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The acoustic far field is the radiation region of an acoustic wave. For wavenumber $k_0=\omega /c_0$, $k_{0}R\gg 1$ places an observer beyond the reactive near field. For a source of extent $\ell$, a range-independent acoustic directivity pattern generally also requires $R\gg \ell$ and $k_0{\ell }^2/R\ll 1$. The last condition controls the quadratic phase error in the Fraunhofer approximation: distance expands as $|R\mathbf{n}-\mathbf{y}|=R-\mathbf{n}\cdot \mathbf{y}+O({\ell }^2/R)$.^[ref]

Outgoing acoustic density perturbations spread as $R^{-1}$ in three dimensions and $R^{-1/2}$ in two. These observer-distance conditions differ from the acoustic compact-source approximation $k_0\ell \ll 1$, which constrains the source itself.

Peter H. Dahl develops physical accounts of acoustic radiation and measurement. His applied-acoustics lectures derive how an extended source's geometry determines its directional sound wave field and distinguish hydrodynamic near-field effects from geometric near/far-field approximations.^[ref]

The far-field approximation drops quadratic path-length phase terms only when their phase error is small. This explains why a receiver being many wavelengths away is not always sufficient for a large aperture: source size also enters the criterion.