THE 



LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[SIXTH SERIES.] 



OCTOBER 1916. 



XXXIX. On Vibration* and Deflexions of Membrane s, Bars, 

 a„<l, Plates. By Lord Rayleigh, O.M., F.B.8.* 



IN 'Theory of Sound,' § 21 1, it was shown that "any con- 

 traction of the fixed boundary of a vibrating membrane 

 mast cause an elevation of pitch, because the now state ofc' 

 things may be conceived to differ from the old merely 

 by the introduction of an additional constraint. Springs, 

 without inertia, are supposed to urge fche line of the proposed 

 boundary towards its equilibrium position, and gradually fco 

 become stiffer. At, each step the vibrations become more 

 rapid, nnt.il they approach a limit corresponding to infinite 

 stiffness of the springs arid absolute fixity of their points of 

 application* It is not necessary that the part cut off should 

 have the same density as the rest, or oven anv density 

 at all." 



From this principle we may infer that the gravest mode 

 of vibration for a membrane of any shape and of any variable 

 density is devoid of internal nodal lines. For suppose that 

 ACDB (fig. 1) vibrating in its longest period (t) has an 

 internal nodal line CB. This requires that a membrane with 

 the fixed boundary AC13 shall also be capable of vibration 

 in period r. The impossibility is easily seen. As ACDB 

 gradually contracts through ACD'B to ACB, the longest 

 period diminishes, so that the longest period of ACB is less 

 than t. No period possible to AT 1 \\ can be equal to t. 



* Communicated by the Author. 



Phil. Ma (/ . S. 6. Vol. 32. No. 190. Oct. 1916. 



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