HARMONIC ANALYSIS. 



If we now suppose a set of waves similar to these, but reversed in position, 

 lo b travelling in the opposite direction, there will be a series of point*. 

 dMtent 4/ from each other, at which there will be no motion of the string; 

 I therefore make no difference to the motion of the string if we suppose 

 UM string frstmnd to fixed supports at any two of these points, and we may 

 ll^g mum+ii the part* of the string beyond these points to be removed, as it 

 taM1 rt afleot the motion of the part which is between them. We have thus 

 arriTjd ^ the caee of a uniform string stretched between two fixed supports, 

 and w conclude that the motion of the string may be completely represented 

 a the reraltant of two sets of periodic waves travelling in opposite direc- 

 tion*, thffir wave-lengths being either twice the distance between the fixed 

 or a submultiple of this wave-length, and the form of these waves, 

 to this condition, being perfectly arbitrary. 



To make the problem a definite one, we may suppose the initial displace- 

 and Telocity of every particle of the string given in terms of its distance 

 from one end of the string, and from these data it is easy to calculate the 

 farm which is common to all the travelling waves. The form of the string 

 at any subsequent time may then be deduced by calculating the positions of 

 the two sets of waves at that time, and compounding their displacements. 



Thus in the wave-method the actual motion of the string is considered 

 as the resultant of two wave-motions, neither of which is of itself, and without 

 the other, consistent with the condition that the ends of the string are fixed. 

 Each of the 'wave-motions is periodic with a wave-length equal to twice the 

 distance between the fixed points, and the one set of waves is the reverse of 

 the other b respect of displacement and velocity and direction of propagation ; 

 hot, subject to these conditions, the form of the wave is perfectly arbitrary. 

 The motion of a particle of the string, being determined by the two waves which 

 pass orer it in opposite directions, is of an equally arbitrary type. 



In the harmonic method, on the other hand, the motion of the string is 

 regarded as compounded of a series of vibratory motions which may be infinite 

 in number, but each of which is perfectly definite in type, and is in fact a 

 particular solution of the problem of the motion of a string with its ends 

 fixed. 



\\ ..-.. 



A simple harmonic motion is thus defined by Thomson and Tait ( 53) : 



in a point Q moves uniformly in a circle, the perpendicular QP, drawn from 



