Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

25.6 Stokes’ equation and Airy integrals

Much of the analysis of situations occurring in physics and engineering is con-

cerned with what happens at a boundary within or surrounding a physical system.

Sometimes the existence of a boundary imposes conditions on the behaviour of

variables describing the state of the system; obvious examples include the zero

displacement at its end-points of an anchored vibrating string and the zero

potential contour that must coincide with a grounded electrical conductor.

More subtle are the effects at internal boundaries, where the same non-vanishing

variable has to describe the situation on either side of the boundary but its

behaviour is quantitatively, or evenqualitatively, different in the two regions. In

this section we will study an equation, Stokes’ equation, whose solutions have

this latter property; as well as solutions written as series in the usual way, we will

find others expressed as complex integrals.

The Stokes’ equation can be written in several forms, e.g.

d^2 y

dx^2

+λxy=0;

d^2 y

dx^2

+xy=0;

d^2 y

dx^2

=xy.

We will adopt the last of these, but write it as

d^2 y

dz^2

=zy (25.32)

to emphasis that its complex solutions are valid for a complex independent

variablez, though this also means that particular care has to be exercised when

examining their behaviour in different parts of the complexz-plane. The other

forms of Stokes’ equation can all be reduced to that of (25.32) by suitable

(complex) scaling of the independent variable.

25.6.1 The solutions of Stokes’ equation

It will be immediately apparent that, even forzrestricted to be real and denoted

byx, the behaviour of the solutions to (25.32) will change markedly asxpasses

throughx= 0. For positivexthey will have similar characteristics to the solutions

ofy′′=k^2 y,wherekis real; these have monotonic exponential forms, either

increasing or decreasing. On the other hand, whenxis negative the solutions

will be similar to those ofy′′+k^2 y= 0, i.e. oscillatory functions ofx.Thisis

just the sort of behaviour shown by the wavefunction describing light diffracted

by a sharp edge or by the quantum wavefunction describing a particle near to

the boundary of a region which it is classically forbidden to enter on energy

grounds. Other examples could be taken from the propagation of electromagnetic

radiation in an ion plasma or wave-guide.

Let us examine in a bit more detail the behaviour of plots of possible solutions

y(z) of Stokes’ equation in the region nearz= 0 and, in particular, what may