357.
ON THE CALCULATION OF CHLADNTS FIGURES FOR A SQUARE PLATE.
[Philosophical Magazine, Vol. xxn. pp. 225—229, 1911.]
IN my book on the Theory of Sound, ch. x. (1st ed. 1877, 2nd ed. 1894) had to speak of the problem of the vibrations of a rectangular plate, whose Iges are free, as being one of great difficulty, which had for the most part ssisted attack. An exception could be made of the case in which /u, (the i,tio of lateral contraction to longitudinal elongation) might be regarded as rancscent. It was shown that a rectangular plate could then vibrate after 10 same law as obtains for a simple bar, and by superposition some of the mpler Chladni's figures for a square plate were deduced. For glass and :etal the value of //, is about £, so that for such plates as are usually experi-.ented on the results could be considered only as rather rough approxi-iations.
I wish to call attention to a remarkable memoir by W. Ritz* in which, nnewhat on the above lines, is developed with great skill what may be igarcled as a practically complete solution of the problem of Chladni's gures on square plates. It is shown that to within a few per cent, all the roper tones of the plate may be expressed by the formulae
Wwtn = um (x) un (y) + um(y) un («),
W'mn = Um (x) Un (y) - Um (y) Un («),
le functions u being those proper to a free bar vibrating transversely. The )ordinate axes are drawn through the centre parallel to the sides of the juare. The first function of the series u0 (x) is; constant; the second i(x}=%. const.; wa (x) is "thus the fundamental vibration in the usual sense, ith two nodes, and so on. Ritz rather implies that I had overlooked the
* "Theorie der Trahsversalschwingungen einer quadratischen Platte rait freien-BSndern,-'' nnalen der Physik, Bd, xxvni, S. 737 (1909). The early death of the talented author-must be scounted a severe loss to Mathematical Physics.p. 451.time is that mso+^^i —^ z Khali remain unchanged. Thus the amplitude which is to be found at #=0 on the screen prevails also behind the screen along the line