418 



HYDRAULICS AND ITS APPLICATIONS 



angular velocity &>, the height of the wave is 2 ;. Let I be its length, 

 measured from crest to crest. Imagine the wave form brought to rest by 

 imposing on it a velocity equal and opposite at every point to that of its 

 normal motion, i.e., imagine the orbital circle concentric with a circle of 

 radius R, where 2 TT R = I, rolling, as shown in Fig. 184D, on a horizontal 

 plane with angular velocity &>, and with a linear velocity V equal to that of 



Then V = <o R = -. 



Zi 7T 



propagation. 



Under these conditions the velocity of a particle at Q is compounded of 

 a velocity <o. K perpendicular to O K, and &>. Q perpendicular to O Q, 



\ 



FIG. 184D. 



and so equals (a. K Q perpendicular to K Q, so that K Q is normal to the 

 surface at Q. If now v c be the velocity of a particle in the crest, and v 

 the velocity at any other point Q in the free surface, and therefore at the 

 same pressure, 



v 2 v 2 



.-. _+,, = ^ + r 



But 



{ Q A' 2 - (R - r) 2 }=*(!- cos 0). 



g 7v a = (R - r cos 0) 2 + (r sin Of 

 = R 2 - 2Rr cos + >- 2 



