TY - JOUR

T1 - Theoretical development of elliptic cross-sectional hyperboloidal harmonics and their application to electrostatics

AU - Sten, J. C.-E.

AU - Fragoyiannis, G.

AU - Vafeas, P.

AU - Koivisto, P. K.

AU - Dassios, G.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - The analytic computation of electric and magnetic fields
near corners and edges is important in many applications
related to science and engineering. However, such
complicated situations are hard to deal with, since they
accumulate charges and consequently they mathematically
represent singularities. In order to model this singular
behavior, we introduce a novel method, which is related
to the geometry and the analysis of the ellipsoidal
coordinate system. Indeed, adopting the benefits of the
corresponding coordinate surfaces, we use a general
non-circular double cone, being the asymptote of a
two-sided hyperboloid of two sheets with elliptic cross
section, which matches almost perfectly the particular
physics and captures the corresponding essential features
in a fully three-dimensional fashion. To this end, our
analytical technique employs the ellipsoidal geometry and
adapts the ellipsoidal functions (solutions of the
well-known Lamé equation) so as to construct a new set of
the so-called elliptic cross-sectional hyperboloidal
harmonics, supplemented by the appropriate orthogonality
rules on every constant coordinate surface. By first
recollecting the key results of the coordinate system and
the related potential functions, including the
indispensable orthogonality results, we demonstrate our
method to the solution of two boundary value problems in
electrostatics. Both refer to a non-penetrable
two-hyperboloid of elliptic cross section and its
double-cone limit, the first one being charged and the
second one scattering off a plane wave. Closed form
expressions are derived for the related fields, while the
already known formulae from the literature are readily
recovered, all cases being followed by the appropriate
numerical implementation.

AB - The analytic computation of electric and magnetic fields
near corners and edges is important in many applications
related to science and engineering. However, such
complicated situations are hard to deal with, since they
accumulate charges and consequently they mathematically
represent singularities. In order to model this singular
behavior, we introduce a novel method, which is related
to the geometry and the analysis of the ellipsoidal
coordinate system. Indeed, adopting the benefits of the
corresponding coordinate surfaces, we use a general
non-circular double cone, being the asymptote of a
two-sided hyperboloid of two sheets with elliptic cross
section, which matches almost perfectly the particular
physics and captures the corresponding essential features
in a fully three-dimensional fashion. To this end, our
analytical technique employs the ellipsoidal geometry and
adapts the ellipsoidal functions (solutions of the
well-known Lamé equation) so as to construct a new set of
the so-called elliptic cross-sectional hyperboloidal
harmonics, supplemented by the appropriate orthogonality
rules on every constant coordinate surface. By first
recollecting the key results of the coordinate system and
the related potential functions, including the
indispensable orthogonality results, we demonstrate our
method to the solution of two boundary value problems in
electrostatics. Both refer to a non-penetrable
two-hyperboloid of elliptic cross section and its
double-cone limit, the first one being charged and the
second one scattering off a plane wave. Closed form
expressions are derived for the related fields, while the
already known formulae from the literature are readily
recovered, all cases being followed by the appropriate
numerical implementation.

UR - http://www.scopus.com/inward/record.url?scp=85019066338&partnerID=8YFLogxK

U2 - 10.1063/1.4982638

DO - 10.1063/1.4982638

M3 - Article

VL - 58

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 5

M1 - 053505

ER -