a(k) = ¦(k) + χ(k) ,

2 ωk

1 i

a— (k) = ¦(’k) ’ χ(’k) . (6.44)

2 ωk

For some years after Fourier™s trigonometric series solution was proposed,

doubts persisted as to whether it was as general as that of d™Alembert. It is,

of course, completely equivalent.

6.2.3 Causal Green Function

We now add a source term:

1 ‚2• ‚2•

’ 2 = q(x, t). (6.45)

c2 ‚t2 ‚x

154 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS

We will solve this by ¬nding a Green function such that

1 ‚2 ‚2

’ G(x, t; ξ, „ ) = δ(x ’ ξ)δ(t ’ „ ). (6.46)

c2 ‚t2 ‚x2

If the only waves in the system are those produced by the source, we should

demand that the Green function be causal , in that G(x, t; ξ, „ ) = 0 if t < „ .

To construct the causal Green function, we integrate the equation over

an in¬nitesimal time interval from „ ’ to „ + and so ¬nd Cauchy data

G(x, „ + ; ξ, „ ) = 0,

d

G(x, „ + ; ξ, „ ) = c2 δ(x ’ ξ). (6.47)

dt

We plug this into d™Alembert™s solution to get

c x+c(t’„ )

G(x, t; ξ, „ ) = θ(t ’ „ ) δ(ζ ’ ξ)dζ

2 x’c(t’„ )

c

θ(t ’ „ ) θ x ’ ξ + c(t ’ „ ) ’ θ x ’ ξ ’ c(t ’ „ ) .

=

2

(6.48)

t

(ξ,„)

x

Support of G(x, t; ξ, „ ) for ¬xed ξ, „ , or the “domain of in¬‚uence”.

Using this we have

c t x+c(t’„ )

•(x, t) = d„ q(ξ, „ )dξ

2 ’∞ x’c(t’„ )

c

= q(ξ, „ )d„ dξ (6.49)

2 „¦

where the domain of integration „¦ is shown in the ¬gure.

6.2. WAVE EQUATION 155

„

(x,t)

x-c(t- „) x+c(t-„ )

(ξ,„)

ξ

The region „¦, or the “domain of dependence”.

We can write the causal Green function in the form of Fourier™s solution of

the wave equation. We claim that

eik(x’ξ) e’iω(t’„ )

dω dk

∞ ∞

2

G(x, t; ξ, „ ) = c , (6.50)

c2 k 2 ’ (ω + i )2

2π 2π

’∞ ’∞

where the i plays the same role in enforcing causality as it does for the

harmonic oscillator in one dimension. This is only to be expected. If we

decompose a vibrating string into normal modes, then each mode is an in-

dependent oscillator of with ωk = c2 k 2 , and the Green function for the PDE

2

is simply the sum of the ODE Green functions for each k mode. Using our

previous results for the single-oscillator Green function to do the integral

over ω, we ¬nd

dk ikx 1

∞

G(x, t; 0, 0) = θ(t)c2 e sin(|k|ct). (6.51)

2π c|k|

’∞

Despite the factor of 1/|k|, there is no singularity at k = 0, so no i is

needed to make the integral over k well de¬ned. We can do the k integral

by recognizing that the integrand is nothing but the Fourier representation,

1

sin ak, of a square-wave pulse. We end up with

k

c

G(x, t; 0, 0) = θ(t) {θ(x + ct) ’ θ(x ’ ct)} , (6.52)

2

the same expression as from our direct construction. We can also write

c dk i

∞

eikx’i|k|ct ’ e’ikx+ic|k|t ,

G(x, t; 0, 0) = t > 0, (6.53)

|k|

2 2π

’∞

156 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS

which is in explicit Fourier-solution form with a(k) = ic/2|k|.

Illustration: Radiation Damping. A bead of mass M slides without friction

on the y axis. It is attached to an in¬nite string which is initially undisturbed

and lying along the x axis. The string has tension T , and a density such that

the speed of waves on the string is c. Show that the wave energy emitted by

the moving bead gives rise to an e¬ective viscous damping force on it.

y

v

T

x

A bead connected to a string.

From the ¬gure we see that M v = T y (0, t), and from the condition of no

™

incoming waves we know that

y(x, t) = y(x ’ ct). (6.54)

Thus y (0, t) = ’y(0, t)/c. But the bead is attached to the string, so v(t) =