17 - Coalescence of coupling points

Summary

Introduction

The first order coupled wave equations (16.22) were derived and studied in § 16.3 for the neighbourhood of a coupling point z = z_p where two roots q₁, q₂ of the Booker quartic equation are equal. If this coupling point is sufficiently isolated, a uniform approximation solution can be used in its neighbourhood and this was given by (16.29), (16.30). It was used in § 16.7 to study the phase integral formula for coupling. The approximations may fail, however, if there is another coupling point near to z_p, and then a more elaborate treatment is needed. When two coupling points coincide this is called ‘coalescence’. This chapter is concerned with coupled wave equations when conditions are at or near coalescence.

A coupling point that does not coincide with any other will be called a ‘single coupling point’. It is isolated only if it is far enough away from other coupling points and singularities for the uniform approximation solution (16.30) to apply with small error for values of |ζ| up to about unity. Thus an isolated coupling point is single, but the reverse is not necessarily true.

Various types of coalescence are possible. The two coupling points that coalesce may be associated with different pairs q₁, q₂ and q₃, q₄ of roots of the Booker quartic. This is not a true coalescence because the two pairs of waves are propagated independently of each other.