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is the general solution of the PDE. We look for a solution of this form which will satisfy
our initial data. This means we need

u(x, 0) = f(x) + g(x) = ϕ(x),

0
0
u t (x, 0) = cf (x) + cg (x) = ψ(x).
Solving this system, we get

1 ψ(x)
0
0
f (x) = ϕ (x) + ,
2 c

1 ψ(x)
0
0
g (x) = ϕ (x) − ,
2 c
which implies
t
Z
1 1
f(t) = ϕ(t) + ψ(t) dt + C 1 ,
2 2c
0
t
Z
1 1
g(t) = ϕ(t) − ψ(t) dt + C 2 .
2 2c
0
Using the fact that
ϕ(x) = f(x) + g(x),
we see that C 1 + C 2 = 0. Therefore, we conclude that
 
x+ct
Z
1 1
u(x, t) = f(x + ct) + g(x − ct) =  ϕ(x + ct) + ψ(z) dz  +
2 2c
0
 
x−ct
1 1 Z
+  ϕ(x − ct) − ψ(z) dz  ,
2 2c
0
which simplifies to
x+ct
Z
1 1
u(x, t) = (ϕ(x + ct) + ϕ(x − ct)) + ψ(z) dz. (6.6)
2 2c
x−ct
This solution formula (6.6) is known as d’Alembert’s formula for the unique solution of
the initial-value problem (6.5) for the wave equation on R.
Example 4.1 Solve the initial value problem (6.5), where c = 2, with the initial data

ϕ(x) = 0, ψ(x) = cos x.

Substituting ϕ(x) and ψ(x) into d’Alambert’s formula, we obtain the solution
x+2t
Z
1 1
u(x, t) = cos z dz = (sin (x + 2t) − sin (x − 2t)) .
4 4
x−2t

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