Discontinuities in aerodynamics and aeroacoustics: The concept and applications of generalized derivatives

Abstract The concept of generalized differentiation is defined for functions of one and several variables. With simple examples it is shown that in finding discontinuous solutions of differential equations, one should write the differential equations in terms of generalized derivatives. The discontinuities in the solution then appear as additional source terms in the resulting equation to which the conventional Green function technique may be applied. The method is very systematic and in many cases, reduces algebraic manipulations. Two examples from aerodynamics and two examples from aeroacoustics are worked out. One example from aerodynamics is a new derivation of the Oswatitsch integral equation of transonic flow. A simple expression for the sound from high speed surfaces in motion is obtained. It is shown that there exists a Poisson equation with time ( t ) as a parameter whose solution is identical to the solution of the acoustic wave equation at only a single point for all values of t . Other well-known results from a new point of view based on generalized differentiation are also derived and presented. In the two appendices the products of delta functions and regularization of a divergent integral appearing in one of the examples are discussed.