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National Advisory Committee for Aeronautics, Report - On Transonic Flow Past a Wave Shaped Wall

The present report is an extension of a previous investigation
(described in NAOA Rep. 1069) concerned with the solution
of the nonlinear difierential equation for transonic flow past
a wavy wall. In the present work several new notions are
introduced which permit the solution of the recursion formulas
arising from the method of integration in series. In addition,
a novel numerical test of convergence, applied to the power
series (in the transonic similarity parameter) representing the
local Mach number distribution at the boundary, indicates
that smooth symmetrical potential flow past the wavy wall is
no longer possible once the critical value of the stream Mach
number has been enceeded.

In NACA Report 1069 (ref. 1) the writer considered the
problem of two—dimensional transonic flow past an infinitely
long sinusoidal solid boundary. The problem was treated
in the physical plane and the purpose was to investigate
whether or not the flow past the wavy wall remains a smooth
symmetrical type of potential flow when the undisturbed-
stream Mach number exceeds its mritical value. By a
smooth type of potential flow is meant one for which the
velocity potential, say, and its first derivatives are single-
valued and continuous; that is, there are no discontinuities
of the nature of shock waves.

The purpose of the present work is to express the solution
of the problem of transonic flow past the wavy wall in a
form more suitable for general considerations and to prove
that the assumed smooth symmetrical type of potential flow
cannot exist at stream Mach numbers beyond the critical
value.