Beamforming for measurements under disturbed propagation conditions using numerically calculated Green's functions

Beamforming methods for sound source localization are usually based on free-field Green's functions to model the sound propagation between source and microphone. This assumption is known to be incorrect for many industrial applications and the beamforming results can suffer from this inconsistency regarding both, accuracy of source power estimation, and accuracy of source localisation. The aim of this paper is to investigate whether the use of numerically calculated Green's functions can improve the results of beamforming measurements. The current test cases of numerical and experimental investigations consists of sources placed in a short rectangular duct. The measurement is performed outside the duct in a semi-anechoic chamber. A typical example for this kind of installation is a fan with a heat exchanger. The Green's functions for this test case are calculated numerically using the boundary element method. These tailored Green's functions are used to calculate the corresponding beamforming steering vectors. The weighting of the Green's functions in the steering vectors has a decisive influence on the beamforming results. A generalization of the common steering vector formulations is given based on two scalars. It is shown that arbitrary differentiable Green's functions can be used to find the correct source position or source power level by using the appropriate steering vector formulations. Beamforming measurements are performed using a loudspeaker as a reference source at different positions in the heat exchanger duct. The measurements are evaluated in the frequency domain and by means of different validation criteria it can be shown that the results with the numerical calculated Green's function are improved compared to free field beamforming especially at low frequencies.