A note on ‘Lifting three-dimensional wings in transonic flow’ by M. S. Cramer

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Abstract
The purpose of this note is to clarify the discrepancy between the results of Cramer (1979), Barnwell (1975) and Cheng & Hafez (1975). These authors have all derived the boundary-value problem governing the flow far from a three-dimensional lifting wing in transonic flow. The results of both Cramer and Barnwell will provide the lowest-order solution in the far field. The results of Cheng & Hafez are not only accurate to lowest order but to first order as well. That is, the theory of Cheng & Hafez is an order-of-magnitude more accurate than those of Barnwell and Cramer. This is the reason for the discrepancy between the boundary condition derived by Cheng & Hafez and that of Cramer and Barnwell. Because Cheng & Hafez correctly derive the first-order results, the lowest-order theories of Cramer or Barnwell cannot correct those of Cheng & Hafez. It should be recognized that the main contribution of Cramer's work was to clarify the structure of the problem and to simplify the analysis rather than correct the basic results.

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