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1911]
CHLADNI'S FIGURES   FOR  A SQUARE   PLATE
49
in which q is Young's modulus, and 2h the thickness of the plate ( 214). Also for the kinetic energy per unit area we have
T=phw~, .....................................(2)
p being the volume-density. From the symmetries of the case w must be an odd function of x and an odd function of y, and it must also be symmetrical between x and y. Thus we may take
w = q^xy + q2xy (a? 4- y2) + qsxy (& + 2/4) + q^y3 +..........(3)
In the actual calculation only the two first terms will be employed.
Expressions (1) and (2) are to be integrated over the square; but it will suffice to include only the first quadrant, so that if we take the side of the square as equal to 2, the limits for x and y are 0 and 1. We find
...........................w
drw d ^e2 d
Thus, if we set
^ = 57-,   -: V',     ...........................(6)
6(1 + /ju)                                                   /
we have                     V^^tf + 2giga + $ql +   _"   ......................(7)
In like manner, if
rp _ *P'1 rnt                                                               /ON
2 g"^ ' .................................W
When we neglect q2 and suppose that ql varies as cosj9i5, bhese expressions give
^ 7,>.......................(10)
if we introduce a as the length of the side of the square.    This is the value found in Theory of Sound,  228, equivalent to Ritz's first approximation.
In proceeding to a second approximation we may omit the factors already accounted for in (10). Expressions (7), (9) are of the standard form if we take
A =1,
= 2,
n  _
6     18
R. VI.n Theory of Sound, "Vol. II. Appendix 1 I had supposed that this treatise abounded in applications of the method i question, see  88, 89, 90, 91, 182, 209, 210, 265 ; but perhaps the mo; explicit formulation of it is in a more recent paper J, where it takes almos exactly the shape employed by Kitz. From the title it will be seen th* I hardly expected the method to be so successful as Ritz made it in the cas of higher modes of vibration.