Aeroacoustics is a branch ofacoustics that studies noise generation via eitherturbulent fluid motion oraerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. A notable example of this phenomenon is theAeolian tones produced by wind blowing over fixed objects.

Although no complete scientific theory of the generation of noise by aerodynamic flows has been established, most practical aeroacoustic analysis relies upon the so-calledaeroacoustic analogy,^[1] proposed by SirJames Lighthill in the 1950s while at theUniversity of Manchester.^[2][3] whereby the governing equations of motion of the fluid are coerced into a form reminiscent of thewave equation of "classical" (i.e. linear) acoustics in the left-hand side with the remaining terms as sources in the right-hand side.

The modern discipline of aeroacoustics can be said to have originated with the first publication of Lighthill^[2][3] in the early 1950s, when noise generation associated with thejet engine was beginning to be placed under scientific scrutiny.

Lighthill^[2] rearranged theNavier–Stokes equations, which govern theflow of acompressibleviscousfluid, into aninhomogeneouswave equation, thereby making a connection betweenfluid mechanics andacoustics. This is often called "Lighthill's analogy" because it presents a model for the acoustic field that is not, strictly speaking, based on the physics of flow-induced/generated noise, but rather on the analogy of how they might be represented through the governing equations of a compressible fluid.

The continuity and the momentum equations are given by

where${\displaystyle \rho }$ is the fluid density,${\displaystyle \mathbf{v}}$ is the velocity field,${\displaystyle p}$ is the fluid pressure and${\displaystyle \mathit{\tau }}$ is the viscous stress tensor. Note that${\displaystyle \mathbf{v}\mathbf{v}}$ is atensor (see alsotensor product). Differentiating the conservation of mass equation with respect to time, taking thedivergence of the last equation and subtracting the latter from the former, we arrive at

${\displaystyle \frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{t}^2}=\mathrm{\nabla }\cdot \left[\mathrm{\nabla }\cdot (\rho \mathbf{v}\mathbf{v})+\mathrm{\nabla }p-\mathrm{\nabla }\cdot \mathit{\tau }\right].}$

Subtracting${\displaystyle c_0^2{\mathrm{\nabla }}^2\rho }$, where${\displaystyle c_0}$ is thespeed of sound in the medium in its equilibrium (or quiescent) state, from both sides of the last equation results in celebratedLighthill equation of aeroacoustics,

${\displaystyle \frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{t}^2}-c_0^2{\mathrm{\nabla }}^2\rho =\mathrm{\nabla }\mathrm{\nabla }:\mathbf{T},\mathbf{T}=\rho \mathbf{v}\mathbf{v}+(p-c_0^2\rho )\mathbf{I}-\mathit{\tau },}$

where${\displaystyle \mathrm{\nabla }\mathrm{\nabla }}$ is theHessian and${\displaystyle \mathbf{T}}$ is the so-calledLighthill turbulence stress tensor for the acoustic field. The Lighthill equation is an inhomogenouswave equation. UsingEinstein notation, Lighthill’s equation can be written as

${\displaystyle \frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{t}^2}-c_0^2{\mathrm{\nabla }}^2\rho =\frac{{\mathrm{\partial }}^2{T}_{ij}}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j},T_{ij}=\rho v_i{v}_j+(p-c_0^2\rho ){\delta }_{ij}-{\tau }_{ij}.}$

Each of the acoustic source terms, i.e. terms in${\displaystyle T_{ij}}$, may play a significant role in the generation of noise depending upon flow conditions considered. The first term${\displaystyle \rho v_i{v}_j}$ describes inertial effect of the flow (or Reynolds' Stress, developed byOsborne Reynolds) whereas the second term${\displaystyle (p-c_0^2\rho ){\delta }_{ij}}$ describes non-linear acoustic generation processes and finally the last term${\displaystyle {\tau }_{ij}}$ corresponds to sound generation/attenuation due to viscous forces.

In practice, it is customary to neglect the effects ofviscosity on the fluid as it effects are small in turbulent noise generation problems such as the jet noise. Lighthill^[2] provides an in-depth discussion of this matter.

In aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation in order to make statements regarding the relevant aerodynamic noise generation mechanisms present. Finally, it is important to realize that Lighthill's equation isexact in the sense that no approximations of any kind have been made in its derivation.

