Mathematical Methods for Physics and Engineering - Matematica.NET

25.6 STOKES’ EQUATION AND AIRY INTEGRALS(b)y(c)(a)(a)(c)z(b)Figure 25.9 Behaviour of the solutions y(z) ofStokes’equationnearz =0for various values of λ = −y ′ (0). (a) with λ small, (b) with λ large and (c) withλ appropriate to the Airy function Ai(z).happen in the region z>0. For definiteness and ease of illustration (see figure25.9), let us suppose that both y and z, and hence the derivatives of y, are real andthat y(0) is positive; if it were negative, our conclusions would not be changedsince equation (25.32) is invariant under y(z) →−y(z). The only difference wouldbe that all plots of y(z) would be reflected in the z-axis.We first note that d 2 y/dx 2 , and hence also the curvature of the plot, has thesame sign as z, i.e. it has positive curvature when z>0, for so long as y(z)remains positive there. What will happen to the plot for z>0 therefore dependscrucially on the value of y ′ (0). If this slope is positive or only slightly negativethe positive curvature will carry the plot, either immediately or ultimately, furtheraway from the z-axis. On the other hand, if y ′ (0) is negative but sufficiently largein magnitude, the plot will cross the y = 0 line; if this happens the sign of thecurvature reverses and again the plot will be carried ever further from the z-axis,only this time towards large negative values.Between these two extremes it seems at least plausible that there is a particularnegative value of y ′ (0) that leads to a plot that approaches the z-axis asymptotically,never crosses it (and so always has positive curvature), and has a slopethat, whilst always negative, tends to zero in magnitude. There is such a solution,known as Ai(z), whose properties we will examine further in the followingsubsections. The three cases are illustrated in figure 25.9.The behaviour of the solutions of (25.32) in the region z