670 BELL SYSTEM TECHNICAL JOURNAL 



It represents a sinusoidal stationary oscillation of the wire, with nodes 

 at the ends and at (^ — 1) points spaced evenly between the ends — a 

 case not difficult to realize with a violin-string, if k be not too great. 

 The constants C and D specify the amplitude of the oscillation, and 

 its phase at any given instant. 



It is of course not necessary that the motion of the wire should 

 conform to a single Eigenfunktion. Any number of Eigenfunktionen, 

 corresponding to different permitted values of m — different integer 

 values of k — might coexist simultaneously, each with its particular 

 values of Ck and D k', the actual distortion of the wire would be the 

 superposition of all. It would in fact be necessary to adjust the initial 

 distortion of the wire and the initial velocities of its points with infinite 

 accuracy, to cause its future motion to conform exactly to a single 

 Eigenfunktion. On the other hand, any choice whatever of initial 

 distortion and initial velocities would entail a future motion com- 

 pounded out of the various Eigenjunktionen with suitable values of 

 Ck and Dk, which could be computed. This process corresponds to 

 that of determining the future orbit of a particle of which the position 

 and the velocity at a gWen instant are preassigned.'-* Both in acoustics 

 and in wave-mechanics it is, as a rule, much more laborious than the 

 determination of natural frequencies; and happily it is often less im- 

 portant, though not always to be neglected. 



Example of the Tensed Membrane 

 The differential equation of the tensed membrane is: 



The coordinate-axes of x and y lie in the equilibrium plane of the 



membrane, and z stands for the displacement of any point of the 



membrane normally from this plane. The symbol u stands for the 



speed of a sine-wave traveling in an infinite membrane of the same 



tension T and surface-density p as the actual one, and is determined 



by the equation : 



u'~ = T/p, (121) 



which is derived by an obvious extension of the method employed in 

 deriving the like equation for a stretched string. In an actual bounded 

 membrane the motion may be tremendously complicated, but it can 



' Inversely, the imposition of quantum-conditions upon orbits corresponds to the 

 determination of natural frequencies; here is the bridge between the atom-models 

 with electron-orbits and the atom-models of wave-mechanics. 



