1.4. Fourier synthesis and rectilinear propagation 21
Fourier transformation with respect to the x variables yields
∂t
2
ˆ(t, u ξ) +
|ξ|2
ˆ(t, u ξ) = 0 , ˆ(0,ξ) u =
ˆ,
f ∂t ˆ(0,ξ) u = ˆ g .
Solve the ordinary differential equations in t to find
ˆ(t, u ξ) =
ˆ(ξ)
f cos t|ξ| + ˆ(ξ) g
sin t|ξ|
|ξ|
.
For t = 0 the function ξ sin(t|ξ|)/|ξ| is a real analytic function on
Rd
with derivatives uniformly bounded on bounded time intervals. Write
cos t|ξ| =
eit|ξ|
+
e−t|ξ|
2
, sin t|ξ| =
eit|ξ|

e−t|ξ|
2i
,
to find
(1.4.2) ˆ(t, u ξ)
eixξ
= a+(ξ)
ei(xξ−t|ξ|)
a−(ξ)
ei(xξ+t|ξ|)
,
with
(1.4.3) a+ :=
1
2
ˆ
f +
ˆ g
i|ξ|
, a− :=
1
2
ˆ
f
ˆ g
i|ξ|
.
For each ξ, the right-hand side of (1.4.2) is a linear combination of the
plane wave solutions of the wave equation
ei(xξ+tτ(ξ))
with dispersion relation
τ = ∓|ξ| and amplitude a±(ξ). The group velocities associated to are
v = −∇ξτ = −∇ξ(∓|ξ|) = ±
ξ
|ξ|
.
The solution is the sum of two terms,
u±(t, x) :=
1
(2π)d/2
a±(ξ)
ei(xξ∓t|ξ|)
.
The conserved energy for the spring equation satisfied by ˆ(t, u ξ) shows
that
1
2
|ˆt(t, u
ξ)|2
+
|ξ|2|ˆ(t,
u
ξ)|2
=
|ξ|2
(
|a+(ξ)|2
+
|a−(ξ)|2
)
= independent of t .
Integrate and use F
(
∂u/∂xj
)
= iξj ˆ, u and Parseval’s theorem to show
that the quantity
|ξ|2
(
|a+(ξ)|2
+
|a−(ξ)|2
)
=
1
2
|ut(t,
x)|2
+ |∇xu(t,
x)|2
dx
is independent of time for the solutions of the wave equation.
The formula for are potentially singular at ξ = 0. The energy for the
wave equation is expressed in terms of the pair of functions |ξ|a±(ξ). They
are given by nonsingular expressions in terms of |ξ|
ˆ
f and ˆ. g
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