THE TENSION IN A LOOP OF CABLE TOWED THROUGH A FLUID 



Introduction 



This work was undertaken in an attempt to find what must be the minimum 

 strength in a loop of wire cable when towed at a given speed by two towboats a given 

 distance apart. To solve the problem it was necessary to determine the tension in 

 the cable at the towboats. Mathematical expressions are derived from which the max- 

 mum tension may be computed for a prescribed loop of cable. The method developed 

 is used to determine the tension in a particular case. 



Analysis 



In the following approximate solution, 

 we suppose that the weight of the cable is 

 negligible compared to its drag so that the 

 cable may be treated as being in a horizontal 

 plane. For the laws of force on the cable, 

 we make the same physical assumptions as in 

 Report 418 (Appendix I); i.e. that the force 

 per unit length normal to the cable is given . 

 by R sin 2 <|>, where <J> is the angle that the FIG. 1. 

 cable makes with the direction of motion, and R is the force per unit length of 

 cable when normal to the stream, and that the force per unit length parallel to the 

 cable is given by a constant F. 



Fig. 1 is a diagram shewing the cable, the towboats and the forces acting. 

 T is the tension in the cable at any point. is taken at the point where the cable 

 is normal to the direction of motion, ox. oy is normal to ox. 



Let s = arc length along the cable measured from o. Then, relating y, s 

 and <)> we have the equation 



dy/ds = sin (ji 



(1) 



The element ds of the cable at P is in equilibrium under the action of the 

 system of forces comprising the force R sin 8 if> ds normal to the cable, the force F ds 

 along the cable, and the tensions T and T + dT. Resolving these forces along the 

 cable we obtain the equation 



dT/ds = F 



(2) 



which may be integrated to give 



T - T = Fs 

 o 



(3) 



where T is the tension of the cable at the origin. Resolving also at right angles 



