The convergence of point-matching techniques

Authors' Reply

“…The same results are valid for the diffraction problems [1], [2]. In this case, the singularities of the fields are defined by the form of the scatterer boundary and the structure of the incident wave.…”

“…Thus, the validity/invalidity of the Rayleigh hypothesis places limitations on the acceptable positioning of the nodes [1], [2]. For example, there are such node distributions that give the exact results only, if the Rayleigh hypothesis is valid and they produce incorrect data in opposite cases.…”

“…For > m, the Rayleigh hypothesis is invalid and for < m , the Rayleigh hypothesis is valid. In our paper, 2 we tried to point out that when < m , point collocation appears to converge for various node distributions but appears to diverge for some distributions, whereas when > m, point collocation appears to diverge for many node distributions but appears to converge for some distributions.…”

“…In this comment I should like to call the readers' attention to the fact that the inferences, based on the rigorous interpolation theory, conflict with the conclusions of the paper. 1 In [1] and [2] we examined the application of the collocation technique to the problem of waves diffraction by different scatterers (the metallic cylinders, gratings, etc. ), when the Rayleigh series are used.…”

Rayleigh Hypothesis

“…This conclusion was supported by a finding of Petit and Cadilhac", who demonstrated that for sinusoidal elevations h == a cos px (12) of sufficient steepness, ap > 0.448, the integral in Eq. (7) (13) kn = ko + pn, qn = JK2-k~, and the divergence of the sum Eq. (13) was proven.…”