A Mathematical Theory of Full-wave Rectification

The results of a systematic asymptotic analysis of two basicfull-wave rectifier circuits are presented in this paper. Thephysical circuits are modelled by two systems of nonlinear differentialequations which are both singularly perturbed in the limit RC.Here is the angular frequency of the source, R is the resistanceof the load, and C is the capacitance of the circuit. The methodof multiple scales is used to construct the asymptotic behaviourof the solutions as RC. The results contain transient, ripple,and steady-state information. The latter two components simplifyand yield the standard results when the diodes' forward impedanceis small. The analysis of more complicated full-wave rectifiersfollows from the procedure described.     

Microwave heating of carbon-coated ceramic fibres: A mathematical model

The microwave heating of a thinly carbon-coated ceramic fibreis modelled and analysed in the small Biot number regime. Theelectric field is assumed known and uniform throughout the cross-section,and constant along the axis of the cylinder. The mathematicalmodel consists of a nonlinear heat equation with an idealizedsource term that models the thin carbon coating and simplifiedreaction kinetics. The analysis yields an asymptotic approximationof the heating process on two time scales. The first capturesthe initial heating of the carbon coating and the ceramic, andthe carbon reaction, while the second determines the long timebehaviour of the sample. The results show a qualitative relationshipbetween the coating thickness and the final temperature of thefibre. If the coating thickness is not uniform along the fibreaxis, then the model explains the mechanism for the formationand propagation of hot-spots.     

Acoustic target reconstruction using geometrical optics phase information

In this note, a simple algorithm which uses the phase informationfrom the geometrical optics limit to construct the shape ofan object is presented. Essentially, the phase in the backscattereddirection determines the equation of the tangent plane at theunique, but unknown, specular point. This plane depends uponthe two spherical angles and , which describe the incidentwave direction. The observation that the tangent plane envelopsthe obstacle as and are varied allows the derivation of anexplicit formula for the euation of the surface in terms ofthe measured scattered phase. An analogous two-dimensional formulais also presented.     

Hot spot formation in microwave heated ceramic fibres

An analysis of microwave heating of a thin ceramic cylinderin a single mode, highly resonant cavity is presented. Realisticassumptions regarding the effective electrical conductivity,thermal parameters, and physical dimensions are adhered to throughout.Consequently, the model developed herein incorporates most ofdetuning and a local electric feild perturbation on the heatingprocess. The model presented takes the form of a one-dimensionalreaction–diffusion equation which contains a functional.The developement of this equation is the product of a systematicmodelling process that involves S-matrix theory, a small-Biot-numberasymptotic analysis, and a matched asymptotic analysis of anon-standard electromagnetic scattering problem. This equationreveals both the mathematical structure and physical mechanismfor the formation of hot spots. An accurate numerical methodwhich approximates the solution of this equation is presented.The results agree qualitatively with experiments and predictobserved trends.