197, 198] WAVE MOTION. 529 



If z is zero when r = <x>, = 0, and the equation of the sur- 

 face is 



The form of the surface is shown in Fig. 167. This is approximately 

 the form taken by the water running out of a circular orifice in the 

 bottom of a tank, although the above investigation takes no account 

 of the vertical motion. 



198. Wave Motion. The case of uniplanar water waves may 

 be dealt with by the method of the preceding section. 1 ) Let us take 

 the XY- plane vertical, the Y-axis pointing vertically upward and 

 the motion as before independent of the coordinate, so that we 

 may use e to denote the complex variable. We shall find that the 

 waves travel with a constant velocity and iib will therefore simplify 

 the problem if we impress upon the whole mass of liquid an equal 

 and opposite velocity so that the waves stand still and the motion 

 is steady. Such still waves are actually seen on the surface of a 

 running stream. 



Let us first consider waves in very deep water. At a great depth 

 the vertical motion will disappear and we shall have only the constant 

 horizontal velocity that we have impressed, so that 



u = a, v = 0, 

 from" which 



cp = ax. 

 The function 



f(z) = a z + Ae~ ik * = - a (x + iy) -f Ae-^+W 

 gives 



(p + ^ = a (% + iy) 4- Ae ky (cos kx i sin kx), 



134) (p = ax + Ae k v cos kx, 

 if/ = ay Ae k y sin kx. 



When y = cx> this makes (p = ax, as required. The free surface 

 of the water being composed of stream lines is represented by one 

 of the lines ^ = const, and if we take the origin in the surface its 

 equation is consequently 



135) ay -f- Ae^sinkx = 0, 



which shows that y is a periodic function of x with the wave-length 

 I = -^- The longer the wave-length, that is the smaller k, the more 



1) Rayleigh, On Waves. Phil. Mag. I, pp. 257 279, 1876. Scientific Papers, 

 Vol. I, p. 261. 



WEBSTER, Dynamics. 34 



