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570 CHAPTER 16 The Wave Equation

0.8

0.4

0
0 0.5 1 1.5 2 2.5 3
x
–0.4

FIGURE 16.2 Wave motion in Example 16.1.

0.1

0.05

0
0 0.5 1 1.5 2 2.5 3
x

–0.05

FIGURE 16.3 Wave profiles in Example 16.2.

EXAMPLE 16.2

We will solve for y(x,t) on [0,π] with zero initial velocity and initial position f (x) =
x cos(5x/2). The coefficients are
2 π 8 n(−1) n+1
c n = ξ cos(5ξ/2)sin(nξ)dξ = .
π 0 π (5 + 2n) (5 − 2n) 2
2
The solution is
∞ n+1
8 n(−1)

y(x,t) = sin(nx)cos(nct).
π (5 + 2n) (5 − 2n) 2
2
n=1
Figure 16.3 shows wave profiles for c = 1 at times t = 0.1,0.3, 0.6, and 0.9, moving
downward over these times.
16.2.2 Zero Initial Displacement
We will solve the initial-boundary value problem for the displacement of the string if there is an
initial velocity but no initial displacement. The problem is

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October 14, 2010 15:23 THM/NEIL Page-570 27410_16_ch16_p563-610