The Decay of Eddy-currents in Thin Sheets and Related Water-wave Problems

摘要

The decay of the eddy-currents that are induced in a thin, uniform, imperfectly-conducting sheet by switching off the source of an external magnetic field is investigated. For the two-dimensional problem of an infinite strip the (non-dimensional) decay constantsλnand eddy-current distributionsin(x) are the eigenvalues and eigenfunctions of the integral equationin(x)+λn/π∫-11in(t)logx-tdt=0,xρ,andKandEare complete elliptic integrals. For both problems the initial eddy-currents have inverse-square-root singularities at the edges but during their decay the eddy currents are finite at the edges and the normal magnetic fields have logarithmic singularities there. Numerical results are given for various initial-value problems.The eddy current problems are closely related to water-wave problems in which there is a strip-shaped or circular aperture in a horizontal rigid dock. Ifλnandψnare the decay constants and magnetic scalar potentials for the strip andσnandψnthe angular frequencies and velocity potentials for the normal modes in the strip-shaped aperture, thenλn=n2andψnandψnare the real and imaginary parts respectively of a holomorphic function. The velocities in the normal modes are deduced from the solution of the eddy-current problem and are found to agree with results given in Miles (1972). For circular geometries the eigenvalues and eigenfunctions of the axisymmetric eddy-current problem are the same as those of the water-wave problem that has angular variationeiϑ; where (ρ,θ,z) are cylindrical polar co-ordinates loc