1 70 Lord Rayleigh on the Nodal Lines of a Square Plate. 



(on the scale adopted). The first curve proceeding outwards is 

 the locus of points at which z—\. The next is the nodal line, 

 separating the regions of opposite displacement. The remaining 

 curves taken in order give the displacements —1, « — 2, —3. 

 The numerically greatest negative displacement does not occur 

 until the corners of the square, where it amounts to 2 x 1*6448 

 = 3-2896. 



The same calculations and constructions give the theoretical 

 solution for another and even better-known mode of vibration of 

 a square plate. We have hitherto supposed that the two com- 

 ponent vibrations were in the same phase. Tf we take them in 

 opposite phases, the curves of equal displacement become 



z x —z y = const., ...... (5) 



which are to be found by crossing the same network of straight 

 lines along the other diagonals. In this mode of vibration the 

 nodal lines are the diagonals of the square ; and the hyperbolic 

 curves in the figure (when completed on the opposite side) are 

 the loci of points where the displacement amounts respectively 

 to 1 and 2. The curves similarly situated in the other two por- 

 tions cut off from the square by the diagonals would correspond 

 to displacements —1 and — 2. The maxima of vibration occur 

 at the middle points of the sides of the square, and amount nu- 

 merically to 1 + 1-6448 = 2-6448. 



Other modes of vibration of a square plate may be obtained 

 by starting from the higher fundamental modes of a bar with 

 three or more nodes. Some of these I hope to examine in my 

 work on Acoustics, now in preparation. 



Strehlke expresses the form of the nodal line as given by his 

 observations by means of polar equations, the origin being taken 

 at the centre of the square. This was a natural course to take 

 in view of the approximate circular form, but does not commend 

 itself to the theorist. The equations representing the results 

 obtained with three different plates are 



r = -40143 + -01 71 cos 4/ + -00127 cos 8/, 



r= -40143 + -0172 cos U + -00127 cos 8/, 



r = -4019 + -0168 cos 4/ + '0013 cos 8/. 



From these we obtain for the radius vector parallel to the sides 

 of the square -41980, -41981, -4200, while the theoretical result 

 is -4154. The radius vector measured along a diagonal is -3856, 

 •3855, -3864, but from theory -3900. 



The agreement between theory and observation is not so good 

 as might have been expected from the case of a circular plate*, 

 for which the theoretical results were calculated by Poisson and 



