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574 CHAPTER 16 The Wave Equation

0.4

0
0 0.5 1 1.5 2 2.5 3
x

–0.4

–0.8

FIGURE 16.6 Wave profiles for different values of
c in Example 16.5.

EXAMPLE 16.5
In Example 16.3, we solved the problem with zero initial displacement and initial velocity given
by g(x) = x(1 + cos(πx/L)).If L = π this solution is
∞ n
3
2(−1)
y(x,t) = sin(x)sin(ct) + sin(nx)sin(nct).
2
2c cn (n − 1)
2
n=2
Figure 16.3 showed wave profiles at various times for c =1. Figure 16.6 shows wave profiles
at time t = 5.3, with c = 1.05, c = 1.1, c = 1.2 and c = 1.65. These increase in amplitude as c
increases.

EXAMPLE 16.6

We will examine the effects of changes in an initial condition on the wave motion with the
problem
∂y 2 ∂y 2
= 1.44 for 0 < x <π,t > 0,
∂t 2 ∂x 2
y(0,t) = y(π,t) = 0for t ≥ 0,
and
∂y
y(x,0) = 0, (x,0) = sin( x) for 0 < x <π
∂t
in which is a positive number that is not an integer.
Use equation (16.6) to write the solution

∞
5
sin(π )(−1) n+1
y(x,t) = sin(nx)sin(1.2t).
3π n − 2
2
n=1
To gauge the effect of on the motion, compare graphs of this solution for different values
of at given times. Figure 16.7 shows the wave profile at t = 0.5for equal to 0.7 (wave above
the x-axis), 1.5 (wave below the axis), and 9.3 (wave oscillating rapidly).

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October 14, 2010 15:23 THM/NEIL Page-574 27410_16_ch16_p563-610