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 a± 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|)

dξ .

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 dξ 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

)

dξ =

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 a± 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