360 



On the Breaking of Waves. 



than for negative values of the exponent, the action at the 

 crests is more violent and sudden than in the troughs. 



Oblique Reflexion. 



When the waves are incident in a direction not at right 

 angles to the barrier, the water has a motion parallel to the 

 obstacle in a horizontal direction. This will also be subject 

 to the drag of the obstacle, and the effect can be investigated 

 on the principles laid down. 



Taking the line of the obstacle as axis of z, suppose the 

 waves incident at an angle a, so that the ordinary reflected 

 wave is reflected at the same angle on the other side of the 

 axis of x. The advancing train is represented by 



77 = sin ^cosa + 2 sin a — ct), 



giving in deep water 



<fi = ce~ k y cos k(x cos a -f ~sin a — ct). 



X 



The ordinary reflected wave is given by 



^> , = ce~ Jcy cos &(#cosa — ,-rsina -\-ct). 



Thus the velocity system to be investigated is 



u' =—ck cos a e~ klJ sin k{cc cos a. — z sin a. + ct), 

 v' = — mck e~* y cos k{x cos a — z sin a. + ct) , 

 iv ; = nck sin ae~ ky sin k[x cos a — z sina-f- ct), 



where m and n are both nearly equal to unity. The dilatation 



cV , <V ~dw' . . . 



^— -t- ^r f- s: — at the origin is thus 



0% oy oz 



ck 2 (m — n sin 2 a — cos 2 a) cos kct, 



and we have at the origin 



- ~^- + ck 2 (m — n sin 2 a — cos 2 a) cos kct = 0. 

 p ot 



Thus the quantity ?n — 1 which occurs in connexion with 



