Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLES25.6 Stokes’ equation and Airy integralsMuch of the analysis of situations occurring in physics and engineering is concernedwith what happens at a boundary within or surrounding a physical system.Sometimes the existence of a boundary imposes conditions on the behaviour ofvariables describing the state of the system; obvious examples include the zerodisplacement at its end-points of an anchored vibrating string and the zeropotential contour that must coincide with a grounded electrical conductor.More subtle are the effects at internal boundaries, where the same non-vanishingvariable has to describe the situation on either side of the boundary but itsbehaviour is quantitatively, or even qualitatively, different in the two regions. Inthis section we will study an equation, Stokes’ equation, whose solutions havethis latter property; as well as solutions written as series in the usual way, we willfind others expressed as complex integrals.The Stokes’ equation can be written in several forms, e.g.d 2 yd+ λxy =0;dx2 2 yd+ xy =0;dx2 2 ydx 2 = xy.We will adopt the last of these, but write it asd 2 y= zy (25.32)dz2 to emphasis that its complex solutions are valid for a complex independentvariable z, though this also means that particular care has to be exercised whenexamining their behaviour in different parts of the complex z-plane. The otherforms 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’ equationIt will be immediately apparent that, even for z restricted to be real and denotedby x, the behaviour of the solutions to (25.32) will change markedly as x passesthrough x = 0. For positive x they will have similar characteristics to the solutionsof y ′′ = k 2 y,wherek is real; these have monotonic exponential forms, eitherincreasing or decreasing. On the other hand, when x is negative the solutionswill be similar to those of y ′′ + k 2 y = 0, i.e. oscillatory functions of x. Thisisjust the sort of behaviour shown by the wavefunction describing light diffractedby a sharp edge or by the quantum wavefunction describing a particle near tothe boundary of a region which it is classically forbidden to enter on energygrounds. Other examples could be taken from the propagation of electromagneticradiation in an ion plasma or wave-guide.Let us examine in a bit more detail the behaviour of plots of possible solutionsy(z) of Stokes’ equation in the region near z = 0 and, in particular, what may888