EXERCISES. 257 



30. If liquids of densities p and p and depths h and h be contained 

 between two fixed horizontal planes at a distance h + h , prove that the 

 velocity v of propagation of waves of small displacement of length X at 

 the common surface is given by 



(v - Fcos a) P coth ~ 4- (v - V cos a ) 2 p coth ^ - 9 ~ (p - p) = 0, 



A A ZTT 



where V and V are the mean velocities of the currents in the liquids, 

 and a and a are the angles the currents make with the direction of pro 

 pagation of the waves, the currents slipping over each other. 



[Greenhill, Math. Trip., 1878.] 



31. A wind of velocity F is blowing horizontally over the surface 

 of deep water ; prove that the velocity of propagation of waves of 

 length X is 



where c = \~- . .. -\ 2 = velocity without wind, 



(27T 1 + o-j 



0- = specific gravity of air, and the upper or lower sign is to be taken 

 according as the waves travel with or against the wind, 



[Thomson.] 



32. Hence shew that the velocity of waves travelling with the 

 wind is greater or less than the velocity of the same waves without 

 wind, according as F &amp;gt; or &amp;lt; 2c; and that waves of length X cannot 

 travel against the wind if 



F &amp;gt; c / . [Thomson.] 



33. Find an expression for the average energy transmitted across a 

 fixed vertical plane parallel to the fronts of an infinite train of irrota- 

 tional harmonic waves, of given small elevation, moving on water of 

 uniform depth. [Math. Trip., 1878.] 



34. A deep rectangular vessel nearly filled with water is continued 

 at one end as a shallow canal of indefinite length ; supposing the water 

 of the vessel thrown into the condition of a stationary undulation, find 

 approximately the rate at which the undulations would subside by com 

 munication to the water of the canal. [Stokes.] 



L. 17 



