Multipole imaging of an elongated magnetic source

This paper describes a mathematical method for modeling the magnetic field from an elongated source, a method that is relevant to the reduction of the magnetic signature of a marine vessel. We seek the magnetic multipole image (MMI), the set of coefficients in a prolate spheroidal harmonic expansion of magnetic scalar potential U, which describes the magnetic source. The MMI is useful for efficient and accurate modeling of the magnetic field outside the source and for developing methods of reducing the magnetic signature. We model an elongated complex source by consideration of elementary internal sources, which consist of concentrated eccentric dipoles (located away from the origin) inside the spheroidal boundary and coils on the spheroidal surfaces (bounded by longitudes and latitudes). For the internal dipole sources, the scalar magnetic potential U is expanded in prolate spheroidal coordinates in terms of associated Legendre functions. The Green function F=1/R, R being the distance between the source and evaluation point, is also expressed In terms of Legendre functions of the first and second kind. Since U of an elementary dipole is proportional to the gradient of F, comparison of the expansion coefficients is used to determine the MMI of the source. In the same way, an integration of the gradient of F over the virtual surface dipoles inside the coil boundaries is used to determine the MMI of each elementary coil.