Wave Propagation Governed by an Infinite Number of Transition Points

The reflection of waves from an isolated transition point isgeneralized to embrace an infinite number of evenly spaced non-isolatedtransition points lying on a straight line perpendicular tothe propagation axis in the complex plane. The comparison functionssuited to the problem are Bessel functions, yielding uniformapproximate solutions along the axis, from which reflectioncoefficients may be calculated. The theory is illustrated bycomparison with an exact analytical solution in terms of Whittakerfunctions, for a model containing two parallel lines of transitionpoints.     

Properties of Some Generalized Hypergeometric Functions Governing Coupling at Transition Points

Understanding of wave propagation problems is enhanced by consideringgeneralizations to differential equations of order 2n. In particular,reflection and coupling of waves at transition points can involvecertain types of generalized hypergeometric functions. In thispaper, properties of _oF_2n-1 functions are considered systematically,when the parameters are specially chosen for application totransition points; a wide range of interesting properties unfolds,which recall the properties of Bessel functions when n = 1.     

Calculation of Transmission Coefficients for Generalized Propagation Problems

Generalized wave propagation takes place through a homogeneousslab governed by a differential equation of arbitrary even order,the elementary physical case being given when this order istwo. Approximation procedures are investigated whereby the transmissioncoefficient is calculated for highly overdense slabs, in sucha way that a correction term can be specified for the unit valueof the modulus of the reflection coefficient.     

Further Exact and Approximate Considerations of the Barrier Problem

When the second order differential equation governing transmissionof waves through a potential barrier is solved approximately,two approximate solutions arise within the barrier, one exponentiallylarge and one exponentially small. When a linear combinationof these solutions is considered, the error involved in theexponentially large solution is much larger than the actualsmaller solution, leading to conceptual difficulties as to howthis small solution is to be interpreted within the linear combination.Here, a new approach is adopted, whereby linear combinationsof approximate solutions are avoided. The reflection coefficientof the barrier is derived and the series expansion of its modulusis obtained before approximations are introduced. The analysisis so arranged that ratios rather than linear combinations enterthis modulus, and error analysis then shows exactly why theerror consists of certain unexpected exponentially small termsrather than expected terms of larger order of magnitude.