Long Waves in a Rectangular Trough. 157 



It' we make the additional assumption that <r/v is large the 

 expression for k becomes very much simpler. Then (12) 

 may be written 



*=Vf v (l+')- 



Substituting in (13) 



p>+2ip K -ghm*+gm>.\/%-(l-i)=0, . (15) 

 and equating the imaginary part to zero, we find 



To a first approximation it may be assumed that the visco- 

 sity has no effect on the period ; then p/m = \/gh and the 

 wave-length \=2?r/m. Hence we may write 



<=tey=©tty («>■ 



To find the effect of the viscosity on the period we equate 

 the real part of equation (15) to zero. Then 



p 2 —gkm 2 +gm 2 A/ — - =0. 



By substitution from (16), this becomes 



p 2 — gh m 2 + 2j)tc = 0. 



Hence when k 2 is neglected in comparison with glmr 



p~—/c+m*/gh (17) 



The period T = 27r/p. T , the value which we obtain when 

 viscosity is neglected, is 27r/(m *Jgh). Hence substituting in 



T=T 0+ £T ' (18) 



We shall next determine the values of the coefficients in 

 the expression for u. 



* I found, after deriving formula (16), that it had already been 

 obtained by S. Hough for progressive waves in an unlimited medium by 

 a longer method (Proc. Lond. Math. Soc. xxviii. p. 276, 1897) ; he finds a 

 general expression for the damping of waves in one horizontal dimension,, 

 and as a special case gives the formula for long waves. 



