10 DYNAMICAL THEORY OF SOUND 



the formula (2), provided it be started at the proper instant. 

 The angle nt+e(=AOQ) is called the "phase"; and the 

 elements a, e are called the " amplitude " and the " initial 

 phase," respectively. The interval Zir/n between two suc- 

 cessive transits through the origin in the same direction is 

 called the " period." In acoustics, where we have to deal with 

 very rapid vibrations, it is usual to specify, instead of the 

 period, its reciprocal the " frequency " (N), i.e. the number of 

 complete vibrations per second ; thus 



In the case of the pendulum, where n = *J(g/l), the period 

 is 2ir^/(l/g). As in the case of all other dynamical systems 

 which we shall have occasion to consider, this is independent 

 of the amplitude so long as the latter is small (. 



The velocity of P in any position is 



............ (3) 



as appears also by resolving the velocity (na) of Q parallel 

 to OA. 



Fig. 3. 



In all cases of rectilinear motion of a point the method of 

 graphical representation by means of a curve constructed with 



* The want of a separate name for the angular velocity n in the auxiliary 

 circle is sometimes felt. In the theory of the tides the term "speed" was 

 introduced by Lord Kelvin. As an alternative term in acoustics the word 

 "rapidity" may perhaps be suggested. 



f This observation was made by Galileo in 1583, the pendulum being a 

 lamp which hangs in the cathedral of Pisa. 



