Predicting the Acoustics of Arbitrarily Shaped Bodies Using an Integral Approach

An integral solution of the Helmoltz equation is developed for predicting the acoustic properties of arbitrarily shaped bodies. With the integral formulation, the acoustic potentials at the surface are solved independently of the internal acoustic field which, effectively, reduces the dimensionality of the problem by one. Considerable reductions in computation time and storage requirements are thus achieved. Efficient numerical techniques for solving the resulting algebraic equations are presented. Numerical results obtained for the two-dimensional problems of a circle and a rectangle agree to within one percent with available exact solutions. The modes of a star-shaped configuration and a duct with a right-angle bend are also determined to demonstrate the applicability of this method to complicated geometries and general boundary conditions. The acoustic properties of a sphere are investigated using an axisymmetric formulation. With the axisymmetric formulation the numerical and exact results agree to three significant figures. I. Introduction T HE prediction of the acoustics of arbitrarily shaped bodies has a variety of applications in aerospace engineering. Among them are the determination of the internal and radiated sound fields from airbreathing propulsion systems and the investigation of the stability limits of rocket combustors. These studies are concerned with obtaining solutions to the Helmholtz equation, which is derived from the wave equation when a sinusoidal time dependence is assumed and which describes the spatial dependence of the oscillations. This equation is included in most standard texts on differential equations of mathematical physics (Ref. 1, Ch. 11) and has been extensively studied in both differential and integral form., The differential form is currently the most widely used.