Foundations of acoustic analogies

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.