The flow of an inviscid, compressible, perfect gas along a sinus-shaped wall is used as a model to shed light into the long-standing transonic controversy. The solution of the small-disturbance approximation for the velocity potential is developed as a formal series in the similarity parameter k. Forty terms in k were obtained, delegating the computational work to a computer. The coefficients of the maximal-speed series turn out to be moments with positive weight on a finite interval of support. This implies
The solution of the classical moment problem, i.e. the recovering of the weight distribution, shows that --a = b = like, with k c as the critical value of the parameter, at which the flow first becomes sonic, k e has been determined as 0.8253, the four digits being regarded as definite. It follows that Urea x as a function of k is analytic on (-k c, k c) and has exactly two singularities located on the real axis at k = -k c and k = k e. It is known from the theory of moments that there does not exist an analytic continuation on the real axis exceeding the interval. This means that the velocity depends analytically on the parameter as long as the local velocity of sound is nowhere reached, but that the exceeding of it is marked by termination of analyticity and, to this degree, is critical Thus, the view that there will not exist a transonic potential flow having neighboar solutions is supported. The smallness of the weight distribution at the interval ends (at least at k = -k c it is even exponentially small, giving an exponentially small singularity of Urea x) is the key to explaining, within the scope of the inviscid model, the shoeldess exceeding of the critical value by some per cent seen in numerical calculations and experiments.