The Acoustics Module has functionality to evaluate the acoustic pressure field in the far-field region. This section gives the relevant definitions and mathematical background as well as some general advice for analyzing the far field. Details about how to use the far-field functionality is described in Far-Field Calculation and in the modeling section Evaluating the Acoustic Field in the Far-Field Region.

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 (which is related to the normal velocity) is known on a closed surface, the acoustic field can be calculated at any point outside.

In general, the solution p to Helmholtz’ equation

in the homogeneous domain exterior to a closed surface, S, can be explicitly expressed in terms of the values of  p and its normal derivative on S:

Here the coordinate vector r parameterizes S. The unit vector n is the outward normal to the exterior infinite domain; thus, n points into  the domain that S encloses. The function G (R, r) is a Green’s function satisfying

This essentially means that the Green’s function, seen as a function of  r, is an outgoing traveling wave excited by a simple source at R. In 3D, the Green’s function is therefore:

For axially symmetric geometries, the full 3D integral must be evaluated. The Acoustics Module uses an adaptive numerical quadrature in the azimuthal direction on a fictitious revolved geometry in addition to the standard mesh-based quadrature in the rz-plane.

To evaluate the full Helmholtz-Kirchhoff integral in Equation 2-13 and Equation 2-14, use the Full integral option in the settings for the far-field variables. See Far-Field Calculation.

The full Helmholtz-Kirchhoff integral gives the pressure at any point at a finite distance from the source surface, but the numerical integration tends to lose accuracy at large distances. At the same time, in many 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.

Taking the limit of Equation 2-13 when | R | goes to infinity and ignoring the rapidly oscillating phase factor, the far field, pfar is defined as

For axially symmetric geometries, the azimuthal integral of the limiting 3D case can be handled analytically, which leads to a rather complicated expression but avoids the numerical quadrature required in the general case. For zero circumferential mode number m = 0, the expression is:

In this integral, r and z are the radial and axial components of r, while R and Z are the radial and axial components of R.

To evaluate the pressure in the far-field limit according to the equations in this section, use the Integral approximation at r →   ∞ option in the Settings window for the far-field variables See Far-Field Calculation.