the longer the string is (i.e., the larger L gets), the closer the
eigenfrequencies become.
A traveling wave can be expressed as the weighted sum of the
string’s normal modes, so it is the sum of the eigenfunctions,
each weighted by the amplitude A n and vibrating at its eigenfrequency ω n ,
u(x, t) =
n =
∞
∑
0
A n U n (x, ω n ) cos (ω n t).
(42)
An important feature of this solution is that the modes are
orthogonal, meaning that the integral over the string of the
product of two different eigenfunctions is zero,
Ύ
0
2
L
mn
m x
L
n x
L
dx
L
sin
sin
,
π
π
δ
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
=
(43)
where δ mn , the Kronecker delta symbol defined in Eqn A.3.37,
is zero unless m = n. Each mode is independent and cannot be
constructed by combining other modes. Thus we can think
of the displacement of the string as a vector in a vector space
(Section A.3.6) whose basis vectors are the eigenfunctions. Any
particular set of waves is given by the amplitudes A n , which are
the weighting factors of the eigenfunctions or the components
of the basis vectors.
The amplitude for each eigenfunction depends on the position of the source that generated the waves and on the behavior
of the source as a function of time. The spatial part of A n has
the same form as U n (Eqn 40), so
A n = sin (nπx s /L)F(ω n ),
(44)
where x s is the position of the source, and F(ω n ) is a weighting
factor describing how different frequencies contribute to the
time history of the source. Thus the normal mode expression
for the displacement (Eqn 42) can be written
u(x, t) =
n =
∞
∑
0
sin (nπx s /L)F(ω n ) sin (nπx/L) cos (ω n t).
(45)
Figure 2.2-8, computed in this way, illustrates how the first
40 modes of a string with fixed ends and a uniform velocity
combine to give traveling waves. The source, at x s = 8, is
described by
F(ω n ) = exp [−(ω n τ) 2 /4]
(46)
with τ = 0.2. The computer program used is similar to that
discussed in Section A.8.1. The mode sum shows two waves,
one propagating to the right and one propagating to the left,
at the expected positions. Hence the mode sum correctly gives
the propagating waves. In addition to the propagating waves,
we see some small oscillations along the string because only the
first 40 modes were summed.
2.2 Waves on a string 37
Fig. 2.2-8 Displacement of a string with fixed ends computed using
the normal mode formulation. The string has length 20, velocity 3,
and was plucked at time 0 by a source at position 8 (triangle). The bottom
trace shows the displacement of the string at time 1.5, computed by
summing the first 40 modes. The mode sum generates both the rightand the left-propagating waves at the appropriate positions. Spatial
eigenfunctions for the individual modes, each of which corresponds to
an integral number of half wavelengths, are also shown above the sum.
The traces are normalized to unit amplitude.
Mode number
0
10
20
30
40
0
Distance
5
1 0
15
20
Sum
We now have two ways to think of the displacement of the
string as a function of time: either as propagating waves or as
normal modes. Neither is more “real” — both are ways of representing how the displacement evolves. Thus comparing the
two gives interesting insights. For example, consider studying
the properties of the string. In the traveling wave formulation,
we measure travel times and thus infer velocity. In the normal
mode formulation, we measure eigenfrequencies and then infer
velocities. Thus the eigenfrequencies are analogous to the travel
times.
eigenfrequencies become.
A traveling wave can be expressed as the weighted sum of the
string’s normal modes, so it is the sum of the eigenfunctions,
each weighted by the amplitude A n and vibrating at its eigenfrequency ω n ,
u(x, t) =
n =
∞
∑
0
A n U n (x, ω n ) cos (ω n t).
(42)
An important feature of this solution is that the modes are
orthogonal, meaning that the integral over the string of the
product of two different eigenfunctions is zero,
Ύ
0
2
L
mn
m x
L
n x
L
dx
L
sin
sin
,
π
π
δ
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
=
(43)
where δ mn , the Kronecker delta symbol defined in Eqn A.3.37,
is zero unless m = n. Each mode is independent and cannot be
constructed by combining other modes. Thus we can think
of the displacement of the string as a vector in a vector space
(Section A.3.6) whose basis vectors are the eigenfunctions. Any
particular set of waves is given by the amplitudes A n , which are
the weighting factors of the eigenfunctions or the components
of the basis vectors.
The amplitude for each eigenfunction depends on the position of the source that generated the waves and on the behavior
of the source as a function of time. The spatial part of A n has
the same form as U n (Eqn 40), so
A n = sin (nπx s /L)F(ω n ),
(44)
where x s is the position of the source, and F(ω n ) is a weighting
factor describing how different frequencies contribute to the
time history of the source. Thus the normal mode expression
for the displacement (Eqn 42) can be written
u(x, t) =
n =
∞
∑
0
sin (nπx s /L)F(ω n ) sin (nπx/L) cos (ω n t).
(45)
Figure 2.2-8, computed in this way, illustrates how the first
40 modes of a string with fixed ends and a uniform velocity
combine to give traveling waves. The source, at x s = 8, is
described by
F(ω n ) = exp [−(ω n τ) 2 /4]
(46)
with τ = 0.2. The computer program used is similar to that
discussed in Section A.8.1. The mode sum shows two waves,
one propagating to the right and one propagating to the left,
at the expected positions. Hence the mode sum correctly gives
the propagating waves. In addition to the propagating waves,
we see some small oscillations along the string because only the
first 40 modes were summed.
2.2 Waves on a string 37
Fig. 2.2-8 Displacement of a string with fixed ends computed using
the normal mode formulation. The string has length 20, velocity 3,
and was plucked at time 0 by a source at position 8 (triangle). The bottom
trace shows the displacement of the string at time 1.5, computed by
summing the first 40 modes. The mode sum generates both the rightand the left-propagating waves at the appropriate positions. Spatial
eigenfunctions for the individual modes, each of which corresponds to
an integral number of half wavelengths, are also shown above the sum.
The traces are normalized to unit amplitude.
Mode number
0
10
20
30
40
0
Distance
5
1 0
15
20
Sum
We now have two ways to think of the displacement of the
string as a function of time: either as propagating waves or as
normal modes. Neither is more “real” — both are ways of representing how the displacement evolves. Thus comparing the
two gives interesting insights. For example, consider studying
the properties of the string. In the traveling wave formulation,
we measure travel times and thus infer velocity. In the normal
mode formulation, we measure eigenfrequencies and then infer
velocities. Thus the eigenfrequencies are analogous to the travel
times.