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XVI. On the Nodal Lines of a Square Plate. 

 By Lord Rayleigh, F.R.S. (Hon. J. W. Strutt)*. 



IN a recent Number of PoggendorfPs Annalenf there is a 

 short paper by Strehlke, complaining of the inaccuracy with 

 which the nodal lines of vibrating square plates are depicted in 

 certain elementary works, and drawing attention to the results 

 of his own careful measurements. His remarks relate princi- 

 pally to that mode of vibration which has for nodal line a closed 

 curve not greatly differing from a circle, but which has been erro- 

 neously regarded by some as a slightly modified form of the 

 inscribed square. 



On the experimental determination of nodal lines there is no 

 greater authority than Strehlke; and if there were no light from 

 theory, his results might be considered to exhaust the question. 

 Strehlke, indeed, quotes Professor Kirchhoff as expressing the 

 opinion that there is at present no prospect at all of a theoretical 

 solution of the problem — an opinion which may be correct if 

 understood to refer to a perfectly general theory, but which is 

 certainly erroneous if intended to apply to the particular mode 

 of vibration under discussion. So long ago as 1833 Wheatstone 

 pointed out the right path J, though he did not follow it cor- 

 rectly. He considers the vibration as resulting from the super- 

 position of two independent but equal and synchronous vibra- 

 tions, in each of which the plate vibrates according to the same 

 law as a simple bar, each line of the plate parallel to one pair of 

 edges being affected by the same motion. On account of the 

 symmetry, the period is the same, whichever pair of edges be 

 taken ; and thus the two vibrations compound into one having 

 the same period, whatever may be the ratio of amplitudes. In 

 the present case the two vibrations must be considered to have 

 the same phase and equal amplitudes, so that the motion at the 

 centre of the square is simply doubled. The nodal line is then 

 the locus of points at which the component vibrations neutralize 

 each other. 



This view is perfectly correct ; but Wheatstone went wrong 

 in his application of it, from not sufficiently considering what 

 the law of vibration of a bar really is. His conclusion that the 

 nodal line coincides with the inscribed square involves the sup- 

 positions that when a bar vibrates freely (in its gravest mode) 

 the amplitudes at its centre and ends are numerically equal, and 

 that the nodes lie midway between these points. As a matter 

 of fact neither of these suppositions is correct; the amplitude 



* Communicated by the Author. 

 t Pogg. Ann. vol. cxlvi. p. 319. 

 % Phil. Trans. 1833. 



