Acoustic far field ($k_{0}R\gg 1$)

The acoustic far field is the radiation region where the observer is many reduced wavelengths from the source and far compared with its geometric extent. Three-dimensional outgoing acoustic density perturbations spread as $R^{-1}$; two-dimensional harmonic fields spread as $R^{-1/2}$. The condition is independent of the acoustic compact-source approximation $k_0\ell \ll 1$. Differentiating a retarded phase produces the radiating part of a Green function integral; derivatives of the spreading factor produce near-field terms.

Acoustic directivity describes the angular dependence of radiation in the acoustic far field. For a time-harmonic source in $d$ spatial dimensions, separating the geometric spreading and oscillatory phase leaves a complex amplitude pattern $\mathcal{D}$. The angular intensity pattern is proportional to $|\mathcal{D}{|}^2$. Directivity can be normalized by a reference direction or total radiated power; a stated convention is needed when comparing absolute directivity factors.