Evaluation of the Mathieu Function Series for Diffraction by a Slot

Diffraction of a wave passing through a slot is a fundamental problem that has applications in many situations. A problem of current interest is the propagation of waves through obstacles such as buildings in an urban environment, where an entrance aperture could be approximated as a slot. Solutions for the diffracted fields of a slot have been obtained by exact eigenfunction expansions as well as various approximate and numerical methods. The eigenfunction solution, in terms of Mathieu functions, has been published by Stamnes and Eide [1] for a slot with several types of converging incident waves. Results of the eigenfunction solution are presented in [2] for slots up to 20 wavelengths in width and compared with Kirchho. and Debye approximations. We have attempted to duplicate the eigenfunction series results in [2]. Little information is given in [2] about the numerical issues in the evaluation. We encountered a number of problems in the evaluation of the Mathieu functions for wide slots that required modifications of the Mathieu function evaluation methods described in [3]. Results were obtained matching those for the largest slot of 20{lambda} in [2] and also for slots up to 40{lambda} width. The evaluation of the eigenfunction solution wasmore » checked against a 2D moment method solution and was in close agreement up to a slot width of 40{lambda}. The eigenfunction solution published by Stamnes and Eide is summarized below, followed by a discussion of the Mathieu function evaluation and problems encountered with large slot widths. Results are then presented for varying slot width and compared with the 2D moment method solutions. A moment method solution for a diverging wave incident on the gap between two square cylinders is also presented and compared with the fields in a slot representing the front aperture.« less