### Abstract

This report presents the best-known acoustic analogies,
and their equations are derived mathematically in detail
to allow their applicability to be extended when
necessary. In the acoustic analogies, the equations
governing the flow-generated acoustic fields are
rearranged in such a way that the field variable
connections (wave operator part) are on the left-hand
side and that which is supposed to form the source
quantities for the acoustic field (source part) is on the
right-hand side. Lighthill's analogy was originally
developed for unbounded flows. The analogy assumes that,
outside the source region, there is no static flow and
the fluid is ideal. The refraction effects are not
included in the wave operator. Powell's analogy is an
approximate version of Lighthill's analogy. The Ffowcs
Williams-Hawkings analogy is such an extension of
Lighthill's analogy that, being based on the same
starting point, it takes into account the effects of
moving boundaries by equivalent Huygens sources. Curle's
analogy is obtained from the Ffowcs Williams-Hawkings
analogy by assuming that the boundaries are not moving.
In Phillips' analogy, the effects of a moving medium are
partially taken into account, and the refraction effects
are included in the wave operator. The fluid outside the
source region is assumed to be ideal. Lilley's analogy is
based on the same starting point as Phillips' analogy,
but all the 'propagation effects' occurring in a
transversely sheared mean flow are inside the wave
operator part of the equation. In Howe's analogy, the
vorticity vector (in the form of Coriolis acceleration)
and the entropy gradients are put in the source part of
the equation, forming the main part of the sources; the
compressibility of the medium is assumed to be constant
and the viscous losses are assumed to vanish. In Doak's
analogy, the compressibility of the medium does not need
to be constant, the vorticity and the entropy gradients
do not need to disappear outside the source region, and
the viscous and thermal losses can be taken into account,
somehow, inside and outside the source region. The four
last-presented analogies assume that the medium is an
ideal gas, so without modifications they cannot be
applied to acoustic fields in liquids.

Original language | English |
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Place of Publication | Espoo |

Publisher | VTT Technical Research Centre of Finland |

Number of pages | 121 |

ISBN (Electronic) | 978-951-38-7725-5 |

ISBN (Print) | 978-951-38-7724-8 |

Publication status | Published - 2011 |

MoE publication type | D4 Published development or research report or study |

### Publication series

Series | VTT Publications |
---|---|

Number | 757 |

ISSN | 1235-0621 |

### Keywords

- noise
- acoustic fields
- source
- flow
- vorticity
- jet
- moving boundaries
- applicability

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## Cite this

Uosukainen, S. (2011).

*Foundations of acoustic analogies*. VTT Technical Research Centre of Finland. VTT Publications, No. 757 http://www.vtt.fi/inf/pdf/publications/2011/P757.pdf