12 DYNAMICAL THEORY OF SOUND 



the type (1). The inertia of the spring itself is here 

 neglected*. 



Again, suppose we have a mass M attached to a wire which 

 is tightly stretched between 

 two fixed points with a ten- 

 sion P. We neglect gravity 

 and the inertia of the wire Flg> 5 ' 



itself; and we further assume the lateral displacement (x) to 

 be so small that the change in tension is a negligible fraction 

 of P. If a, b denote the distances of the attached particle 

 from the two ends, we have 



which is of the same form as 4 (3), with n? = P (a + b)/Mab. 

 The frequency is therefore 



ab 



This case is of interest because acoustical frequencies can 

 easily be realized. Thus if the tension be 10 kilogrammes, 

 and a mass of 5 grammes be attached at the middle, the 

 wire being 50 cm. long, we find N = 63. 



7. Dynamics of a System with One Degree of Freedom. 

 Free Oscillations. 



The above examples are all concerned with the rectilinear 

 motion of a particle, but exactly the same type of vibration 

 is met with in every case of a dynamical system of one degree 

 of freedom oscillating freely, through a small range, about 

 a configuration of stable equilibrium. 



A system is said to have "one degree of freedom" when 

 the various configurations which it can assume can all be 

 specified by assigning the proper values .to a single variable 

 element or "coordinate." Thus, the position of a cylinder 

 (of any form of section) rolling on a horizontal plane is defined 

 by the angle through which it has turned from some standard 

 position. A system of two particles attached at different points 

 of a string whose ends A, B are fixed has one degree of freedom 



* A correction on this account is investigated in 7. 



