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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
116:. 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.
1847:
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298:{\displaystyle {\begin{aligned}{\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)&=0,\\{\frac {\partial }{\partial t}}\left(\rho \mathbf {v} \right)+\nabla \cdot (\rho \mathbf {v} \mathbf {v} )&=-\nabla p+\nabla \cdot {\boldsymbol {\tau }},\end{aligned}}}
887:
644:
2063:{\displaystyle {\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}p}{\partial t^{2}}}-\nabla ^{2}p=\rho _{0}{\frac {\partial ^{2}{\hat {T}}_{ij}}{\partial x_{i}\partial x_{j}}},\quad {\text{where}}\quad {\hat {T}}_{ij}=v_{i}v_{j}.}
1771:{\displaystyle {\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}p}{\partial t^{2}}}-\nabla ^{2}p={\frac {\partial ^{2}{\tilde {T}}_{ij}}{\partial x_{i}\partial x_{j}}},\quad {\text{where}}\quad {\tilde {T}}_{ij}=\rho v_{i}v_{j}.}
553:
1127:{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho ={\frac {\partial ^{2}T_{ij}}{\partial x_{i}\partial x_{j}}},\quad T_{ij}=\rho v_{i}v_{j}+(p-c_{0}^{2}\rho )\delta _{ij}-\tau _{ij}.}
807:{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho =\nabla \nabla :\mathbf {T} ,\quad \mathbf {T} =\rho \mathbf {v} \mathbf {v} +(p-c_{0}^{2}\rho )\mathbf {I} -{\boldsymbol {\tau }},}
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The modern discipline of aeroacoustics can be said to have originated with the first publication of
Lighthill in the early 1950s, when noise generation associated with the
45:
Although no complete scientific theory of the generation of noise by aerodynamic flows has been established, most practical aeroacoustic analysis relies upon the so-called
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are the (characteristic) density and pressure of the fluid in its equilibrium state. Then, upon substitution the assumed relation between pressure and density into
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on the fluid as it effects are small in turbulent noise generation problems such as the jet noise. Lighthill provides an in-depth discussion of this matter.
38:
forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. A notable example of this phenomenon is the
437:
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of "classical" (i.e. linear) acoustics in the left-hand side with the remaining terms as sources in the right-hand side.
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However, even after the above deliberations, it is still not clear whether one is justified in using an inherently
1167:, may play a significant role in the generation of noise depending upon flow conditions considered. The first term
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in the medium in its equilibrium (or quiescent) state, from both sides of the last equation results in celebrated
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If one is to allow for approximations to be made, a simpler way (without necessarily assuming the fluid is
373:
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derive an aeroacoustic equation analogous to
Lighthill's (i.e., an equation for sound generated by "
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59:. whereby the governing equations of motion of the fluid are coerced into a form reminiscent of the
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2221:. The answer is affirmative, if the flow satisfies certain basic assumptions. In particular, if
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as the textbooks on the subject show: e.g., Naugolnykh and
Ostrovsky and Hamilton and Morfey.
2646:
2558:, Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1.
2521:
M. J. Lighthill, "On Sound
Generated Aerodynamically. II. Turbulence as a Source of Sound,"
2446:
M. J. Lighthill, "On Sound
Generated Aerodynamically. II. Turbulence as a Source of Sound,"
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16:
2482:, Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1.
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of the fluid. Furthermore, unlike
Lighthill's equation, Landau and Lifshitz's equation is
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1841:) everywhere, then we obtain exactly the equation given in Landau and Lifshitz, namely
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427:). Differentiating the conservation of mass equation with respect to time, taking the
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in the sense that no approximations of any kind have been made in its derivation.
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548:{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}=\nabla \cdot \left.}
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2ed., Course of
Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75.
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of the last equation and subtracting the latter from the former, we arrive at
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2469:
2ed., Course of
Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75.
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describes non-linear acoustic generation processes and finally the last term
874:
105:
60:
39:
2577:, eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3.
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92:
2507:
M. J. Lighthill, "On Sound
Generated Aerodynamically. I. General Theory,"
2495:, eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3.
2427:
M. J. Lighthill, "On Sound Generated Aerodynamically. I. General Theory,"
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describes inertial effect of the flow (or Reynolds' Stress, developed by
35:
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1402:) to obtain an approximation to Lighthill's equation is to assume that
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Of course, one might wonder whether we are justified in assuming that
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Williams, J. E. Ffowcs, "The Acoustic Analogy—Thirty Years On"
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fluid. The inhomogeneous wave equation that they obtain is for the
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corresponds to sound generation/attenuation due to viscous forces.
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And for the case when the fluid is indeed incompressible, i.e.
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wave equation. Nevertheless, it is a very common practice in
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In practice, it is customary to neglect the effects of
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The continuity and the momentum equations are given by
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we obtain the equation (for an inviscid fluid, Ď = 0)
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was beginning to be placed under scientific scrutiny.
