Newman and Tuck 



between the body and the undisturbed incident wave height is of principal im- 

 portance; in this way the theory accounts for the diffraction effects, or the cor- 

 rection to the Froude-Krylov exciting force due to the presence of the ship. It 

 is important to note that both double integrals contain truly three-dimensional 

 effects, with the disturbance at one station of the ship (i) affecting the force at 

 another (x). 



ni. FORCED OSCILLATIONS AT FINITE SPEED 

 OF ADVANCE 



Introduction 



In this portion of the paper we suppose that the ship is being forced to make 

 small oscillations about an equilibrium fixed position, while an otherwise uni- 

 form stream u flows past in the positive x direction. These forced oscillations, 

 which need not in general be sinusoidal or even periodic, will be described by 

 given functions of time ^i(t), ^2(^)5 ^3^^) for the linear displacements in 

 surge, sway and heave, and C^it), CjCt), CgCt) for the roll, pitch, and yaw an- 

 gles. Similarly we denote the resulting hydrodynamic force component in the 

 ith mode by F.(t). 



Under the usual control theory assumptions of linearity and causality there 

 will be a linear relationship between F- and all C , j = 1, ... 6, which we may 

 write symbolically as 



Fi = E C,. ^, (3.1) 



i = l 



for some set of linear operators C-. Alternatively (3.1) may be interpreted 

 literally as a linear algebraic relationship between the Fourier transforms of 

 the variables F^, ^j , with coefficients C^j = Cjj(a)) called "transfer functions." 

 Here the Fourier transform of i-^(t) is defined as 



^j(^) = [ dte^^^^jCt) (3.2) 



•'-co 



(the use of the same symbol for a function of time and for its Fourier transform 

 is common and convenient, and will not cause confusion). Clearly C. -(a;) is the 

 Fourier transform of the force C- -(t) in the ith mode due to a unit impulse set) 

 of displacement in the jth mode, the actual relationship between C (t) and F.(t) 

 being thus a convolution integral with C- ( t) as kernel. 



In the case of sinusoidal motion at a real radian frequency w, the real and 

 imaginary parts of the functions Ci j(w) define the frequency response of the 

 forces to sinusoidal displacements of vinit amplitude. Historically these quan- 

 tities as used in ship problems have been calculated in the form of "added 

 masses" 



144 



