226-227] SIMPLE-HARMONIC APPLICATION OF PRESSURE. 395 



If we write K = g/c 2 , so that 2?r//c is the wave-length of the 

 free waves which could maintain their position in space against 

 the flow of the stream, the last formula may be written 



P (k K) cos kx fa sin lex 



- - - 



where ^ = //,/c. 



This shews that if p. be small the wave-crests will coincide in position with 

 the maxima, and the troughs with the minima, of the applied pressure, when 

 the wave-length is less than 27r/ ; whilst the reverse holds in the opposite 

 case. This is in accordance with a general principle. If we impress on 

 everything a velocity c parallel to #, the result obtained by putting ^ = in 

 (13) is seen to be merely a special case of Art. 165 (12). 



In the critical case of = *, we have 



P 



y=--- sm ^ 



shewing that the excess of pressure is now on the slopes which face down the 

 stream. This explains roughly how a system of progressive waves may be 

 maintained against our assumed dissipative forces by a properly adjusted 

 distribution of pressure over their slopes. 



227. The solution expressed by (13) may be generalized, in 

 the first place by the addition of an arbitrary constant to x t and 

 secondly by a summation with respect to k. In this way we may 

 construct the effect of any arbitrary distribution of pressure, say 



using Fourier's expression 



/(#)=!( dk( f(\)cosk(x-\)d\ ........ (15). 



7T J o J -oo * 



It will be sufficient to consider the case where the imposed 

 pressure is confined to an infinitely narrow strip of the surface, 

 since the most general case can be derived from this by in- 

 tegration. We will suppose then that /"(A,) vanishes for all but 

 infinitely small values of X, so that (15) becomes 



x)=- 



7T J 



(16)*. 



* The indeterminateness of this expression may be avoided by the temporary use 

 of Poisson's formula 



-00 ,.00 



/(a) = Lt - I e~ ak dk f (\) cos k (x - \) d\ 



a = * J o J - 



in place of (15). 



