REPRESENTATION THEORY
M M
M
Figure 29.6 A circular drumskin loaded with three symmetrically placed
masses.A circular drumskin has three equal masses placed on it at the vertices of an equilateral
triangle, as shown in figure 29.6. Determinewhich degenerate normal modes of the drumskin
can be split in frequency by this perturbation.When no masses are present the normal modes of the drum-skin are either non-degenerate
or two-fold degenerate (see chapter 21). The degenerate eigenfunctions Ψ of thenth normal
mode have the forms
Jn(kr)(cosnθ)e±iωt or Jn(kr)(sinnθ)e±iωt.Therefore, as explained above, we need to consider the two-dimensional vector space
spanned by Ψ 1 =sinnθand Ψ 2 =cosnθ. This will generate a two-dimensional representa-
tion of the group 3m(orC 3 v), the symmetry group of the perturbation. Taking the easiest
element from each of the three classes (identity, rotations, and reflections) of group 3m,
we have
IΨ 1 =Ψ 1 ,IΨ 2 =Ψ 2 ,
AΨ 1 =sin[
n(
θ−^23 π)]
=
(
cos^23 nπ)
Ψ 1 −
(
sin^23 nπ)
Ψ 2 ,
AΨ 2 =cos[
n(
θ−^23 π)]
=
(
cos^23 nπ)
Ψ 2 +
(
sin^23 nπ)
Ψ 1 ,
CΨ 1 = sin[n(π−θ)] =−(cosnπ)Ψ 1 ,
CΨ 2 =cos[n(π−θ)] = (cosnπ)Ψ 2.The three representative matrices are therefore
D(I)=I 2 , D(A)=
(
cos^23 nπ −sin^23 nπ
sin^23 nπ cos^23 nπ)
, D(C)=
(
−cosnπ 0
0cosnπ)
.
The characters of this representation areχ(I)=2,χ(A)=2cos(2nπ/3) andχ(C)=0.
Using (29.18) and table 29.1, we find that
mA 1 =^16(
2+4cos^23 nπ)
=mA 2
mE=^16(
4 −4cos^23 nπ)
.
Thus
D={
A 1 ⊕A 2 ifn=3, 6 , 9 ,...,
Eotherwise.Hence the normal modesn=3, 6 , 9 ,...each transform under the operations of 3m