412 Chapter 14Partial differential equations
in space of the first few of these normal modes are illustrated in Figure 14.7 (see also
Figure 12.6 for the particle in a box).
The nth mode has n 1 − 11 nodesbetween the end points; that is, points of zero
displacement that do not move; the solutions (14.98) represent standing waves.
The complete solution
Each of the normal modes contains two constants,A
nandB
n, that are to be determined
by the initial conditions (14.88); that is, by the way the motion is initiated. It is possible
to choose the initial conditions so that the actual motion is one of the normal modes;
for example, the nth mode is produced if
(14.99)
In general however, a single mode will not satisfy the initial conditions; the motion is
not a pure normal mode but is a mixture or superposition of normal modes.
To obtain the solution for the general case, we invoke the principle of superposition
discussed in Section 12.2; ify
1, 1 y
2, 1 y
3,1=are solutions of a homogeneous linear
equation then any linear combination of them is also a solution. Thus, for each
normal mode we have
(14.100)
so that, if
y 1 = 1 c
1y
11 + 1 c
2y
21 + 1 c
3y
31 +1- (14.101)
then
(14.102)
∂
∂
=
∂
∂
22222y 1
x
y
v t
∂
∂
=
∂
∂
222221
y
x
y
t
nnv
tnny
t
gx B
nx
l
=∂
∂
==
0() ω sin
π
yx f x A
nx
l
n() () sin,= 0 =
π
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................................Figure 14.7