175-176] WAVES IN A FINITE CANAL. 285 



horizontal displacement of the water is always in the same 

 phase with the force, so long as the period is greater than 

 that of the slowest free mode, or <rl/2c < \TT. If the period be 

 diminished until it is less than the above value, the phase is 

 reversed. 



When the period is exactly equal to that of a free mode of 

 odd order (s=l, 3, 5,...), the above expressions for f and 77 

 become infinite, and the solution fails. As pointed out in 

 Art. 165, the interpretation of this is that, in the absence of 

 dissipative forces (such as viscosity), the amplitude of the motion 

 becomes so great that the foregoing approximations are no longer 

 justified. 



If, on the other hand, the period coincide with that of a free 

 mode of even order (s = 2, 4, 6,...), we have sino-//c = 0, 

 coso-//c = l, and the terminal conditions are satisfied indepen- 

 dently of the value of D. The forced motion may then be 

 represented by 



............ (10)* 



The above example is simpler than many of its class in that it is possible 

 to solve it without resolving the impressed force into its normal components. 

 If we wish to effect this resolution, we must calculate the work done during 

 an arbitrary displacement 



. 



L 



Since X denotes the force on unit mass, we have 



whence Q s = C 8 cos(<Tt+) (i), 



11 n 1 COSS7T 07 ,. .... 



provided C a = . pSlf (11). 



STT 



This vanishes, as we should expect, for all even values of s. The solution of 

 our problem then follows from the general formulae of Art. 165. The 

 identification (by Fourier's Theorem) of the result thus obtained with that 

 contained in the formulae (8) is left to the reader. 



* In the language of the general theory, the impressed force has here no 

 component of the particular type with which it synchronizes, so that a vibration 

 of this type is not excited at all. In the same way a periodic pressure applied at 

 any point of a stretched string will not excite any fundamental mode which has a 

 node there, even though it synchronize with it. 



