3.2 Solution of the Vibrating String Problem 223
Figure 3 On the left are the graphs off ̄o(x+ct)(solid) and ̄fo(x−ct)(dashed)
for the given value ofct. On the right is the graph ofu(x,t)for 0<x<a,made
by averaging the graphs on the left.
The function values can be read directly from a graph off(x). The manipula-
tions with the series solution in the example can be done for anyf(x).There-
fore the formula of Eq. (13) is a solution of Eqs. (1)–(4) for anyf(x),provided
thatg(x)≡0. We will generalize in later sections.
Frequencies of Vibration
The product solutions in Eq. (8) provide important information about pos-
sible frequencies of vibration. The multipliersλncin the sines and cosines of