In mathematics, and specifically partial differential equations, d'Alembert's formula is the general solution to the one-dimensional wave equation:
u
t
t
−
c
2
u
x
x
=
0
,
u
(
x
,
0
)
=
g
(
x
)
,
u
t
(
x
,
0
)
=
h
(
x
)
,
for
−
∞
<
x
<
∞
,
t
>
0
. It is named after the mathematician Jean le Rond d'Alembert.
The characteristics of the PDE are
x
±
c
t
=
c
o
n
s
t
, so use the change of variables
μ
=
x
+
c
t
,
η
=
x
−
c
t
to transform the PDE to
u
μ
η
=
0
. The general solution of this PDE is
u
(
μ
,
η
)
=
F
(
μ
)
+
G
(
η
)
where
F
and
G
are
C
1
functions. Back in
x
,
t
coordinates,
u
(
x
,
t
)
=
F
(
x
+
c
t
)
+
G
(
x
−
c
t
)
u
is
C
2
if
F
and
G
are
C
2
.
This solution
u
can be interpreted as two waves with constant velocity
c
moving in opposite directions along the x-axis.
Now consider this solution with the Cauchy data
u
(
x
,
0
)
=
g
(
x
)
,
u
t
(
x
,
0
)
=
h
(
x
)
.
Using
u
(
x
,
0
)
=
g
(
x
)
we get
F
(
x
)
+
G
(
x
)
=
g
(
x
)
.
Using
u
t
(
x
,
0
)
=
h
(
x
)
we get
c
F
′
(
x
)
−
c
G
′
(
x
)
=
h
(
x
)
.
Integrate the last equation to get
c
F
(
x
)
−
c
G
(
x
)
=
∫
−
∞
x
h
(
ξ
)
d
ξ
+
c
1
.
Now solve this system of equations to get
F
(
x
)
=
−
1
2
c
(
−
c
g
(
x
)
−
(
∫
−
∞
x
h
(
ξ
)
d
ξ
+
c
1
)
)
G
(
x
)
=
−
1
2
c
(
−
c
g
(
x
)
+
(
∫
−
∞
x
h
(
ξ
)
d
ξ
+
c
1
)
)
.
Now, using
u
(
x
,
t
)
=
F
(
x
+
c
t
)
+
G
(
x
−
c
t
)
d´Alembert's formula becomes:
u
(
x
,
t
)
=
1
2
[
g
(
x
−
c
t
)
+
g
(
x
+
c
t
)
]
+
1
2
c
∫
x
−
c
t
x
+
c
t
h
(
ξ
)
d
ξ
.
