On the Deficiency of the First-Order OttClemmow Saddle Point Method as Applied to the Sommerfeld Half-Space Problem

The accuracy of the first-order multiplicative (Ott-Clemmow) and additive (van der Waerden) modified saddle point integration methods for the Sommerfeld problem of a vertical Hertzian dipole over a lossy half-space is investigated using two theoretically equivalent formulations with either a positive or a negative image term extracted. It is demonstrated that whereas the additive method leads to the same asymptotic field representation irrespective of the sign of the image, the multiplicative variant yields distinct results in these two cases, both of which differ from the unique result of the additive method. It is further found that the positive-image Ott-Clemmow method yields the well-known Norton formula, but the negative-image variant is inaccurate and predicts a zero field in the on-surface transmitter-receiver configuration. It is also demonstrated that the first-order multiplicative method yields an incomplete representation of the first order in the inverse distance.