and 



Joosen 



A i 

 P(x) = i-e^ 



|P(x)|' = ^ 



k- 



The final expression for f^ becomes then 



F, .£^f lal^ A2(k,x)dx, (4) 



where A is related to the two-dimensional damping by 



n = A^ ^ . (5) 



The result obtained by Havelock is 



— 1 OJ^ 



^^ " T T^^o ^0 ^'" ^ + Pq^o sin y') 



where B^ is the amplitude of the exciting force for heave, ^^ the amplitude of 

 the heave motion, and y the phase lag. Pq , 0^, and y' are the same quantities 

 corresponding to the pitch motion. Using Havelock's uncoupled equations of 

 motion the result can also be written as 



i4(N^„^. N'^,^). (6) 



2 g 



where N and N' are the damping values. This expression is exactly the same 

 as Eq. (4) except for the coupling term of heave and pitch. 



ADDED RESISTANCE 



In order to obtain the added resistance of a ship moving in waves, it seems 

 necessary to derive a theory in which the forward speed effect is included. 



Let us consider, however, the case of a slender, sharply pointed body which 

 oscillates in short waves, the frequency parameter k being of order unity. If 

 the body is moving forward at speed v, we have to assume the order of magni- 

 tude of the Froude number in regard to the slenderness parameter in order to 

 develop a correct asymptotic expansion with respect to e. 



The speed V appears in the first order term of the velocity potential if we 

 assume the Froude number to be of order unity or the parameter "jV g to be 

 close to 0.25. When considering practical values, however, it seems reasonable 

 to suppose the Froude number to be of order v'^. From (5) we know that this 



642 



