Spectral Analysis of Acoustic Sources Moving Along Elliptical Orbits

Predicting the frequency spectrum from sources on non-circular trajectories is challenging because closed-form representations are scarce. This study derives Mathieu-function-based solutions for harmonic sources moving on elliptical paths. First, the two-dimensional sound field of line sources traveling along an ellipse is analyzed for a single revolution and for periodic motion over infinitely many revolutions. The resulting frequency spectrum is expressed in terms of Mathieu functions. As the eccentricity of the ellipse decreases and the orbit approaches circularity, the solution converges to that of the corresponding circular orbit. The discrete line spectrum of the periodic case coincides with the continuous spectrum obtained for a single revolution. In the second part, a monopole moving along an elliptical helix is investigated. The three-dimensional spectral sound field generated by the source is represented by a series expansion in Mathieu functions. The series solution shows good agreement with that derived from the integral formulation of the Cartesian Convolution Integral over a range of source frequencies and angular velocities. As the cross section of the elliptical helix approaches a circular shape, the computed sound field converges to the known solution for the circular helix. The results enable sound-field prediction for imperfect rotating sources and propellers.