y(0, t), and therefore

™

T

Mv = ’

™ v. (6.55)

c

The e¬ective viscosity coe¬cient is thus · = T /c. Note that we need an

in¬nitely long string for this formula to be true for all time. If the string has

a ¬nite length L, then, after a period of 2L/c, energy will be re¬‚ected back

to the bead and will complicate matters.

We can also derive the radiation damping from the Caldeira-Leggett anal-

ysis of chapter 5. Our bead-string contraption has Lagrangian

M ρ2 T 2

L

[y(0, t)]2 ’ V [y(0, t)] + y ’ y dx.

L= ™ ™ (6.56)

2 2 2

0

Here V [y] is some potential energy for the bead. Introduce a function φ0 (x)

such that φ0 (0) = 1 and φ0 (x) decreases rapidly to zero as x increases.

6.2. WAVE EQUATION 157

’ φ™(x)

1 0

φ (x)

0

x

The function φ0 (x) and its derivative.

We therefore have ’φ0 (x) ≈ δ(x). Expand y(x, t) in terms of φ0 (x) and the

normal modes of a string with ¬xed ends as

2

y(x, t) = y(0, t)φ0(x) + qn (t) sin kn x. (6.57)

Lρ

n

Here kn L = nπ. Because y(0, t)φ0(x) describes the motion of only an in-

¬nitesimal length of string, y(0, t) makes a negligeable contribution to the

string kinetic energy, but it provides a linear coupling of the bead to the string

normal modes, qn (t), through the T y 2 /2 term. Plugging the expansion into

L, and after about half a page of arithmetic, we end up with

2

M 12 1 fn

[y(0)]2 ’V [y(0)]+y(0) 22

y(0)2,

qn ’ ωn qn ’

L= ™ fn qn + ™ 2

2 2 2 ωn

n n n

(6.58)

where ωn = ckn , and

2

fn = T

kn . (6.59)

Lρ

This is exactly the Caldeira-Leggett Lagrangian, including the frequency-

shift counter-term. When L becomes large, the eigenvalue density of states

δ(ω ’ ωn )

ρ(ω) = (6.60)

n

becomes

L

ρ(ω) = . (6.61)

πc

The Caldeira-Leggett spectral function

2

π fn

δ(ω ’ ωn ),

J(ω) = (6.62)

2 ωn

n

158 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS

is therefore

π 2T 2 k 2 1 L T

J(ω) = · · · = ω, (6.63)

2 Lρ kc πc c

where we have used c = T /ρ. Comparing with Caldeira-Leggett™s J(ω) =

·ω, we see that the e¬ective viscosity is given by · = T /c, as before. The

necessity of having an in¬nitely long string here translates into the require-

ment that we must have a continuum of oscillator modes. It is only after the

sum over discrete modes ωi is replaced by an integral over the continuum of

ω™s that no energy is ever returned to the system being damped.

This formalism can be extended to other radiation damping problems.

For example we may consider1 the drag forces induced by the emission of

radiation from accelerated charged particles. We end up with a deeper un-

derstanding of the traditional, but pathological, Abraham-Lorentz equation,

M (v ’ „ v) = Fext ,

™ ¨ (6.64)

which is plagued by runaway solutions. (Here

2 e2 1 1

„= , (6.65)

3 c3 M 4π 0

the factor in square brackets being needed for SI units. It is absent in Gaus-

sian units.)

Odd vs. Even Dimensions

6.2.4

Consider the wave equation for sound in the three dimensions. We have a

velocity potential φ which obeys the wave equation

‚2φ ‚2φ ‚2φ 1 ‚2φ

+ 2 + 2 ’ 2 2 = 0, (6.66)

‚x2 ‚y ‚z c ‚t

and from which the velocity, density, and pressure ¬‚uctuations can be ex-

tracted as

v1 = φ,

ρ0 ™

= ’ 2 φ,

ρ1

c

c2 ρ1 .

P1 = (6.67)

1

G. W. Ford, R. F. O™Connell, Phys. Lett. A 157 (1991) 217.

6.2. WAVE EQUATION 159

In three dimensions, and considering only spherically symmetric waves,

the wave equation becomes

‚ 2 (rφ) 1 ‚ 2 (rφ)

’2 = 0, (6.68)

‚r2 c ‚t2

with solution

1 r 1 r

φ(r, t) = f t ’ + g t+ . (6.69)

r c r c

Consider what happens if we put a point volume source at the origin (the

sudden conversion of a negligeable volume of solid explosive to a large volume

of hot gas, for example). Let the rate at which volume is being intruded be

q. The gas velocity very close to the origin will be

™

q(t)