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.