On the issue of resonance in an unsteady supersonic cascade

M a recent paper, Verdon and McCune1 presented a inearized analysis of an unsteady supersonic cascade with subsonic axial velocity. It is an interesting advancement on Ver don's previous paper.2 To achieve its objective of computing the pressure distribution, the analysis of Ref. 1 starts out to add the contributions from unsteady disturbances generated at all the oscillating airfoils below the reference airfoil in the cascade. The partial sums of the series were found to oscillate about apparent limiting values in general but, unfortunately, convergence was not proved. Aside from the problem of convergence, Ref. 1 also reports that under certain circumstances the numerical scheme broke down rather inexplicably. Thus it states in the concluding remarks that further work is needed to resolve these questions. Several years ago, the author of the present Note encountered essentially the same series and found that it diverges at a certain number of discrete points, although these results were never published. The appearance of Ref. 1, therefore, seems a fitting opportunity to point out the divergence of the series, to offer it as a possible explanation for the aforementioned breakdown of the numerical scheme and to discuss its physical implications in regard to resonance and other salient points. Consider, for example, the following kernel function K(x) , which appears in the integral equation (23) of Ref. 1 : where S (x) is given by -7