ON VIBRATIONS AND DEFLEXIONS OF MEMBRANES, BARS, AND PLATES.
[Philosophical Magazine, Vol. xxxn. pp. 353—364, 1916.]
IN Theory of Sound, § 211, it was shown that "any contraction of the fixed boundary of a vihrating membrane must cause an elevation of pitch, because the new state of things maybe conceived to differ from the old merely •by the introduction, of an additional constraint. Springs, without inertia, are supposed to urge the line of the proposed boundary towards its equilibrium position, and gradually to become stiffer. At each step the vibrations become more rapid, until they approach a limit corresponding to infinite stiffness of the springs and 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 even any 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
(r) has an internal nodal line CB. This requires that a membrane with the fixed boundary ACB shall also be capable of vibration in period r. The impossibility is easily seen. As ACDB gradually contracts through ACD'B to ACS, the longest period diminishes, so that the longest period of AGB is less than T. No period possible to ACB can be equal to r.lid obstacle between the source of sound and the listener.]now concerned. The problem of the grating is trea Theory of Sound, 2nd edition, § 272 a.