The Decay of Eddy-currents in Thin Sheets and Related Water-wave Problems |
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Authors: | HURLEY D G; SIEW P F |
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Institution: |
University of Western Australia, Department of Mathematics Nedlands, Western Australia 6009
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Abstract: | The decay of the eddy-currents that are induced in a thin, uniform,imperfectly-conducting sheet by switching off the source ofan external magnetic field is investigated. For the two-dimensionalproblem of an infinite strip the (non-dimensional) decay constants n and eddy-current distributions in(x) are the eigenvalues andeigenfunctions of the integral equation
with the constraint . For the circular disc the corresponding equation is
where
and K and E are complete elliptic integrals. For both problemsthe initial eddy-currents have inverse-square-root singularitiesat the edges but during their decay the eddy currents are finiteat the edges and the normal magnetic fields have logarithmicsingularities there. Numerical results are given for variousinitial-value problems. The eddy current problems are closely related to water-waveproblems in which there is a strip-shaped or circular aperturein a horizontal rigid dock. If n and n are the decay constantsand magnetic scalar potentials for the strip and n and n theangular frequencies and velocity potentials for the normal modesin the strip-shaped aperture, then n =n2 and n and n are thereal and imaginary parts respectively of a holomorphic function.The velocities in the normal modes are deduced from the solutionof the eddy-current problem and are found to agree with resultsgiven in Miles (1972). For circular geometries the eigenvaluesand eigenfunctions of the axisymmetric eddy-current problemare the same as those of the water-wave problem that has angularvariation ei ; where ( , , z) are cylindrical polar co-ordinateslocated at the centre of the basin. |
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