104 DYNAMICAL THEORY OF SOUND 



reduced to rest. In the excepted case the conditions (4) are 

 satisfied independently of the values of A s and B s whenever 

 sin (STTX/I) = 0, i.e. those normal modes remain unaffected 

 which have a node at the point touched. 



A question arises as to the effect of non-periodic forces on 

 a dynamical system. For the reason already so often insisted 

 upon, it is convenient, whenever possible, to resolve the force 

 into a series of terms of the type 



A cos pt + B sin pt ................... (5) 



Each element A cospt or Bsmpt then produces throughout 

 the system its own effect, viz. an oscillation of the same type 

 and period, the configuration and its amplitude depending on 

 the speed p. In some cases the resolution presents itself quite 

 naturally, as for example in the theory of the tides. The 

 disturbing effect of the sun and moon, when account is taken 

 of their varying declinations, and of the inequalities in their 

 orbital motions, can be sufficiently represented by a series of 

 terms of the type (5). It follows that the tide-height at any 

 particular place must be expressed by a series of like character, 

 in which the values of p are known. The theoretical determina- 

 tion of the coefficients is out of the question for the actual 

 ocean, with its variable depth and irregular boundaries, but 

 their values can be inferred a posteriori with more or less 

 accuracy from a comparison of the formula with observation, 

 and when once ascertained can be used for prediction*. 



When the disturbing force is perfectly arbitrary in character, 

 without any obvious periodic elements, the question is more 

 complicated. There is a form of Fourier's theorem specially 

 appropriate to this case, but its application is usually difficult, 

 and it is simpler to have recourse, as in 38, to the formula (12) 

 of 8. The objection that this implies a knowledge of the 

 whole previous history of the system is met if we introduce 

 the consideration of damping, which is in reality always present. 

 The equation 



.................. (6) 



* For an elementary account of the matter see Sir G. H. Darwin, The Tides, 

 London, 1898. 



