CHAPTER II 



STRINGS 



22. Equation of Motion. Energy. 



We proceed to the more or less detailed study of the 

 vibrations of various types of continuous systems. Amongst 

 these the first place must for many reasons be assigned to 

 the transverse vibrations of a uniform tense string. Historically, 

 this was the first problem of the kind to be treated theoretically. 

 The mathematical analysis is simple, and various points of the 

 general theory sketched in the preceding chapter receive 

 interesting illustrations, which are moreover easily verified 

 experimentally. Again, the sequence of the natural periods 

 of free vibration has the special "harmonic" relation which 

 has long been recognized as in some way essential to good 

 musical quality, although the true reason, which is ultimately a 

 matter of physiology, has only in recent times been investigated. 

 The mathematical theory has further suggested some remarkable 

 theorems, as to the resolution of a vibration of arbitrary type 

 into simple-harmonic constituents, which are of far-reaching 

 significance. Finally it is to be noted that in the propagation 

 of a disturbance along a uniform string we have the first and 

 simplest type of wave-motion. 



The string is supposed to be of uniform line-density p, 

 and to be stretched with a tension P. The axis of x is taken 

 along the equilibrium position, and we denote by y the trans- 

 verse deflection at the point x, at time t. It is assumed that 

 the gradient dy/dx of the curve formed by the string at any 

 instant is so small that the change of tension may be 



