HARMONIC ANALYSIS. 799 



its position at any instant , to a fixed diameter A A' of the circle, intersects the 

 diameter in a point P whose position changes by a simple harmonic motion. 



The amplitude of a simple harmonic motion is the range on one side or the 

 other of the middle point of the course. 



The period of a simple harmonic motion is the time which elapses from 

 any instant until the moving-point again moves in the same direction through 

 the same position. 



The phase of a simple harmonic motion at any instant is the fraction of 

 the whole period which has elapsed since the moving point last passed through 

 its middle position in the positive direction. 



In the case of the stretched string, it is only in certain particular cases 

 that the motion of a particle of the string is a simple harmonic motion. In 

 these particular cases the form of the string at any instant is that of a curve 

 of sines having the line joining the fixed points for its axis, and passing 

 through these two points, and therefore having for its wave-length either twice 

 the length of the string or some submultiple of this wave-length. The ampli- 

 tude of the curve of sines is a simple harmonic function of the time, the 

 period being either the fundamental period or some submultiple of the funda- 

 mental period. Every one of these modes of vibration is dynamically possible 

 by itself, and any number of them may coexist independently of each other. 



By a proper adjustment of the initial amplitude and phase of each of these 

 modes of vibration, so that their resultant shall represent the initial state of 

 the string, we obtain a new representation of the whole motion of the string, 

 in which it is seen to be the resultant of a series of simple harmonic vibra- 

 tions whose periods are the fundamental period and its submultiples. The 

 determination of the amplitudes and phases of the several simple harmonic 

 vibrations so as to satisfy the initial conditions is an example of harmonic 

 analysis. 



We have thus two methods of solving the partial differential equation of 

 the motion of a string. The first, which we have called the wave-method, ex- 

 hibits the solution in the form containing an arbitrary function, the nature of 

 which must be determined from the initial conditions. The second, or harmonic 

 method, leads to a series of terms involving sines and cosines, the coefficients 

 of which have to be determined. The harmonic method may be defined in a 



