© 1949 by Oxford University Press
FLOW PATTERNS AND THE METHOD OF CHARACTERISTICS NEAR A SONIC LINE
( Department of Mathematics, The University Manchester )
The work described has two aims: (i) to investigate the geometry of the flow pattern near a sonic line, and (ii) to develop a method for the numerical integration of the equations of motion near a sonic line, where the usual method based on characteristics (ref. 1) breaks down. Only isentropic irrotational steady flow with two independent variables is considered, and attention is mainly confined to plane flow.
(i) The directions of the Mach lines coincide at a sonic line, and, in general, their curvatures are infinite; the Mach lines are approximately semi-cubical parabolas (eqn. (21)), the isoclines approximately parabolas (eqn. (15)), and the equipotential lines approximately cubical parabolas (eqn. (14)). The state of affairs is different if the curvature of the streamline vanishes on the sonic line; the curvatures of the Mach lines at the sonic point are then discontinuous but bounded; other results for this special case are set out in 
6.
(ii) Two methods are described for extending computations near a sonic line. In one method double Taylor series for the velocity components are used; in the other method expansions in the neighbourhood of a sonic point are used for the coordinates of a Mach line through the sonic point in powers of the complement of the Mach angle. The methods are developed for plane flow; the necessary modifications are indicated for flows with axial symmetry.