Generalized functions and acoustic analogy

The aeroacoustic analogy is reviewed. The importance of generalized functions is demonstrated deriving Gauss like theorems in three and two dimensions. It is shown how the Dirac delta function plays an essential part in the theory of fundamental solutions (Green’s function). The concept of generalized derivatives as introduced by Farassat is derived in a consistent way for the time derivative and vector operations, for example gradient, divergence and curl. Applied to the wave equation, the impact of these operators when acting on surface distributions is presented. The compatibility conditions which apply at static and moving surfaces are derived in a unique manner. New source terms are found for the FW-H equation which steam from second order generalized derivatives, whereas all terms with the mass flux vector cancel out due to symmetry conditions. This procedure is validated by the derivation of the same result from the Lighthill equation directly and a term-by-term comparison with the Kirchhoff formula. Finally, it is shown how terms, containing the curvature of the integration surface, appear in the final solutions. The role of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>δ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> factor in generating dipole as well as monopole sources is outlined. Essential knowledge provided from electromagnetic research is transferred to aeroacoustic theory. The vector wave equation is proposed instead of the scalar version to mitigate spurious noise effects.