530 XI. HYDRODYNAMICS. 



nearly does the exponential reduce to unity and the more nearly is 

 the profile a curve of sines. The velocity is given hy 



136) 2 = u 2 + v*, 

 where 



u = fl = v = ~ a-Ake^smJsx, 



137) a * 



I? = = - > = Ake**coakx, 



138) # 2 = a 2 + ^fcV** + 2^afce** sin lex. 



So far all our work has been kinematical. The relation to 

 dynamics is given by introducing the equation 33) for steady motion, 





and at the surface putting jp = 0, and making use of the equation 135), 



140) gy + i {a 2 + A*k*#*y - 2a*ky} = C. 



Since the surface passes through the origin, putting y = we obtain 



C= 1 -{a* + A*V}, 

 inserting which gives 



141) (g - a 2 fy y + AW (e^ - 1) = 0. 



This equation can be only approximately fulfilled, but if the height 

 of the waves is small compared with the wave-length, so that 2 ky 

 is small, developing the exponential and neglecting terms of higher 

 order than the first in ky we have 



giving the equation connecting the velocity and wave-length 



142) g - a 2 k + AW = 0. 



If ky is small the equation of the surface 135) is approximately 



143) y = -- sin kx 



so that the maximum height of the waves above the origin is T$ = - 



Inserting the values of the height and wave-length in equation 142) 

 it becomes 



8 f2/. 4:* 2 .BM 



144 ) a {TV --^1 ~ * 



an equation connecting the wave-length, height and velocity. For 



