3. Solution of the Taylor-Maccoll equations

Eqs. (5) and (6) are simultaneous, first-order, total differential equations that can be solved by standard methods, such as in Ralston and Wilf [45], for the two Mach numbers u and v in terms of θ. The Mach number M is then found from <sup>M</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>p</sup> and other thermodynamic values follow from isentropic relations. The shape of the intake surface can also be integrated within the integration routine to give r in terms of θ, r ¼ fð Þθ so that the Cartesian coordinates of the axisymmetric Busemann surface shape are found from x ¼ r cos θ and y ¼ rsin θ.
