Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.6 STOKES’ EQUATION AND AIRY INTEGRALS

(a)

(a)

(b)

(b)

(c)

(c)

y

z

Figure 25.9 Behaviour of the solutionsy(z)ofStokes’equationnearz=0

for various values ofλ=−y′(0). (a) withλsmall, (b) withλlarge and (c) with

λappropriate to the Airy function Ai(z).

happen in the regionz>0. For definiteness and ease of illustration (see figure

25.9), let us suppose that bothyandz, and hence the derivatives ofy, are real and

thaty(0) is positive; if it were negative, our conclusions would not be changed

since equation (25.32) is invariant undery(z)→−y(z). The only difference would

be that all plots ofy(z) would be reflected in thez-axis.

We first note thatd^2 y/dx^2 , and hence also the curvature of the plot, has the

same sign asz, i.e. it has positive curvature whenz>0, for so long asy(z)

remains positive there. What will happen to the plot forz>0 therefore depends

crucially on the value ofy′(0). If this slope is positive or only slightly negative

the positive curvature will carry the plot, either immediately or ultimately, further

away from thez-axis. On the other hand, ify′(0) is negative but sufficiently large

in magnitude, the plot will cross they= 0 line; if this happens the sign of the

curvature reverses and again the plot will be carried ever further from thez-axis,

only this time towards large negative values.

Between these two extremes it seems at least plausible that there is a particular

negative value ofy′(0) that leads to a plot that approaches thez-axis asymptot-

ically, never crosses it (and so always has positive curvature), and has a slope

that, whilst always negative, tends to zero in magnitude. There is such a solu-

tion, known as Ai(z), whose properties we will examine further in the following

subsections. The three cases are illustrated in figure 25.9.

The behaviour of the solutions of (25.32) in the regionz<0 is more straight-