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M. F. Hamilton and C. L. Morfey, "Model Equations,"
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M. F. Hamilton and C. L. Morfey, "Model Equations,"
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A similar approximation [in the context of equation
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2214:{\displaystyle p-p_{0}=c_{0}^{2}(\rho -\rho _{0})}
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1137:Each of the acoustic source terms, i.e. terms in
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2593:Aeroacoustics at the University of Mississippi
1264:{\displaystyle (p-c_{0}^{2}\rho )\delta _{ij}}
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2299:). In fact, the approximate relation between
42:produced by wind blowing over fixed objects.
2138:{\displaystyle T\approx \rho _{0}{\hat {T}}}
873:. The Lighthill equation is an inhomogenous
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2598:Aeroacoustics at the University of Leuven
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2089:
1545:
881:, Lighthillâs equation can be written as
596:{\displaystyle c_{0}^{2}\nabla ^{2}\rho }
30:that studies noise generation via either
412:{\displaystyle \mathbf {v} \mathbf {v} }
392:is the viscous stress tensor. Note that
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2603:International Journal of Aeroacoustics
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2291:theory of sound waves (see, e.g., the
108:, thereby making a connection between
2556:Nonlinear Wave Processes in Acoustics
2480:Nonlinear Wave Processes in Acoustics
1320:LandauâLifshitz aeroacoustic equation
385:{\displaystyle {\boldsymbol {\tau }}}
2615:Examples in Aeroacoustics from NASA
2456:
13:
2247:{\displaystyle \rho \ll \rho _{0}}
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869:Lighthill turbulence stress tensor
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2535:L. D. Landau and E. M. Lifshitz,
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2465:L. D. Landau and E. M. Lifshitz,
2554:K. Naugolnykh and L. Ostrovsky,
2478:K. Naugolnykh and L. Ostrovsky,
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1807:{\displaystyle \rho =\rho _{0}}
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1395:exact; it is an approximation.
1200:{\displaystyle \rho v_{i}v_{j}}
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833:{\displaystyle \nabla \nabla }
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1340:" fluid motion), but for the
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859:{\displaystyle \mathbf {T} }
343:{\displaystyle \mathbf {v} }
7:
2385:Computational aeroacoustics
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1324:In their classical text on
10:
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2514:(1952) pp. 564â587.
2339:that we assumed is just a
2293:linearized Euler equations
2280:{\displaystyle p\ll p_{0}}
1294:{\displaystyle \tau _{ij}}
1211:) whereas the second term
370:is the fluid pressure and
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55:in the 1950s while at the
1834:{\displaystyle \rho _{0}}
1495:{\displaystyle \rho _{0}}
83:Lighthill rearranged the
57:University of Manchester
2357:relation to simplify a
350:is the velocity field,
85:NavierâStokes equations
2528:(1954) pp. 1â32.
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2297:acoustic wave equation
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1160:{\displaystyle T_{ij}}
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871:for the acoustic field
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2530:This article on JSTOR
2523:Proc. R. Soc. Lond. A
2516:This article on JSTOR
2509:Proc. R. Soc. Lond. A
2448:Proc. R. Soc. Lond. A
2429:Proc. R. Soc. Lond. A
2334:
2332:{\displaystyle \rho }
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2093:{\displaystyle (*)\,}
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1809:
1773:
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1549:{\displaystyle (*)\,}
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1522:{\displaystyle p_{0}}
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1384:{\displaystyle \rho }
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623:{\displaystyle c_{0}}
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321:{\displaystyle \rho }
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19:
2434:(1952) pp. 564-587.
2409:(1984) pp. 113-124.
2341:linear approximation
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79:Lighthill's equation
48:aeroacoustic analogy
2587:Preview from Google
2575:Nonlinear Acoustics
2568:Preview from Google
2549:Preview from Amazon
2493:Nonlinear Acoustics
2363:nonlinear acoustics
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1342:incompressible flow
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2627:Aeroacoustics.info
2620:2016-03-04 at the
2608:2005-10-30 at the
2404:IMA J. Appl. Math.
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51:, proposed by Sir
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363:{\displaystyle p}
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2453:(1954) pp. 1-32.
