162 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS

Let us now repeat the explosive sheet calculation for an exploding wire.

r

s

P

x

Line-source geometry.

Using

√ r dr

r 2 ’ x2 = √

ds = d , (6.80)

r 2 ’ x2

and combining the contributions of the two parts of the wire that are the

same distance from p, we can write

1 r 2r dr

∞

√

f t’

φ(x, t) =

r c r 2 ’ x2

x

r dr

∞

√

f t’

=2 , (6.81)

c r 2 ’ s2

x

with f (t) = ’q(t)/4π, where now q is the volume intruded per unit length.

™ ™

We may approximate r2 ’ x2 ≈ 2x(r ’ x) for the near parts of the wire where

r ≈ x, since these make the dominant contribution to the integral. We also

set „ = t ’ r/c, and then have

2c dr

(t’x/c)

φ(x, t) = √ f („ ) ,

2x (ct ’ x) ’ c„

’∞

1 2c d„

(t’x/c)

=’ q („ )

™ . (6.82)

2π x (t ’ x/c) ’ „

’∞

The far-¬eld velocity is the x gradient of this,

1 2c d„

(t’x/c)

v1 (r, t) = q („ )

¨ , (6.83)

2πc x (t ’ x/c) ’ „

’∞

6.3. HEAT EQUATION 163

and is therefore proportional to the 1/2-derivative of q(t ’ r/c).

™

v v

r r

Near field Far field

In two dimensions the far-¬eld pulse has a long tail.

The far-¬eld pulse never completely dies away to zero, and this long tail

means that one cannot use digital signalling in two dimensions.

Moral Tale: A couple of years ago one of our colleagues was performing

numerical work on earthquake propagation. The source of his waves was a

long deep linear fault, so he used the two-dimensional wave equation. Not

wanting to be troubled by the actual creation of the wave-pulse, he took as

initial data an outgoing ¬nite-width pulse. After a short propagation time

his numerics always went crazy. He wasted several months in vain attempt to

improve the stability of his code before it was pointed out him that what he

was seeing was real. The lack of a long tail on his pulse meant that it could

not have been created by a well-behaved line source. The numerical craziness

was a consequence of the source striving to do the impossible. Moral : Always

check that a solution actually exists before you waste your time trying to

compute it.

6.3 Heat Equation

Fourier™s heat equation

‚2φ

‚φ

=κ 2 (6.84)

‚t ‚x

is the archetypal parabolic equation. It often comes with initial data φ(x, t = 0),

but this is not Cauchy data, as the curve t = const. is a characteristic.

The heat equation is also known as the di¬usion equation.

164 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS

6.3.1 Heat Kernel

If we Fourier transform the initial data

dk ˜

∞

φ(k)eikx ,

φ(x, t = 0) = (6.85)

2π

’∞

and write

dk ˜

∞

φ(k, t)eikx ,

φ(x, t) = (6.86)

2π

’∞

we can plug this into the heat equation and ¬nd that

˜

‚φ ˜

= ’κk2 φ. (6.87)

‚t

Hence,

dk ˜

∞

φ(k, t)eikx

φ(x, t) =

2π

’∞

dk ˜

∞ 2

φ(k, 0)eikx’κk t .

= (6.88)

2π