NEW PROPERTIES OF THE BUSEMANN ELLIPSE

A combination of the Busemann ellipse, the inscribed unit circle and a circle of radius root 2 about the same centre is considered. For supe rsonic two-dimensional potential gas flows, it is shown that the incli nations of the velocity vector in motion along an arbitrary characteri stic, the characteristic itself and the characteristic of the other fa mily have values equal to, respectively: the difference between the ar eas of the elliptical and circular (R = 1) sectors, the difference bet ween the areas of the elliptical and circular (R = root 2) sectors, an d the area of the elliptical sector, apart from unimportant multiplica tive and additive constants. The straight sides of the sectors in ques tion are the semiminor axis of the ellipse and the radius vector of th e velocity. The obvious analogy with one of Kepler's laws is pointed o ut. The existence of a point of intersection of the ellipse and the se cond circle illustrates a well-known result of Khristianovich concerni ng the points of inflexion of characteristics with a monotone velocity distribution. It is shown how the combination of the ellipse and the inscribed circle illustrates the simplification of the compatibility c onditions sind the Darboux equation for trans-and hypersonic flows. (C ) 1998 Elsevier Science Ltd. All rights reserved.