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16.6 Characteristics and d’Alembert’s Solution 597

where
1 1 x
ϕ(x) = f (x) − g(w)dw
2 2c 0
and
1 1 x
β(x) = f (x) + g(w)dw.
2 2c 0

We call ϕ(x − ct) a forward wave. Its graph is the graph of ϕ(x) translated ct units to the right,
and so may be thought of as a wave moving to the right with speed c. We call β(x + ct) a back-
ward wave. Its graph is the graph of β(x) translated ct units to the left, and may be thought
of as a wave moving to the left with speed c. This allows us to think of the wave profile
y = u(x,t) at any time t as a sum of a wave moving to the right and a wave moving to the
left.

EXAMPLE 16.13
Suppose g(x) = 0,c = 1 and

4 − x 2 for −2 ≤ x ≤ 2
f (x) =
0 for |x| > 2.
The solution is
1
u(x,t) = ϕ(x + ct) + β(x + ct) = ( f (x − t) + f (x + t)).
2
At any time t the wave profile consists of the initial position function translated t units to
the right, superimposed on the initial position function translated t units to the left. Figures 16.13
through 16.19 show this profile at increasing times.

4
3.5

3
3
2.5
2
2
1.5

1 1
0.5
0 0
–4 –2 0 2 4 –4 –2 0 2 4
x x
FIGURE 16.13 Initial position in FIGURE 16.14 Wave in Example 16.13
Example 16.13. at time t = 0.5.

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October 14, 2010 15:23 THM/NEIL Page-597 27410_16_ch16_p563-610