Wave equations and solutions for split-step numerical modelling of propagation in urban areas

Radiowave propagation in built-up areas such as in terrestrial mobile and urban fixed radio link services is limited to short distances. For this case the scalar wave equation is reduced in different forms suitable for computer implementation by usage of FFT routines. Two wave equations are considered related to what is known as the wide and the low angle solutions the last one valid for propagation angles close to the horizontal. The solutions are described in term of direct and inverse Fourier transforms. This allows to obtain a discrete step-by-step representation which can be solved by the Fourier split-step technique. The above formulation is reported both for the two and for the three dimensional case. For testing purposes the simulated case of diffraction by a slit in a 2D space is treated. Results obtained by wide angle and low angle solutions using FFT algorithms are compared with results obtained by approximated analytical methods. In both cases a good agreement is reached inside certain limits of validity. Some considerations on computer implementation problems are also reported.