Lord Rayleigh on the Nodal Lines of a Square Plate. 1G7 



of vibration at the ends is decidedly (more than in the ratio of 

 3 : 2) greater than at the centre, from which it follows that the 

 nodal line does not pass through the centres of the sides, and 

 the nodes of a vibrating bar deviate sensibly from the positions 

 assigned to them. It will be observed that the four points in 

 which the required locus intersects the diagonals of the square 

 are determined by a knowledge of the nodes of the component 

 vibrations; for a point which lies on both these systems neces- 

 sarily satisfies the condition for the nodal line of the resultant 

 vibration. Further it is obvious that no part of the nodal line 

 can lie in those compartments of the plate where the component 

 vibrations have the same sign, whether positive or negative; 

 that is to say, the curve passes through the rectangles, and not 

 through the squares, into which the disk is divided by the two 

 primary systems of nodes. The reproduction of Wheatstone's 

 argument in a work so well known as Tyndall's e Lectures on 

 Sound ; relieves me from the necessity of stating it at greater 

 length here. 



The algebraical calculation of the form of a bar vibrating 

 freely was given originally by Euler; but I do not find that the 

 result has been reduced to numbers further than was necessary 

 for calculating the position of the nodes. If the distance of any 

 point from the end, expressed as a fraction of the total length, 

 be w, and z denote the transverse displacement, we have the fol- 

 lowing expression, giving the relative displacements of the dif- 

 ferent points of the bar* : 



+e « sin (l|M) +e - Mcos (l|M) ; . . . (1) 



where, approximately, 



m=4*7300, /3='0176 (2) 



v 

 For numerical computation put #= — ; then on reduction we 



have z expressed as the sum of three terms, the first of which is 

 - >/2sin (r x 13° 33'-45° 30' 15"), and the logarithms of the 

 two others rx '10271 +3-9444 and -rx '10271 + 1-99998 re- 

 spectively. 



From this formula z was calculated for integral values of r 

 from to 10, and the results reduced in such a proportion as to 

 make the last term (that corresponding to the middle of the rod) 



* Donkin's ' Acoustics/ p. 190. More exact values are m=4'73004, 

 /3=-01765, 



