Jump conditions for fields that have infinite, integrable singularities at an interface

Abstract

Jump conditions are derived for a basic set of first order partial differential equations whose fields have infinite, integrable singularities, as well as finite discontinuities, at a moving and deforming interface. These basic formulas are then applied to Maxwell’s electrodynamic equations and yield the jump conditions that hold when the electric and magnetic fields have such singularities. These jump conditions are shown to be generalizations of formulas previously derived for double charge layers in electrostatics and for an interface with surface magnetization density in magnetostatics. The basic formulas are also used to obtain jump conditions for the wave equation and other second order partial differential equations whose fields have finite discontinuities at an interface. A similar application to second order vector identities yields a new set of jump identities. These identities show that the normal and tangential components of the jump in a vector field are kinematically interdependent; e.g., shock waves and vortex sheets are kinematically linked—a fact that may be significant for shock-slip flows in aerodynamics. The jump identities also indicate the fields that must be measured at an epoch in order to calculate the instantaneous growth/decay rate of a propagating discontinuity, such as an atmospheric front. A feature of the derivation is that the field jumps and surface densities are mathematically defined as continuous and differentiable functions of three-dimensional space and time that assume physical values on the physical interface. This approach is simpler and more general than previous approaches that define jumps and surface densities only on the physical interface because the brackets (jumps) now commute with all derivatives, instead of with only the tangential and displacement derivatives. Boundary value problems with propagating infinite singularities in the electrodynamic fields are presented as examples.