On acoustic and electric Faraday cages

Abstract

Two-dimensional problems involving many identical small circles are considered; the circles are the cross sections of parallel wires, modelling a cage or a grating. Both electrostatic and acoustic fields are considered. The main emphasis is on periodic configurations of N circles distributed evenly around a large circle (a ring). Foldy's theory is used for acoustic problems and then adapted for electrostatic problems. In both cases, circulant matrices are encountered: the fields can be calculated explicitly. Then, the limit N → ∞ is studied. A connection between the N -circle problem and the limiting problem (fields exterior to the ring) is established, using known results on the convergence of a defective form of the trapezoidal rule, defective in that endpoint contributions are ignored, because the integrand has logarithmic singularities at those points. This shows that the solution of the limiting problem is approached very slowly, as N − 1 log N as N → ∞ .