The Acoustics Module has functionality to evaluate the acoustic pressure field in the exterior or far-field region outside of the computational domain. This is the Exterior Field Calculation feature available for pressure acoustics problems. This section first gives a definition of the far- and near-field and then some general advice for analyzing the exterior field.

The solution domain for a scattering or radiation problem can be divided into two zones, reflecting the behavior of the solution at various distances from objects and sources. In the far-field region, scattered or emitted waves are locally planar, velocity and pressure are in phase with each other, and the ratio between pressure and velocity approaches the free-space characteristic specific impedance of a plane wave rc.

Moving closer to the sources into the near-field region, pressure and velocity gradually slide out of phase. This means that the acoustic field contains energy that does not travel outward or radiate. These evanescent wave components are effectively trapped close to the source. Looking at the sound pressure level, local maxima and minima are apparent in the near-field region.

Naturally, the boundary between the near-field and far-field regions is not sharp. A general guideline is that the far-field region is that beyond the last local energy maximum, that is, the region where the pressure amplitude drops monotonously at a rate inversely proportional to the distance R from any source or object.

A similar definition of the far-field region is the region where the radiation pattern — the locations of local minima and maxima in space — is independent of the distance to the wave source. This is equivalent to the criterion for Fraunhofer diffraction in optics, which occurs for Fresnel numbers, F = a2/λ R, much smaller than 1. For engineering purposes, this definition of the far-field region can be applied:

In Equation 2-5, a is the radius of a sphere enclosing all objects and sources, λ is the wavelength, and k is the wave number. Another way to write the expression leads to the useful observation that the size of the near-field region expressed in source-radius units is proportional to the dimensionless number k a, with a prefactor slightly larger than one. This relation is known as the Rayleigh radius R0 = S/λ, where S is the source area, for example for a piston.

In many cases, solving the acoustic Helmholtz equation everywhere in the domain where results are requested is neither practical nor necessary. For homogeneous media, the solution anywhere outside a closed surface containing all sources and scatterers can be written as a boundary integral in terms of quantities evaluated on the surface. To evaluate this Helmholtz-Kirchhoff   integral, it is necessary to know both Dirichlet and Neumann values on the surface. Applied to acoustics, this means that if the pressure and its normal derivative is known on a closed surface, the acoustic field can be calculated at any point outside, including amplitude and phase. This functionality is included in the Exterior Field Calculation feature. The feature has two options for the evaluation, one full integral (the default) and one that only looks in the extreme far field. See the section Theory for the Exterior Field Calculation: The Helmholtz-Kirchhoff Integral for further details.

To evaluate the full Helmholtz-Kirchhoff integral, use the default Full integral option in the Type of integral options. The full Helmholtz-Kirchhoff integral gives the pressure (including phase) at any point at a finite distance from the source surface. This is necessary in many acoustic applications, for example, when analyzing the sensitivity of transducers. Note that numerical integration tends to lose accuracy at very large distances. See Exterior Field Calculation.

In many scattering applications, the quantity of interest is the far-field radiation pattern, which can be defined as the limit of r | p | when r goes to infinity in a given direction. To evaluate the pressure in the far-field limit set the Type of integral to Integral approximation at r → ∞ in the settings for the exterior field variables. With this option, only the direction of the evaluation point r is of interest not the distance |r|. See Exterior Field Calculation.