APPENDIX A 



FRICTION ON THE CABLE 



In order to derive an expression for the friction on the cable by the 

 surrounding water, we will consider the mathematical model of an infinite 

 straight cable surrounded by a viscous fluid of infinite extent. Furthermore, 

 we will assume that the cable moves longitudinally like a rigid body with a spe- 

 cified velocity. This rigid-body approximation is valid as long as the distance 

 along the cable required for an appreciable change in the velocity is much 

 greater than the radius of the cable. This requirement is easily met in the 

 present case . 



The governing equations of the motion of the fluid reduce to: 



8w 1 B a /. ,v 



-^-=v-^r^- w (A-1) 



5t r 3r 9r ^ ' 



where w is the velocity of the fluid along the cable, v is the kinematic vis- 

 cosity of the fluid, and r is the radial coordinate. 



If the velocity of the cable is sinusoidal of amplitude Wq and fre- 

 quency U) , then the solution for the amplitude of the velocity of the fluid W is: 



,(1) ,„ r— i"^ 



H^ ' (r / -i=) 



O V V 



W = W —— ■ (A-2) 



° h(1) (a/T^) 



O ^ V 



where a is the radius of the cable and H^^^ is the Hankel function giving out- 

 going waves . 



Therefore, the amplitude of the force F exerted by the fluid on the 

 cable per unit length is given by: 



, h(/) (aTT^) 



F = -2napvW V-i- — ■ (A-3) 



°^ ^ H<1) (aTT^) 



V 



where p is the density of the fluid. 



32 



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