156.] WAVES IN DEEP WATER. 187 



and in this we may suppose y equated to zero, for the error in the 

 value of y will only introduce an error of the second order in this 

 condition. Substituting the value (25) of &amp;lt;, we find that in order 

 that (26) may hold for all values of x we must have F (t) = 0, and 



e^ h )P=0 (27), 



with an equation of the same form for Q. The solution of (27) is 



P = A cos kct + B sin kct, 

 where 



ft okh _ p ~ kh 



&amp;lt;f = g f^^ (28), 



and A, B now denote absolute constants. Combining our results 

 we get for &amp;lt; a series of terms of the form 



a{e*(*+v + e-*&amp;lt;* + ]k(xct) ........... (29), 



where a is a constant. 



Let us examine the motion represented by one of these terms 

 alone, taking, say, in the last factor the cosine with the minus sign 

 in its argument. The form of the free surface (p = const.) is given 

 by (22), viz. it is 



where in the last term we suppose y equated to zero, for the 

 reason already given. Hence if the origin be taken at the mean 

 level, the equation of the free surface is 



y = a sin k (x - ct) ........................ (30) , 



where 



-) ..................... (31). 



j 



The wave-profile is therefore the curve of sines, and it advances 

 without change of form in the direction of x positive, with uniform 

 velocity c given by (28). The wave-length, i.e. the distance 

 between two successive crests or hollows is 



