On the general connection between wave impedance and complex sound intensity

This paper presents the rigorous definition of acoustic reactivity as a tensor of rank 2, given on the basis of the unique decomposition of particle velocity through the orthogonalization process of acoustic pressure and particle velocity in the Hilbert space of signals. Its spatial representation on the orthonormal basis (Serret-Frenet frame) co-moving with the average energy along any power streamline allows to introduce the concept of "flat" acoustic space, when the streamline has a negligible curvature and torsion. The complex sound intensity has been accordingly defined as an element S-(x) of the complex vector space C3 tangent to any power streamline at each point x. From this definition a general spectral equation linking the frequency distribution of complex intensity to the wave impedance times the particle velocity autospectrum, has been formulated and called the acoustic energy-mass equation. The graphical proof of the energy-mass equation has been given for two "flat" and one "curved" acoustic spaces built up with quasi-stationary plane waves and divergent spherical waves. This grand result allowed to develop a precision device for measuring 3D sound intensity and its active and reactive spectral components as reported in the companion paper 2aSPb3