In their classical text onfluid mechanics,Landau andLifshitz^[4] derive an aeroacoustic equation analogous to Lighthill's (i.e., an equation for sound generated by "turbulent" fluid motion), but for theincompressible flow of aninviscid fluid. The inhomogeneous wave equation that they obtain is for thepressure${\displaystyle p}$ rather than for the density${\displaystyle \rho }$ of the fluid. Furthermore, unlike Lighthill's equation, Landau and Lifshitz's equation isnot exact; it is an approximation.

If one is to allow for approximations to be made, a simpler way (without necessarily assuming the fluid isincompressible) to obtain an approximation to Lighthill's equation is to assume that${\displaystyle p-p_0=c_0^2(\rho -{\rho }_0)}$, where${\displaystyle {\rho }_0}$ and${\displaystyle p_0}$ are the (characteristic) density and pressure of the fluid in its equilibrium state. Then, upon substitution the assumed relation between pressure and density into${\displaystyle (\ast )}$ we obtain the equation (for an inviscid fluid, σ = 0)

${\displaystyle \frac{1}{c_0^2}\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^{2}p=\frac{{\mathrm{\partial }}^2{\tilde{T}}_{ij}}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j},\text{where}{\tilde{T}}_{ij}=\rho v_i{v}_j.}$

And for the case when the fluid is indeed incompressible, i.e.${\displaystyle \rho ={\rho }_0}$ (for some positive constant${\displaystyle {\rho }_0}$) everywhere, then we obtain exactly the equation given in Landau and Lifshitz,^[4] namely

${\displaystyle \frac{1}{c_0^2}\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^{2}p={\rho }_0\frac{{\mathrm{\partial }}^2{\hat{T}}_{ij}}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j},\text{where}{\hat{T}}_{ij}=v_i{v}_j.}$

A similar approximation [in the context of equation${\displaystyle (\ast )}$], namely${\displaystyle T\approx {\rho }_0\hat{T}}$, is suggested by Lighthill^[2] [see Eq. (7) in the latter paper].

Of course, one might wonder whether we are justified in assuming that${\displaystyle p-p_0=c_0^2(\rho -{\rho }_0)}$. The answer is affirmative, if the flow satisfies certain basic assumptions. In particular, if${\displaystyle \rho \ll {\rho }_0}$ and${\displaystyle p\ll p_0}$, then the assumed relation follows directly from thelinear theory of sound waves (see, e.g., thelinearized Euler equations and theacoustic wave equation). In fact, the approximate relation between${\displaystyle p}$ and${\displaystyle \rho }$ that we assumed is just alinear approximation to the genericbarotropicequation of state of the fluid.

However, even after the above deliberations, it is still not clear whether one is justified in using an inherentlylinear relation to simplify anonlinear wave equation. Nevertheless, it is a very common practice innonlinear acoustics as the textbooks on the subject show: e.g., Naugolnykh and Ostrovsky^[5] and Hamilton and Morfey.^[6]

• Acoustic theory
• Aeolian harp
• Computational aeroacoustics

• M. J. Lighthill, "On Sound Generated Aerodynamically. I. General Theory,"Proc. R. Soc. Lond. A211 (1952) pp. 564–587.This article on JSTOR.
• M. J. Lighthill, "On Sound Generated Aerodynamically. II. Turbulence as a Source of Sound,"Proc. R. Soc. Lond. A222 (1954) pp. 1–32.This article on JSTOR.
• L. D. Landau and E. M. Lifshitz,Fluid Mechanics 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75.ISBN 0-7506-2767-0,Preview from Amazon.
• K. Naugolnykh and L. Ostrovsky,Nonlinear Wave Processes in Acoustics, Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1.ISBN 0-521-39984-X,Preview from Google.
• M. F. Hamilton and C. L. Morfey, "Model Equations,"Nonlinear Acoustics, eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3.ISBN 0-12-321860-8,Preview from Google.
• Aeroacoustics at the University of Mississippi
• Aeroacoustics at the University of Leuven
• International Journal of AeroacousticsArchived 2005-10-30 at theWayback Machine
• Examples in Aeroacoustics from NASAArchived 2016-03-04 at theWayback Machine
• Aeroacoustics.info

• Acoustics
• Aerodynamics
• Fluid dynamics
• Sound

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