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2622:Wayback Machine
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110:fluid mechanics
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53:James Lighthill
26:is a branch of
12:
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2652:Fluid dynamics
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2502:External links
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1400:incompressible
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2583:0-12-321860-8
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2564:0-521-39984-X
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938:
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875:wave equation
872:
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843:
801:
793:
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742:
739:
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674:
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611:
590:
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520:
514:
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492:
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357:
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107:
106:wave equation
104:
103:inhomogeneous
100:
97:
94:
90:
86:
76:
74:
64:
62:
61:wave equation
58:
54:
50:
49:
43:
41:
40:Aeolian tones
37:
33:
29:
25:
24:Aeroacoustics
18:
2647:Aerodynamics
2574:
2555:
2536:
2525:
2522:
2511:
2508:
2492:
2487:
2479:
2474:
2466:
2450:
2447:
2431:
2428:
2406:
2403:
2398:
2380:Aeolian harp
2358:
2354:
2352:
2288:
2147:
2072:
1780:
1397:
1392:
1349:
1323:
1313:
1310:
1303:
1136:
867:
816:
558:Subtracting
557:
307:
118:
93:compressible
82:
70:
46:
44:
23:
22:
36:aerodynamic
2636:Categories
2391:References
2345:barotropic
2100:], namely
429:divergence
423:(see also
101:, into an
73:jet engine
2642:Acoustics
2359:nonlinear
2327:ρ
2265:≪
2236:ρ
2232:≪
2229:ρ
2200:ρ
2196:−
2193:ρ
2159:−
2130:^
2115:ρ
2111:≈
2084:∗
2020:^
1987:∂
1974:∂
1957:^
1941:∂
1928:ρ
1912:∇
1908:−
1892:∂
1878:∂
1823:ρ
1796:ρ
1789:ρ
1743:ρ
1725:~
1692:∂
1679:∂
1662:~
1646:∂
1627:∇
1623:−
1607:∂
1593:∂
1540:∗
1484:ρ
1454:ρ
1450:−
1447:ρ
1413:−
1379:ρ
1338:turbulent
1306:viscosity
1280:τ
1250:δ
1243:ρ
1225:−
1175:ρ
1110:τ
1106:−
1094:δ
1087:ρ
1069:−
1037:ρ
1001:∂
988:∂
964:∂
954:ρ
945:∇
926:−
910:∂
905:ρ
896:∂
828:∇
825:∇
798:τ
794:−
783:ρ
765:−
743:ρ
720:∇
717:∇
711:ρ
702:∇
683:−
667:∂
662:ρ
653:∂
591:ρ
582:∇
534:τ
530:⋅
527:∇
524:−
518:∇
499:ρ
493:⋅
490:∇
482:⋅
479:∇
460:∂
455:ρ
446:∂
379:τ
316:ρ
285:τ
281:⋅
278:∇
269:∇
266:−
243:ρ
237:⋅
234:∇
218:ρ
204:∂
200:∂
168:ρ
160:⋅
157:∇
145:∂
140:ρ
137:∂
114:acoustics
32:turbulent
28:acoustics
2618:Archived
2606:Archived
2369:See also
2295:and the
1475:, where
1350:pressure
1346:inviscid
1334:Lifshitz
877:. Using
603:, where
842:Hessian
840:is the
630:is the
96:viscous
67:History
2581:
2562:
2543:
2355:linear
2289:linear
1344:of an
1330:Landau
817:where
421:tensor
308:where
2657:Sound
2008:where
1713:where
1314:exact
419:is a
99:fluid
91:of a
2579:ISBN
2560:ISBN
2541:ISBN
2319:and
2254:and
1502:and
1332:and
844:and
112:and
89:flow
2526:222
2512:211
2451:222
2432:211
1393:not
1312:is
2638::
2585:,
2566:,
2547:,
2458:^
2439:^
2414:^
2407:32
1328:,
2589:.
2570:.
2551:.
2532:.
2518:.
2307:p
2273:0
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2204:0
2190:(
2185:2
2180:0
2176:c
2172:=
2167:0
2163:p
2156:p
2127:T
2119:0
2108:T
2087:)
2081:(
2058:.
2053:j
2049:v
2043:i
2039:v
2035:=
2030:j
2027:i
2017:T
2003:,
1995:j
1991:x
1982:i
1978:x
1967:j
1964:i
1954:T
1945:2
1932:0
1924:=
1921:p
1916:2
1900:2
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1858:c
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1827:0
1800:0
1792:=
1766:.
1761:j
1757:v
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1740:=
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1722:T
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1700:j
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1636:p
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1537:(
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1426:=
1421:0
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1219:(
1193:j
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1063:(
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1034:=
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1022:T
1017:,
1009:j
1005:x
996:i
992:x
981:j
978:i
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957:=
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939:2
934:0
930:c
918:2
914:t
900:2
853:T
802:,
790:I
786:)
778:2
773:0
769:c
762:p
759:(
756:+
752:v
747:v
740:=
736:T
731:,
727:T
723::
714:=
706:2
696:2
691:0
687:c
675:2
671:t
657:2
616:0
612:c
586:2
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567:c
543:.
539:]
521:p
515:+
512:)
508:v
503:v
496:(
486:[
476:=
468:2
464:t
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272:p
263:=
256:)
252:v
247:v
240:(
231:+
227:)
222:v
214:(
207:t
191:,
188:0
185:=
177:)
172:v
164:(
154:+
148:t
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