34 DYNAMICAL THEORY OF SOUND 



unaltered. The intensity sinks to one-half its maximum when 

 a? = fp, or 



-ll (5) 



n nr 



Thus if the damping be such that a free vibration would have 

 its amplitude diminished in the ratio l/e in 10, 100, 1000 

 periods*, respectively, the corresponding values of the interval 

 p/n at which the dissipation would be reduced to one-half the 

 maximum would be 1 '016, 1 '0016, 1 '00016. The curve 

 in (4) is shewn in Fig. 14. 



The above argument deals with the dissipation, which is the 

 most important feature. The consideration of the square of the 

 amplitude, or of the energy stored in the system, leads to very 

 similar results, especially when the damping is slight. 



14. Systems of Multiple Freedom. Examples. The 

 Double Pendulum. 



We approach the consideration of systems having any finite 

 number of degrees of freedom. A system is said to have ra 

 such degrees when m independent variables, or " coordinates," 

 are required and are sufficient to specify the various configura- 

 tions which it can assume. The notion, first brought into 

 formal prominence by Lord Kelvin f, has a wide application 

 in mechanism and in theoretical mechanics. In the case of 

 the telescope of an altazimuth instrument or of an equatorial 

 we have m = 2; in the gyroscope, or (more generally) in any 

 case of a rigid body free to turn about a fixed point, m = 3 ; 

 for a rigid structure or frame movable in two dimensions 

 m = 3; for a rigid structure freely movable in space m = 6. 

 The choice of the coordinates in any particular case can be 

 made in an endless variety of ways, but the number is always 

 determinate. Thus in technical mechanics we have the pro- 

 position that a rigid frame movable in one plane can be fixed by 



* In an experiment by Lord Eayleigh, the number of periods for a particular 

 tuning fork of 256 v.s. was about 5900. When a resonator was used the number 

 fell to 3300. Theory of Sound, vol. n., p. 436. 



t William Thomson, afterwards Lord Kelvin (18241907), Professor of 

 Natural Philosophy at Glasgow 184699. The matter is explained in Thomson 

 and Tait's Natural Philosophy, 2nd ed., 195201 (1879). 



