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Using the trigonometric identities for the sine of a sum and difference of two angles, we
will simplify the above solution

1
u(x, t) = cos x sin 2t.
2

You should verify that this indeed solves the wave equation and satises the given initial
conditions.

p
Example 6.1 The Plucked String. For a vibrating string the speed is c = T/ρ.
Consider an infinitely long string with initial position
(
b|x|
b − , for |x| < a,
φ(x) = a
0, for |x| > a
and initial velocity φ(x) = 0 for all x. This is a ”three-finger” pluck, with all three
1
fingers removed at once. A ”movie” of this solution u(x, t) = [φ(x + ct) + φ(x − ct)]
2
is shown in Figure 6.1. Each of these pictures is the sum of two triangle functions,
one mowing to the right and one to the left, as is clear graphically. To write down the
formulas that correspond to the pictures is a lot more work. The formulas depend on
the relationships among the five numbers 0, ±a, x ± ct. For instance, let t = a/2c.
Then x ± ct = x ± a/2.
First, if x < −3a/2, then x ± a/2 < −a and u(x, t) ≡ 0. Second, if −3a/2 < x <
−a/2, then

1
1 1 b|x + a| 3b bx
1
u(x, t) = φ(x + a) = b − 2 = + .
2 2 2 a 4 2a
Third, if lxl < a/2, then

1
1
1
u(x, t) = φ(x + a) + φ(x − a) =
2 2 2
1 1
1 b(x + a) b( a − x) b
= b − 2 + b − 2 = .
2 a a 2
and so on [see Figure 6.1].

6.3 The Wave Equation for an Infinite Domain

Consider the Cauchy problem for the wave equation on −∞ < x < ∞.

2
u tt = c u xx , −∞ < x < ∞, t > 0
u(x, 0) = f(x), u t (x, 0) = g(x)

We know that the solution is the sum of right-moving and left-moving waves.

u(x, t) = F(x − ct) + G(x + ct) (6.7)

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