Inverse aeroacoustic problem for a streamlined body. I - Basic formulation

The inverse aeroacoustic problem associated with a flat-plate airfoil in unsteady compressible flow is formulated in terms of a Fredholm integral equation of the first kind. The feasibility of determining the unsteady pressure along the airfoil surface from the radiated acoustic signal is demonstrated. It is shown that the Hadamard conditions of existence and uniqueness of the inverse solution can be satisfied. The third Hadamard condition for the continuous dependence of the solution on the input acoustic data shows the problem to be ill-posed. The singular value decomposition method with regularization is used to treat the associated ill-conditioned algebraic problem. Discretization and collocation techniques are used to represent the unsteady pressure on the airfoil. Both methods give very accurate reconstructions when perfect input data are used. The collocation method requires only a small amount of input data either from the far field or the near field, thus making it more suitable for applications.