Vector Wave Equation of Aeroacoustics and Acoustic Velocity Formulations for Quadrupole Source

An analytical acoustic vector formulation is meaningful to visualize acoustic energy radiating from sources and to predict sound scattered by surfaces. This paper presents a vector wave equation of aeroacoustics, which is an exact rearrangement of the continuity and momentum equations, by following the idea of the acoustic analogy. Based on the vector wave equation, an alternative method is proposed to deduce the acoustic velocity formulations for thickness and loading sources. Moreover, both the time-domain and frequency-domain acoustic velocity formulations for the quadrupole source are deduced as well. The formulation solved by the time-domain numerical method is suitable for the quadrupole source in arbitrary motion, whereas the formulation related to the frequency-domain numerical method is valid for the rotating source. A numerical test case is carried out to validate the proposed acoustic velocity formulations for the quadrupole source. Up to now, all the time-domain and frequency-domain acoustic pressure and acoustic velocity formulations for the thickness, loading, and quadrupole sources have been developed, indicating that both the acoustic scalar and vector quantities can be computed directly by employing the acoustic analogy. Further research on the acoustic vector is discussed as well.