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while the mode of oscillation corresponding to the functions f2,1 and g2,1 will
vibrate with frequency
−λ2,1
α2,1
=
= .81736.
2π
2π
A general vibration of the vibrating membrane will be a superposition of
these modes of oscillation. The fact that the frequencies of oscillation of a
circular drum are not integral multiples of a single fundamental frequency (as
in the case of the violin string) limits the extent to which a circular drum can
be tuned to a speciﬁc tone.
Proof that J0 (x) has inﬁnitely many positive zeros: First, we make a change
of variables x = ez and note that as z ranges over the real numbers, the corresponding variable x ranges over all the positive real numbers. Since
dx = ez dz, d
1d
=z
dx
e dz and hence x 1d
d
d
= ez z
=
.
dx
e dz
dz Thus Bessel’s equation (5.30) in the case where n = 0 becomes
d2 y
+ e2z y = 0.
dz 2 (5.31) Suppose that z0 > 1 and y (z ) is a solution to (5.31) with y (z0 ) = 0. We
claim that y (z ) must change sign at some point between z0 and z0 + π ....
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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