The term was initially used for a low Mach number approach (Expansion of the acoustic perturbation field about an incompressible flow) as it is described under EIF.
Later in the beginning 1990s the growing CAA community picked up the term and extensively used it for any kind of numerical method describing the noise radiation from an aeroacoustic source or the propagation of sound waves in an inhomogeneous flow field.
With the rapid development of the computational resources this field has undergone spectacular progress during the last three decades.
Both steady state (RANS, SNGR (Stochastic Noise Generation and Radiation), ...) and transient (DNS, LES, DES, URANS, ...) fluid field solutions can be used.
When applying Lighthill's theory [6][7] to the Navier Stokes equations of Fluid mechanics, one obtains volumetric sources, whereas the other two analogies provide the far field information based on a surface integral.
By damping the source gradually to zero at the exit of the domain or adding some additional terms to correct this end-effect, these cut-off errors can be minimized.
To obtain Lighthill's aeroacoustic analogy the governing Navier-Stokes equations are rearranged.
As Lighthill's analogy follows directly from the Navier-Stokes equations without simplification, all sources are present.
The wave operator of Lighthill's analogy is limited to constant flow conditions outside the source zone.
Different mean flow conditions are identified as strong sources with opposite sign by the analogy, once an acoustic wave passes it.
Several modifications to Lighthill's original theory have been proposed to account for the sound-flow interaction or other effects.
The major difficulty with the acoustic analogy, however, is that the sound source is not compact in supersonic flow.
Furthermore, an accurate account of the retarded time-effect requires keeping a long record of the time-history of the converged solutions of the sound source, which again represents a storage problem.
For realistic problems, the required storage can reach the order of 1 terabyte of data.
Kirchhoff and Helmholtz showed, that the radiation of sound from a limited source region can be described by enclosing this source region by a control surface - the so-called Kirchhoff surface.
A modification of the method allows even to calculate the pressure on the surface based on the normal velocity only.
However, the modification to avoid the acoustic pressure on the surface to be known leads to problems, when considering an enclosed volume at its resonant frequencies, which is a major issue of the implementations of their method.
Variations of the average flow field (speed of sound, density and velocity) can be taken into account by a similar method (e.g. dual reciprocity BEM).
The integration method of Ffowcs Williams and Hawkings is based on Lighthill's acoustic analogy.
Different from the Kirchhoff method, these sources follow directly from the Navier-Stokes equations through Lighthill's analogy.
[12] For high Mach number flows in compressible regimes, the acoustic propagation may be influenced by non-linearities and the LEE may no longer be the appropriate mathematical model.
A Fourier pseudospectral time-domain method can be applied to wave propagation problems pertinent to computational aeroacoustics.
The original algorithm of the Fourier pseudo spectral time domain method works for periodical problems without the interaction with physical boundaries.
A slip wall boundary condition, combined with buffer zone technique to solve some non-periodical aeroacoustic problems has been proposed.