SIMPLIFIED INTERPRETATION OF FIRST-ORDER 

 FORCES AND MOMENTS 



These expressions for the forces and moments can be viewed from 

 a simple point of view. Consider, for example, the x-component of force, 

 Equation [l4a], which when written out becomes 



X^ = p I S(z.'')'<ZgkAe 1 cos(kaQCOs9--ojt) 



- ig + Y aQ sin e. - z.' (3 I dz." [I4a'] 



From Equation [5] -we see that 



U , « kz^ 



at ax 



gkAe 1 cos (kaQ cos Q- - cot) 



on the equilibrium position of the i spar. Thus the first term in the 

 bracket in Equation [I4a'] is just tw^ice the local acceleration that the 

 water would have at the mean position of the spar axis if the spar w^ere 

 not present. The terms, — Xq + y an sin 9 . - z-'' p , give the negative ac- 

 celeration (in the x-direction) of the point on the spar axis. The quantity 

 pS(z-'') is the added mass per unit length of a cylinder accelerating nor- 

 mally to its axis. Thus the x-component of force is the integral over the 

 mean spar length of 



(added mass per unit length) times (2 times local 

 water acceleration on spar axis due to incident wave 

 alone minus acceleration of point on axis of spar). 



It may appear strange that the water particle acceleration is doubled 

 in this formula. However, the cause is seen on examination of Equations 

 [8] and [10]. In the latter equation, the terms containing the factor 

 (cos X-) give rise to x-components of force. Here the term due to the 

 incident waves (the second term) is already doubled. Half of this con- 

 tribution comes directly from the first term of Equation [8] (i.e., direct- 

 ly from the incident wave potential) and half comes from the term 



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