Waves near Stokes lines

The large-k asymptotics of d2u(z)/dz2 = k2R2(z) u(z) are studied near a Stokes line (ω ≡ ∫z z0 R dz real, where z0 is a zero of R2(z), of any order), on which there is greatest disparity between the dominant and subdominant exponential waves in the phase-integral (WKB) approximations. The aim is to establish precisely how the multiplier b_ of the subdominant wave varies across the Stokes line. Although b_ always has a total change proportional to i times the multiplier of the dominant wave (the Stokes phenomenon), the form of the change depends on the convention used to define the two waves. The optimal convention, for which the variation is maximally compact and smooth, is to define them by the phase-integral approximation truncated at its least term, whose order is proportional to k and therefore large (‘asymptotics of asymptotics’). Then the variation of b_ is proportional to the error function of the natural Stokes-crossing variable Im ω √(k/Re ω). This result is obtained without resumming divergent series (thereby avoiding ‘asymptotics of asymptotics of asymptotics’). An application is given, to the birth of exponentially weak reflected waves in media with smoothly varying refractive index.