Extension of the angular spectrum approach to curved radiators

The angular spectrum approach (ASA) is conventionally applied to the evaluation of acoustic fields from planar radiators because it is usually based on the 2-D Fourier transform (or the zero-order Hankel transform in the axisymmetrical case) which is implemented only in a plane. The present paper is intended to extend the ASA to more general cases where radiators have curved surfaces. For this purpose, two approaches are developed. The first one is the extended ASA and is derived in a general way. From this approach, the angular spectrum of a curved radiator is given by a double integral that does not take the 2-D Fourier transform form, and thus cannot be implemented using 2-D fast Fourier transform (FFT) but by numerical integration. The second approach is the indirect ASA that gives the angular spectrum via 2-D Fourier transforming an initial field pre-calculated in a plane. The method for calculating the initial field is proposed based on the method developed by Ocheltree and Frizzell for planar sources. An example is given of a linear array with a cylindrically concave surface, and in this case, the angular spectrum (the double integral) from the extended ASA reduces to a single integral. The angular spectra of the array are calculated using both approaches and compared. The comparison has shown that the angular spectra obtained from both approaches are in excellent agreement. The accuracy and efficiency of the two approaches are studied in the numerical implementation. In this example, the extended ASA has been shown to be more efficient and more accurate than the latter approach. Both approaches can be applied to arbitrarily curved transducers. In the general case where the double integral cannot be reduced to a single integral, the latter approach can be more efficient.