Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.5 THE DIFFUSION EQUATION

term is a little less obvious. It can be viewed as representing the accumulated

transverse displacement at positionxdue to the passage pastxof all parts of

the initial motion whose effects can reachxwithin a timet, both backward and

forward travelling.

The extension to the three-dimensional wave equation of solutions of the type

we have so far encountered presents no serious difficulty. In Cartesian coordinates

the three-dimensional wave equation is

∂^2 u

∂x^2

+

∂^2 u

∂y^2

+

∂^2 u

∂z^2

−

1

c^2

∂^2 u

∂t^2

=0. (20.32)

In close analogy with the one-dimensional case we try solutions that are functions

of linear combinations of all four variables,

p=lx+my+nz+μt.

It is clear that a solutionu(x, y, z, t)=f(p) will be acceptable provided that
(
l^2 +m^2 +n^2 −

μ^2

c^2

)

d^2 f(p)

dp^2

=0.

Thus, as in the one-dimensional case,fcan be arbitrary provided that

l^2 +m^2 +n^2 =μ^2 /c^2.

Using an obvious normalisation, we takeμ=±candl,m,nas three numbers

such that

l^2 +m^2 +n^2 =1.

In other words (l, m, n) are the Cartesian components of a unit vectornˆthat

points along the direction of propagation of the wave. The quantitypcan be

written in terms of vectors as the scalar expressionp=nˆ·r±ct, and the general

solution of (20.32) is then

u(x, y, z, t)=u(r,t)=f(nˆ·r−ct)+g(ˆn·r+ct), (20.33)

wherenˆisanyunit vector. It would perhaps be more transparent to writenˆ

explicitly as one of the arguments ofu.

20.5 The diffusion equation

One important class of second-order PDEs, which we have not yet considered

in detail, is that in which the second derivative with respect to one variable

appears, but only the first derivative with respect to another (usually time). This

is exemplified by the one-dimensional diffusion equation

κ

∂^2 u(x, t)

∂x^2

=

∂u

∂t

, (20.34)