MATH 220: PROPERTIES OF SOLUTIONS OF SECOND ORDER
PDE
ANDRAS VASY
We have solved the initial value problem for the wave equation
(
∂
2
t
−
c
2
∂
2
x
)
u
= 0
, u
(
x,
0) =
φ
(
x
)
, u
t
(
x,
0) =
ψ
(
x
)
,
namely we showed that the solution is
u
(
x,t
) =
1
2
(
φ
(
x
+
ct
) +
φ
(
x
−
ct
)) +
1
2
c
integraldisplay
x
+
ct
x
−
ct
ψ
(
σ
)
dσ.
There are a few facts that can be read off from this expression immediately. We
consider
t >
0 here;
t <
0 is similar.
First, for
t
0
>
0,
u
(
x
0
,t
0
) depends on the
initial data
φ
just at the two points
x
0
±
ct
0
, while it depends on the values of
ψ
in
the whole interval [
x
0
−
ct
0
,x
0
+
ct
0
]. Thus, we call the interval [
x
0
−
ct
0
,x
0
+
ct
0
]
the
domain of dependence
of (
x
0
,t
0
):
if the initial conditions vanish there, the
solution vanishes at (
x
0
,t
0
).
Note that the straight lines
x
−
ct
=
x
0
−
ct
0
and
x
+
ct
=
x
0
+
ct
0
which go through (
x
0
,t
0
) and (
x
0
±
ct
0
,
0) are characteristics.
In fact, it is convenient (for reasons that will be more clear when we solve the
inhomogeneous
wave equation,
square
u
=
f
) to consider the domain of dependence of
(
x,t
) to be the whole region
D
−
x
0
,t
0
=
{
(
x,t
) :
t
≤
t
0
,

x
−
x
0
 ≤
c
(
t
0
−
t
)
}
.
This is the
backward characteristic triangle
from (
x
0
,t
0
): its sides are the charac
teristics
x
−
ct
=
x
0
−
ct
0
and
x
+
ct
=
x
0
+
ct
0
. With this definition, if the initial
data are imposed at
t
=
T
instead, where
T <t
0
, then the solution
u
at (
x
0
,t
0
)
depends on the initial data in the interval [
x
0
−
c
(
t
0
−
T
)
,x
0
+
c
(
t
0
−
T
)], which
is just the segment in which the line
t
=
T
intersects the backward characteristic
triangle. Indeed, a change of variables (replacing
t
by
τ
=
t
−
T
) shows that the
solution of
(
∂
2
t
−
c
2
∂
2
x
)
u
= 0
, u
(
x,T
) =
φ
T
(
x
)
, u
t
(
x,T
) =
ψ
T
(
x
)
,
is
u
(
x,t
) =
1
2
(
φ
T
(
x
+
c
(
t
−
T
)) +
φ
T
(
x
−
c
(
t
−
T
))) +
1
2
c
integraldisplay
x
+
c
(
t
−
T
)
x
−
c
(
t
−
T
)
ψ
T
(
σ
)
dσ.
On the flipside, we may ask at which points (
x
0
,t
0
) does, or does not, the initial
data at (
x,
0) influence the solution.
More precisely, we can ask at which points
(
x
0
,t
0
) is the solution guaranteed to be unaffected if we change the initial data at
(
x,
0)? This happens exactly if (
x,
0) is
not
in the backward characteristic triangle
from (
x
0
,t
0
),
D
−
x
0
,t
0
, i.e. (keeping in mind
t
0
>
0) if

x
−
x
0

>ct
0
. Conversely, if

x
−
x
0
 ≤
ct
0
, then the solution may change (and in general does change) if
φ
and
ψ
are changed at (
x,
0). Thus, we call
the forward characteristic triangle
D
+
x,
0
=
{
(
x
0
,t
0
) :
t
0
≥
0
,

x
−
x
0
 ≤
ct
0
}
1
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ANDRAS VASY
the
domain of influence
of (
x,
0).
More generally, for initial data at
t
=
T
, the
(forward) domain of influence of (
x,T
) is
D
+
x,T
=
{
(
x
0
,t
0
) :
t
0
≥
T,

x
−
x
0
 ≤
c
(
t
0
−
T
)
}
.
That the size of the intersection of the domain of influence with the line
t
=
t
0
increases at speed
c
is called
Huygens’ principle
, and is also valid for wave equations
with variable coefficients: waves propagate (at most) as fast as
c
, so
c
is reasonably
called the speed of waves.
Although the solution at (
x
0
,t
0
) depends on the initial data everywhere inside its
domain of dependence, its dependence on the data may not be very significant. One
way to think about this is the following. ‘Information’ is carried by singularities of
solutions to the PDE, e.g. one flips a switch, and gets a jump in the amplitude of
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 Fall '10
 Vasy
 dx, Boundary conditions, Dirichlet boundary condition, Neumann boundary condition, ANDRAS VASY

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