Kotik and Lurye 



body when at rest.) Now suppose the body executes a small time-harmonic mo- 

 tion of angular frequency a in one of the six modes: surge, sway, heave, roll, 

 pitch, or yaw. These modes are denoted respectively by the index i = 1, 2, ... 6 

 with i = 1, 2, 3 representing translations parallel to the x, y, z axes respectively 

 and i = 4, 5, 6 representing rotations about those axes. If F^j e'^"^* is the com- 

 plex hydrodynamic force or moment exerted by the fluid on the body in the j th 

 mode when the body has a complex linear or angular velocity e'^°^* in the ith 

 mode with all other velocities zero, then the complex hydrodynamic cross- 

 coupling coefficient H-. is defined by 



Hij(a) = -Fi3(cr) (2.1) 



where the dependence on frequency has been indicated. 



It is a familiar fact that a knowledge of the H- j together with the inertial 

 and hydrostatic properties of the body sxiffices to determine the steady state re- 

 sponse of the body to an arbitrary time-harmonic set of exciting forces or mo- 

 ments applied simultaneously in all six modes. 



Writing ' 



Hij(a) = Hi-Ccr) + Hj.(cr) (2.2) 



we now outline the proof that H. j and h{ • satisfy the Kramers-Kronig relations. = 



General Equations for Transient Problem 



Consider the transient disturbance that results when the body, initially at 

 rest in the steady flow, is given at t = a small displacement which is an ar- 

 bitrary function of time in the ith mode. We characterize this displacement by 

 a vector function of position and time a.(x, y, z, t) defined only on the undis- 

 placed body surface (call it s^), such that a.(x,y, z, t) is the displacement at 

 time t in the ith mode of a body surface point whose coordinates were (x,y, z) 

 at t = 0. Let Ml? ^2? M3 be unit vectors in the x,y, and z directions respec- 

 tively, Xj(t), X2(t), X3(t) the instantaneous magnitudes of the translational 

 displacements in the first three modes, and x^(t), x^ct) , Xg(t) the instantane- 

 ous magnitudes of the angular displacements in the last three modes. 



Then 



a.(t) = Xi(t) Ai i = 1, 2, 3 (2.3) 



a.(x,y,z,t) = Xi(t);i..3xf i = 4, 5, 6 (2.4) 



'''Strictly, only after certain terms have been subtracted from the Hj j , do the 

 real and imaginary parts of the remainder satisfy the Kramers-Kronig rela- 

 tions. See Eqs. (2.24) ff. 



408 



