Ogilvie 



^35= "^35 - <U/ia>) T^j; (2-79) 



T53=T53^ +(U/la)) T^^l, (2-80) 



c) A term proportional to U . This occurs only in Tggi 



^55=^55 +(U/lco)' Tfl^ (2-81) 



Even at zero forward speed, there is coupling between the 

 heave and pitch modes, unless the body is symmetrical fore -and-, 

 aft. If the body _i£ symmetrical, one can show that T^J and T53 

 are zero. But even in this case, the existence of forward speed 

 causes a loss of symmetry, and so a pure-heave motion causes a 

 pitch moment, and a pure-pitch motion causes a heave force. The 

 symmetry between T__ and T-^ should be noted: The speed- 

 independent parts are equal, wnereas the speed-dependent parts are 

 exactly opposite. 



One remarkable fact is that there is no interaction between 

 the oscillatory motion and the perturbation of the uniform stream 

 by the steady forward motion. If the above formulas are derived 

 from the kinetic-energy formula by use of the Lagrange equations, 

 this fact is perhaps obvious. When we derive expressions for force 

 and moment on an oscillating ship, it is anything but obvious. 



For the sake of completeness, I write out here the final 

 formulas for the Tj; 's for a slender body in an infinite fluid. We 

 note first that, by the same procedures used in the steady-forward- 

 motion problem, the following is true to a first approximation: 



<^j + <^J2z = 0» iri the near field. 

 From (2-72'), it is rather obvious that, for a slender body, 



ng = - xnj 1 + 0(e^)] . 

 and thus, from (2-73): 



<),,= - x<(,Jl +0(e2)]. 



Now let: 



m(x) = p \ din,<|>, = added mass per unit length, (2-82) 



730 



