') d^ 



(77/ 2) + S 



2/Xi .(-/2) + S 



ttV 



J 



(77/ 2) + S "•0 



exp(-ma))m 

 k 2 sec^e + m' 



kg sec 2 (9 sin(mZj) - m cos (mz j ) 



dmd^ . (9) 



683 



