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National Advisory Committee for Aeronautics, Technical Notes - Transformations of the Hodograph Flow Equation and the Introduction of Two Generalized Potential Functions

It has been shown that the hodograph equations of motion can be
derived in a symmetrical form by the choice of the velocity and the mass
velocity as independent variables. The equations obtained by the use
of the velocity potential, the stream function, or their transforms as
the unknown function are of the same general form and therefore can be
treated in the same manner.

Particular sets of solutions have been studied independently of the
gas law adopted and. some properties of the series obtained by means of
these sets have been discussed. Approximate gas laws for which the
solutions of the hodograph equations can be easily found have been
briefly discussed.

The equations have been further transformed so as to have as
independent variables the complex velocity and the complex mass velocity.
Two new generalized potential functions can then be introduced that
satisfy very compact equations. From these functions, all the
quantities concerned with the representation of the motion can be
derived by means of formulas independent of the gas law adopted. By
means of the generalized potential functions some developments have been
performed with the apnoximate Chaplygin—Von Karman—Tsien law.

An approximate transonic method has also been suggested.
From a purely mathematical point of view, the ordinary hodograph
equations for the stream function or for the velocity-potential function
and the equations relating them to the physical coordinates are
sufficient for the study of two—dimensional isentropic flows. However,
from a more physical point of view they are not very elegant because of
their lack of symmetry in contrast with the symmetry of the corresponding
relations for the incompressible case.