2c

Now,

‚ ‚x ‚ ‚t ‚ 1 ‚ 1‚

= + = + . (6.25)

‚ξ ‚ξ ‚x ‚ξ ‚t 2 ‚x c ‚t

Similarly

‚ 1 ‚ 1‚

’

= . (6.26)

‚· 2 ‚x c ‚t

Thus

‚2 1 ‚2 ‚2

‚ 1‚ ‚ 1‚

’ ’

= + =4 . (6.27)

‚x2 c2 ‚t2 ‚x c ‚t ‚x c ‚t ‚ξ‚·

The characteristics of the equation

‚2•

4 =0 (6.28)

‚ξ‚·

6.2. WAVE EQUATION 151

are ξ = const. or · = const. There are two characteristics curves through

each point, so the equation is hyperbolic.

With lightcone coordinates it is easy to see that a general solution to

‚2 1 ‚2 ‚2•

’ •=4 =0 (6.29)

‚x2 c2 ‚t2 ‚ξ‚·

is

• = f (ξ) + g(·) = f (x + ct) + g(x ’ ct). (6.30)

The curve t = 0 is not a characteristic, so we can propagate a solution

from Cauchy data •(x, t = 0) ≡ •0 (x) and •(x, t = 0) ≡ v0 (x). We use this

™

data to ¬t f and g in

•(x, t) = f (x + ct) + g(x ’ ct). (6.31)

We have

f (x) + g(x) = •0 (x),

c(f (x) ’ g (x)) = v0 (x), (6.32)

so

1 x

f (x) ’ g(x) = v0 (ξ) dξ + A. (6.33)

c 0

Therefore

1 1 1

x

f (x) = •0 (x) + v0 (ξ) dξ + A,

2 2c 2

0

1 1 1

x

•0 (x) ’ v0 (ξ) dξ ’ A.

g(x) = (6.34)

2 2c 2

0

Thus

1 1 x+ct

•(x, t) = {•0 (x + ct) + •0 (x ’ ct)} + v0 (ξ) dξ. (6.35)

2 2c x’ct

This is called d™Alembert™s solution of the wave equation.

152 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS

t

(x,t)

x’ct x+ct

x

Range of Cauchy data in¬‚uencing •(x, t).

The value of • at x, t, is determined by only a ¬nite interval of the initial

Cauchy data. In more generality, •(x, t) depends only on what happens in

the past line-cone of the point, which is bounded by pair of characteristic

curves.

We can bring out the role of characteristics in the d™Alembert solution by

writing the wave equation as

‚2• 1 ‚2• ‚ 1‚ ‚• 1 ‚•

’2 2 ’

0= = + . (6.36)

‚x2 c ‚t ‚x c ‚t ‚x c ‚t

This tells us that

‚ 1‚

(u ’ v) = 0,

+ (6.37)

‚x c ‚t

where

‚• 1 ‚•

u= , v= . (6.38)

‚x c ‚t

Thus the quantity u ’ v is constant along the curve

x ’ ct = const, (6.39)

which is a characteristic. Similarly u + v is constant along the characteristic

x + ct = const. (6.40)

This provides another route to the construction of d™Alembert™s solution.

6.2. WAVE EQUATION 153

6.2.2 Fourier™s Solution

Starting from the same Cauchy data as d™Alembert, Fourier proposed a com-

pletely di¬erent approach to solving the wave equation. He sought a solution

in the form

∞ dk

a(k)eikx’iωk t + a— (k)e’ikx+iωk t ,

•(x, t) = (6.41)

’∞ 2π

where ωk ≡ c|k| is the positive root of ω 2 = c2 k 2 . The terms being summed

by the integral are all individually of the form f (x ’ ct), or f (x + ct), and so

•(x, t) is indeed a solution of the wave equation. The positive-root convention

means that positive k corresponds to right-going waves, and negative k to

left-going waves.

We ¬nd the amplitudes a(k) by ¬tting to the Fourier transforms of the

initial data

∞ dk

¦(k)eikx ,

•(x, t = 0) =

’∞ 2π

∞ dk

χ(k)eikx ,

•(x, t = 0) =

™ (6.42)

’∞ 2π

so

¦(k) = a(k) + a— (’k),

χ(k) = iωk a— (’k) ’ a(k) . (6.43)

Solving, we ¬nd