On the efficient evaluation of Sommerfeld integrals over an impedance plane: exact and asymptotic expressions

In this work, the efficient evaluation of Sommerfeld integrals (SIs) above an impedance plane is addressed. Started from Weyl’s expression of SIs, using the coordinate transformation and steepest descent path approach, an exact single image representation to SIs is derived. This single image representation image eliminates oscillating and slow-decay integrand in traditional SIs, and efficient to calculate. Moreover, the far-field asymptotic behavior of SIs in this case is considered and is represented by the Fresnel-integral related function. A high-order approximation based on series expansion of Fresnel integral is provided for fast evaluation. Finally, the validity of the proposed expressions is verified by numerical examples.