ON THE LOSS OF ENERGY BY FRICTION 

 OF THE TIDES. 



In this paper the waters of the ocean will be conceived as concentrated 

 in a basin, rectangular in shape, the width of which will be taken as 2,860 

 miles and the length as 60,000 miles. The bottom will have a uniform 

 slope from the surface to a depth of 600 feet at 100 miles from the shore, 

 dropping then to a depth of 9,000 feet at a distance of 150 miles from the 

 shore, and then parallel to the surface out to the middle of the basin, the 

 opposite side having the same shape. The tide will be supposed to rise 

 4yt feet in 6 hours — falling at the same rate. 



; Sec. a ! Sec. 6 ! Sec. c ! 



• 100 mi. I SOmi. I i2aomi. I 



■ i i 



! I 



„...j 



Fio. 8. Cross-section of basin (showing one-half). 



The rigorous determination of the motion of the water in such a basin 

 on the principles of hydrodynamics seems to be unattainable at the pres- 

 ent time. It is true that, to start with, we have the equations of motion of 

 a viscous incompressible liquid, but I have not succeeded in finding a solu- 

 tion for them with the assigned boundary conditions, and therefore am not 

 able to give an exact statement of the rate at which energy is dissipated. 

 We may, however, approach the problem through some of the formulae 

 of hydraulics and obtain an approximation, which, even though it be 

 rough, will permit us to form some idea of the order of magnitude of dissi- 

 pation. If we liken the ebb and flow of the tide to the flow of water in a 

 canal we can use the formulae of engineers for the loss of head due to fric- 

 tion and viscosity, and consequently the loss of energy. 



Weisbach ^ gives us the following formula : 



h = e Xlength X ^"'"^^ '^'^•^' X ^^^^f perimeter 



2g area of cross-section 



where h is the total fall of water in the canal necessary to maintain the flow, 

 f is the coefficient of friction, and g is the acceleration of gravity. We will 



* Theoretical Mechanics. Translated by E. B. Coxe. 8th Amer. ed., see. 475. 



71 



