Computation of Spherical Harmonic Representations of Source Directivity Based on the Finite-Distance Signature

The measurement of directivity for sound sources that are not electroacoustic transducers is fundamentally limited because the source cannot be driven with arbitrary signals. A consequence is that directivity can only be measured at a sparse set of frequencies-for example, at the stable partial oscillations of a steady tone played by a musical instrument or from the human voice. This limitation prevents the data from being used in certain applications such as time-domain room acoustic simulations where the directivity needs to be available at all frequencies in the frequency range of interest. We demonstrate in this article that imposing the signature of the directivity that is obtained at a given distance on a spherical wave allows for all interpolation that is required for obtaining a complete spherical harmonic representation of the source's directivity, i.e., a representation that is viable at any frequency, in any direction, and at any distance. Our approach is inspired by the far-field signature of exterior sound fields. It is not capable of incorporating the phase of the directivity directly. We argue based on directivity measurement data of musical instruments that the phase of such measurement data is too unreliable or too ambiguous to be useful. We incorporate numerically-derived directivity into the example application of finite difference time domain simulation of the acoustic field, which has not been possible previously.