Constitutive Relations and Subsidiary Conditions in the Eikonal Approximation

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Abstract
The eikonal approximation is considered for propagation in an inhomogeneous, time-dependent, anisotropic environment. The usual eikonal analysis is generalized to take into account constitutive relations which contain space and time derivatives of the field quantities. The field equations are obtained for all orders of eikonal approximation. These consist of homogeneous equations for the zeroth-order, and recursion relations which connect arbitrary consecutive higher orders of approximation. An apparent inconsistency between the zeroth-order equations and the higher order ones is avoided by the development of a set of subsidiary conditions on the field approximations. The lowest order condition in this set is considered in detail, and a general relation is derived between a certain vector quantity appearing in this condition, and the tangent vector to a ray path. For an environment which is stationary and free of spatial dispersion, this relation is shown to have, as one of its consequences, a new normalization condition for the zeroth-order fields in anisotropic propagation. The relation is applied to propagation in a cold magnetoplasma, and a number of results in ionospheric propagation are recovered and extended.

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