Evaluation of Motions of Marine Craft in Irregular Seas 



z(t) = ^ J zic-)e'-'dc.. (23) 



One then multiplies Eqs. (1) and (2) through by e"i^* and integrates pver all 

 time from -oo to +oo under the assumption of vanishing z, z, 6 and h at ico and 

 the satisfaction of integrability conditions by F(t) and M(t). The solution for 

 heave is, as an example, given by 



00 00 



z=Zf+z =J-f ^i^ Yico) e'^-' dc - ^ \ ^^ M(a;)e--*dc. (24) 



f '" 2-TT J T(a;) 27t J T( w) 



where one next replaces the transforms F and M by 



r^ F(T)I 



Ice M(^)J 



^1 ^^^^^^' --dr (25) 



M " 



and, upon interchange of the orders of integration, obtains the familiar result 



Z = Zr + Z 



00 00 CD CO 



-co -CD V ' _m - m *■ ' t<M^ 



f -^ ^m - 2 



which leads to the definition of the kernel functions 



(26) 



K,(u) . 1 f ^^ e^"- dco (27) 



- 00 



K.(u) = f f ^^e^^-da.. (28) 



m"^ >" 277 J X(aj) 



- CO ^ ' 



These are defined as the impulsive response function for the body in the fluid. 

 It is to be noted that they are dependent only on the body coefficients obtained 

 from either the impulsive response in calm water or from the response of the 

 body to regular waves. The final expression for the heave is then 



z = z, + z^ (29) 



with 



CO 



Zf = j F(r) Kf(t - r) dr (30) 



- 03 



CO 



z^ = - I M(T) KJt-T) dr (31) 



467 



