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604 CHAPTER 16 The Wave Equation

J (x) y = J (x)
0
0
x = 0 x = j 1
x = 0 x = j 1 x = j 2
FIGURE 16.22 First normal mode of
vibration. FIGURE 16.23 Second normal mode.

16.7.1 Normal Modes of Vibration
The numbers ω n = j n c/R are the frequencies of normal modes of vibration of the membrane with
periods 2π/ω n = 2π R/j n c.The normal modes of vibration are the functions z n (r,t), which are
often written in phase angle form as

j n r

z n (r,t) = A n J 0 cos(ω n t + δ n )
R

in which ω n = j n c/R, A n = a + b and δ n = arctan(−b n /a n ) if a n = 0.
2
2
n n
The first normal mode is
j 1 r

z 1 (r,t) = A 1 J 0 cos(ω 1 t + δ 1 ).
R
As r varies from 0 to R, j 1 r/R varies from 0 to j 1 , the first positive zero of J 0 .Atany time
t, a radial section through the membrane takes the shape of the graph of J 0 (x) for 0 ≤ x ≤ j 1
(Figure 16.22).
The second normal mode is

j 2 r
cos(ω 2 t + δ 2 ).
z 2 (r,t) = A 2 J 0
R
As r varies from 0 to R, j 2 r/R varies from 0 to j 2 , passing through j 1 along the way. Since
J 0 ( j 2 r/R) = 0 when j 2 r/R = j 1 , this mode has a nodal circle (fixed in the motion) at radius
r = j 1 R/j 2 . A section through the membrane takes the shape of the graph of J 0 (x) for 0 ≤ x ≤ j 2
(Figure 16.23).
Similarly, the third normal mode is
j 3 r

z 3 (r,t) = A 3 J 0 cos(ω 3 t + δ 3 )
R
and this mode has two nodes, one at r = j 1 R/j 3 and the second at r = j 2 R/j 3 . Now a radial section
has the shape of a graph of J 0 (x) for 0 ≤ x ≤ j 3 (Figure 16.24).
In general, the nth normal mode has N − 1 nodes (fixed circles in the motion of the
membrane), occurring at j 1 R/j n , j 2 R/j n , ··· , j n−1 R/j n .

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October 14, 2010 15:23 THM/NEIL Page-604 27410_16_ch16_p563-610