[From the Encyclopaedia Britannica.} 



CI. Harmonic Analysis. 



HARMONIC ANALYSIS is the name given by Sir William Thomson and 

 Professor Tait in their treatise on Natural Philosophy to a general method of 

 investigating physical questions, the earliest applications of which seem to have 

 been suggested by the study of the vibrations of strings and the analysis of 

 these vibrations into their fundamental tone and its harmonics or overtones. 



The motion of a uniform stretched string fixed at both ends is a periodic 

 motion ; that is to say, after a certain interval of time, called the fundamental 

 period of the motion, the form of the string and the velocity of every part 

 of it are the same as before, provided that the energy of the motion has not 

 been sensibly dissipated during the period. 



There are two distinct methods of investigating the motion of a uniform 

 stretched string. One of these may be called the wave method, and the other 

 the harmonic method. The wave method is founded on the theorem that in 

 a stretched string of infinite length a wave of any form may be propagated 

 in either direction with a certain velocity, V, which we may define as the 

 " velocity of propagation." If a wave of any form travelling in the positive 

 direction meets another travelling in the opposite direction, the form of which 

 is such that the lines joining corresponding points of the two waves are all 

 bisected in a fixed point in the line of the string, then the point of the 

 string corresponding to this point will remain fixed, while the two waves pass 

 it in opposite directions. If we now suppose that the form of the waves 

 travelling in the positive direction is periodic, that is to say, that after the 

 wave has travelled forward a distance I, the position of every particle of the 

 string is the same as it was at first, then Z is called the wave-length, and the 

 tune of travelling a wave-length is called the periodic time, which we shall 

 denote by T, so that 



l=VT. 



