2.4. WAVE EQUATION 69
(ii) We see from (8) that if g ∈
Ck
and h ∈
Ck−1,
then u ∈
Ck
but is not
in general smoother. Thus the wave equation does not cause instantaneous
smoothing of the initial data, as does the heat equation.
A reflection method. To illustrate a further application of d’Alembert’s
formula, let us next consider this initial/boundary-value problem on the
half-line R+ = {x 0}:
(9)
⎧
⎨
⎩
utt − uxx = 0 in R+ × (0, ∞)
u = g, ut = h on R+ × {t = 0}
u = 0 on {x = 0} × (0, ∞),
where g, h are given, with g(0) = h(0) = 0.
We convert (9) into the form (3) by extending u, g, h to all of R by odd
reflection. That is, we set
˜(x, u t) :=
u(x, t) (x ≥ 0, t ≥ 0)
−u(−x, t) (x ≤ 0, t ≥ 0),
˜(x) g :=
g(x) (x ≥ 0)
−g(−x) (x ≤ 0),
˜(x)
h :=
h(x) (x ≥ 0)
−h(−x) (x ≤ 0).
Then (9) becomes
˜tt u = ˜xx u in R × (0, ∞)
˜ u = ˜ g, ˜t u =
˜
h on R × {t = 0}.
Hence d’Alembert’s formula (8) implies
˜(x, u t) =
1
2
[˜(x g + t) + ˜(x g − t)] +
1
2
x+t
x−t
˜(y)
h dy.
Recalling the definitions of ˜ u, ˜ g,
˜
h above, we can transform this expression
to read for x ≥ 0, t ≥ 0:
(10) u(x, t) =
1
2
[g(x + t) + g(x − t)] +
1
2
x+t
x−t
h(y) dy if x ≥ t ≥ 0
1
2
[g(x + t) − g(t − x)] +
1
2
x+t
−x+t
h(y) dy if 0 ≤ x ≤ t.
If h ≡ 0, we can understand formula (10) as saying that an initial dis-
placement g splits into two parts, one moving to the right with speed one
and the other to the left with speed one. The latter then reflects off the
point x = 0, where the vibrating string is held fixed.
Note that our solution does not belong to
C2,
unless g (0) = 0.
