Radial cancellation in spinning sound fields

The radiating part of a circular acoustic source is determined on the basis of an exact analysis of the radiation properties of a source with angular dependence \exp \J n\theta and arbitrary radial dependence. It is found that the number of degrees of freedom in the radiated field is no greater than $k-n$, where $k$ is the wavenumber. The radiating part of the source at low frequency is the wavenumber. The radiating part of the source at low frequency is explicitly stated and used to analyze noise cancellation. The results are applied to the identification of sources in jet noise and an explanation for the low order structure of jet noise fields is proposed.Comment: Submitted to Journal of Fluid Mechanic

Moving least squares via orthogonal polynomials

A method for moving least squares interpolation and differentiation is presented in the framework of orthogonal polynomials on discrete points. This yields a robust and efficient method which can avoid singularities and breakdowns in the moving least squares method caused by particular configurations of nodes in the system. The method is tested by applying it to the estimation of first and second derivatives of test functions on random point distributions in two and three dimensions and by examining in detail the evaluation of second derivatives on one selected configuration. The accuracy and convergence of the method are examined with respect to length scale (point separation) and the number of points used. The method is found to be robust, accurate and convergent.Comment: Extensively revised in response to referees' comment

Information in spinning sound fields

The information content of a spinning sound field is analyzed using a combination of exact and asymptotic results, in order to set limits on how accurately source identification can be carried out. Using a transformation of the circular source to an exactly equivalent set of line source modes, given by Chebyshev polynomials, it is found that the line source modes of order greater than the source wavenumber generate exponentially small fields. Asymptotic analysis shows that the remaining, lower order, modes radiate efficiently only into a region around the source plane, with this region shrinking as the mode order is increased. The results explain the ill-conditioning of source identification methods; the successful use of low order models in active noise control; and the low radiation efficiency of subsonic jets.Comment: Submitted to Journal of the Acoustical Society of Americ

Quadrature for second-order triangles in the Boundary Element Method

A quadrature method for second-order, curved triangular elements in the Boundary Element Method (BEM) is presented, based on a polar coordinate transformation, combined with elementary geometric operations. The numerical performance of the method is presented using results from solution of the Laplace equation on a cat's eye geometry which show an error of order $P^{-1.6}$, where $P$ is the number of elements.Comment: 14 pages; 6 figures; submitted to International Journal for Numerical Methods in Engineerin

Extrapolation of rotating sound fields

A method is presented for the computation of the acoustic field around a tonal circular source, such as a rotor or propeller, based on an exact formulation which is valid in the near and far fields. The only input data required are the pressure field sampled on a cylindrical surface surrounding the source, with no requirement for acoustic velocity or pressure gradient information. The formulation is approximated with exponentially small errors and appears to require input data at a theoretically minimal number of points. The approach is tested numerically, with and without added noise, and demonstrates excellent performance, especially when compared to extrapolation using a far-fieldassumption

Shielding of rotor noise by plates and wings

A method of noise reduction proposed for the next generation of aircraft is to shield noise from the propulsion system, by positioning the noise source over a wing or another surface. In this paper, an approximate analysis is developed for the acoustic field far from a circular source placed near the edge of a semi-infinite plate, a model problem for shielding of noise by a wing and for scattering by a trailing edge. The approximation is developed for a source of small radius and is found to be accurate when compared to full numerical evaluation of the field

Closed-form evaluation of potential integrals in the Boundary Element Method

A method is presented for the analytical evaluation of the singular and near-singular integrals arising in the Boundary Element Method solution of the Helmholtz equation. An error analysis is presented for the numerical evaluation of such integrals on a plane element, and used to develop a criterion for the selection of quadrature rules. The analytical approach is based on an optimized expansion of the Green's function for the problem, selected to limit the error to some required tolerance. Results are presented showing accuracy to tolerances comparable to machine precision