The hodograph method is used to formulate several design problems in t
ransonic flow using small-disturbance theory. Analytical and numerical
methods give solutions to several optimum critical airfoil designs wi
th different constraints on the tail angle. Special airfoil shapes fly
ing at free-stream Mach number one are designed. The problem of constr
ucting a shock-free body of revolution at subsonic speed but having a
supersonic zone is formulated in the hodograph and solved numerically.