356 Lord Rayleigh on Vibrations and 



by a rise of pitch. But ('Theory of Sound/ § 230) the state- 

 ment does not hold good when the boundary is free. 



When a localised transverse force acts upon the plate, we 

 may inquire whether the displacement is at all points in 

 the same direction as the force. This question was 

 considered in a former paper * in connexion with Fig. 2. 

 a hy drodynamical analogue, and it may be convenient 

 to repeat the argument. Suppose that the plate 

 (fig. 2), clamped at a distant boundary, is almost 

 divided into two independent parts by a straight 

 partition CD extending across, but perforated by a 

 narrow aperture AB ; and that the force is applied 

 at a distance from CD on the left. If the partition , 

 were complete, w and div/dn would be zero over the 

 whole (in virtue of the clamping), and the displace- |B 

 ment in the neighbourhood on the left would be 

 simple one-dimensional bending, with w positive 

 throughout. On the right w would vanish. In 

 order to maintain this condition of things a certain Iq 

 couple acts upon the plate in virtue of the supposed 

 constraints along CD. 



Along the perforated portion AB the couple required to 

 produce the one-dimensional bending fails. The actual 

 deformation accordingly differs from the one-dimensional 

 bending by the deformation that would be produced by a 

 couple over AB acting upon the plate, as clamped along CA, 

 BD, but otherwise free from force. This deformation is 

 evidently symmetrical with change of sign upon the two 

 sides of CD, w being positive on the left, negative on the 

 right, and vanishing on AB itself. Thus upon the whole a 

 downward force acting on the left gives rise to an upward 

 motion on the right, in opposition to the general rule 

 proposed for examination. 



If we suppose a load attached at the place where the force 

 acted, but that otherwise the plate was devoid of mass, we 

 see that a clamped plate vibrating freely in its gravest 

 mode may have internal nodes in the sense that w is there 

 evanescent, but of course not in the full sense of places 

 which behave as if they were clamped. 



In the case of a plate whose boundary is merely supported, 

 i. e. acted upon by a force (without couple) constraining w to 

 remain zerof, it is still easier to recognize that a part of 



* Phil. Mag. vol. xxxvi. p. oo4 (1893) ; Scientific Papers, vol. iv. 

 p. 88. 



t It may be remarked that the substitution of a supported for a 

 clamped boundary is equivalent to the abolition of a constraint, and is 

 in consequence attended by a fall in the frequency of free vibrations. 



