664 BELL SYSTEM TECHNICAL JOURNAL 



a tuned circuit— each of these vibrates in a wave-pattern of "nodes" 

 and "loops" if the frequency of vibration imposed upon it conforms to 

 one of its own "natural frequencies" or "resonance frequencies." 

 To each of these natural frequencies corresponds a particular pattern 

 of loops and nodes; when one of them is impressed upon the medium, 

 its corresponding wave-pattern springs into existence, and would con- 

 tinue forever were it not for friction internal or external. When any 

 frequency not agreeing with one of the resonances is imposed upon the 

 bounded medium, the resulting motion is very much more compli- 

 cated. The calculation of these natural frequencies, the mapping 

 of these vibration-patterns, is performed by using the methods of one 

 of the great divisions of mathematical physics — the methods under- 

 lying the Theory of Acoustics. 



May the Stationary States, then, of a natural atomic system be 

 visualized as stationary wave-patterns such as these, and their energy- 

 values as the products of the natural frequencies by the constant of 

 Planck? Are the problems of atomic theory to be solved by devising 

 atom-models imitated after familiar resonant bodies or tuned circuits, 

 and applying to these "acoustic models" the mathematical technique 

 of the Theory of Acoustics? This idea was developed by E. Schroe- 

 dinger.'' 



Familiar Examples of Stationary Wave-Patterns 

 To display the laws governing wave-patterns, I will develop three 

 examples: the stretched string, the tensed membrane, the ball of 

 fluid confined in a spherical shell. The first of these is the simplest 

 and most familiar of all instances; excursions into the theory of 

 vibrating systems commence always at the wire of the piano and the 

 string of the violin. Physically, this is a case of one dimension (dis- 

 tance, measured along the length of the string) ; mathematically, it is 

 a case of two variables (that distance, and the time). The example 

 of the tensed membrane is not unfamiliar in the practice of telephony, 

 though many of the diaphragms of actual instruments are too thick 

 to be considered such ; for a membrane is, by definition, infinitely thin. 

 It is a case of two dimensions and three variables. It will reveal to 

 us the desirability of choosing for each specific problem its appropriate 

 set of coordinates; and we shall observe what happens when one of 

 the chosen coordinates is cyclic, being an angle which for all practical 

 purposes returns to its original value when increased by lir; and we 



^ Since the present article is based henceforth chiefly on Schroedinger's pubhca- 

 tions, I wish to make particular reference here to works embodying de BrogHe's 

 contributions: his own Ondes et mouvements (Paris, Gauthier-Villars, 1926) and 

 article in Jour, de Pliys. (6), 7, pp. 321-337 (1926); L. Brillouin, ibid., pp. 353-368. 



