following table, we display a list of all relative vibrational frequencies (12.158) that are
<
6.
Once the lowest frequency
ω
0
,
1
has been determined — either theoretically, numerically
or experimentally — all the higher overtones
ω
m,n
=
ρ
m,n
ω
0
,
1
are simply obtained by
rescaling.
Relative Vibrational Frequencies of a Circular Disk
n
backslashbigg
m
0
1
2
3
4
5
6
7
8
9
. . .
1
1
.
000
1
.
593
2
.
136
2
.
653
3
.
155
3
.
647
4
.
132
4
.
610
5
.
084
5
.
553
. . .
2
2
.
295
2
.
917
3
.
500
4
.
059
4
.
601
5
.
131
5
.
651
.
.
.
.
.
.
.
.
.
3
3
.
598
4
.
230
4
.
832
5
.
412
5
.
977
.
.
.
.
.
.
4
4
.
903
5
.
540
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Nodal Curves
When a membrane vibrates, its individual atoms move up and down in a quasi
periodic manner.
As such, there is little correlation between their motion at different
locations. However, if the membrane is set to vibrate in a pure eigenmode, say
u
n
(
t, x, y
) = cos(
ω
n
t
)
v
n
(
x, y
)
,
(12
.
159)
then all points move up and down at a common frequency
ω
n
=
radicalbig
λ
n
, which is the square
root of the eigenvalue corresponding to the eigenfunction
v
n
(
x, y
). The exceptions are the
points where the eigenfunction vanishes:
v
n
(
x, y
) = 0
,
(12
.
160)
which remain stationary. The set of all points (
x, y
)
∈
Ω that satisfy (12.160) is known
as the
n
th
nodal set
of the domain Ω. Scattering small particles (e.g., fine sand) over a
membrane vibrating in an eigenmode will enable us to visualize the nodal set, because the
particles will tend to accumulate along the stationary nodal curves.
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Normal mode, Peter J. Olver, nodal curves

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