encountered by the towed sea sled and the subsequent calculation of wave 

 forces on the components of the platform. 



The total force exerted by waves on a unit section, dz, of an object 

 is composed of the drag force and the virtual mass force: 



d f = (f D + f ± ) dz , (1) 



where the drag force is proportional to the square of the orbital velocity 

 in the horizontal plane: 



f D = 1/2 C D pD (|u|u) , (2) 



and the virtual mass force is proportional to the horizontal acceleration 

 force exerted on a mass of water displaced by the object: 



fi = 1/4 C M p*D 2 |H. . (3) 



In equations (2) and (3) , D is the diameter of the cylindrical 

 object, p is the mass density of seawater, u is the horizontal compo- 

 nent of the orbital wave velocity, and Co an d Cm are the coefficients 

 of drag and inertia, respectively. 



The horizontal components of velocity and acceleration can be approx- 

 imated using: 



and 



itH cosh k (y + d) „ . r/n 



u = — — y, -. cos cot , (4 J 



T cosh kd 



3u = _ 2^H coshkd (y + d) sin (5) 



3t T2 sinh kd ' 



where k = 2ir/L and co = 2tt/T are the wave numbers, y is the vertical 

 coordinate measured upward from the bottom, d is the water depth, H 

 is the wave height, T is the wave period, and L is the wavelength. 



Expanding equation (4), the force exerted on a section of a pipe, dz, 

 at any position z becomes: 



df = T^pDH^ t _ f ± f 2 ut] . (6) 



dz 2T2 L 1 u J 



The overturning moment about the bottom according to Morison, et al., (1950) 

 is: 



M = J (d + z) df = P T2 l^p, • C M K lS ino)t ± C D K 2 cos 2 u)tl. (7) 

 •' o 

 Because the small -amplitude theory underestimates u in shallow water, 

 substitute the empirical stream function theory of Dean (1965) using cases 

 5-B and 3-B (Dean, 1975) for illustration (Figs. 2 to 6) . Case 5-B repre- 

 sents the deepest water in which the sled is operated; case 3-B represents 

 the breaking wave conditions. 



12 



