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.
In Equation 2-6,
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
Far-field integral approximation for 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.
The exterior field pressure can be evaluated in a given point (x0,y0,z0) by simply writing
pext(x0,y0,z0). To evaluate the sound pressure level in the same point, it is advantageous to use the
subst() operator and, for example, write
subst(acpr.efc1.Lp_pext,x,x0,y,y0,z,z0).