Aeroacústica - Wikipedia, la enciclopedia libre

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Disponible en 11 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-11" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">11 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%88%D8%AA%D9%8A%D8%A7%D8%AA_%D9%87%D9%88%D8%A7%D8%A6%D9%8A%D8%A9" title="صوتيات هوائية (árabe)" lang="ar" hreflang="ar" data-title="صوتيات هوائية" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Aeroakustik" title="Aeroakustik (alemán)" lang="de" hreflang="de" data-title="Aeroakustik" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%B5%CF%81%CE%BF%CE%B1%CE%BA%CE%BF%CF%85%CF%83%CF%84%CE%B9%CE%BA%CE%AE" title="Αεροακουστική (griego)" lang="el" hreflang="el" data-title="Αεροακουστική" data-language-autonym="Ελληνικά" data-language-local-name="griego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Aeroacoustics" title="Aeroacoustics (inglés)" lang="en" hreflang="en" data-title="Aeroacoustics" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Aeroakustika" title="Aeroakustika (estonio)" lang="et" hreflang="et" data-title="Aeroakustika" data-language-autonym="Eesti" data-language-local-name="estonio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%88%D8%A7%D8%B5%D9%88%D8%AA%E2%80%8C%D8%B4%D9%86%D8%A7%D8%B3%DB%8C" title="هواصوت‌شناسی (persa)" lang="fa" hreflang="fa" data-title="هواصوت‌شناسی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/A%C3%A9roacoustique" title="Aéroacoustique (francés)" lang="fr" hreflang="fr" data-title="Aéroacoustique" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Aeroacustica" title="Aeroacustica (italiano)" lang="it" hreflang="it" data-title="Aeroacustica" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Aeroakustyka" title="Aeroakustyka (polaco)" lang="pl" hreflang="pl" data-title="Aeroakustyka" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Aeroac%C3%BAstica" title="Aeroacústica (portugués)" lang="pt" hreflang="pt" data-title="Aeroacústica" data-language-autonym="Português" data-language-local-name="portugués" 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class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><p><b>Aeroacústica</b> es la rama de la <a href="/wiki/Ac%C3%BAstica" title="Acústica">acústica</a> y <a href="/wiki/Mec%C3%A1nica_de_fluidos" title="Mecánica de fluidos">mecánica de fluidos</a> que estudia la generación de ruido debida a flujos <a href="/wiki/Turbulencia" title="Turbulencia">turbulentos</a> o fuerzas aerodinámicas interactuando con fronteras sólidas. La generación de ruido también es asociada con fluctuaciones de <a href="/wiki/Vorticidad" title="Vorticidad">vorticidad</a>. Un ejemplo de este fenómeno son las <a href="/wiki/Calle_de_v%C3%B3rtices_de_von_K%C3%A1rm%C3%A1n" class="mw-redirect" title="Calle de vórtices de von Kármán">resonancias eólicas</a> producidas por el viento cuando fluye alrededor de cuerpos sumergidos en este, por ejemplo <a href="/wiki/Calle_de_v%C3%B3rtices_de_von_K%C3%A1rm%C3%A1n" class="mw-redirect" title="Calle de vórtices de von Kármán">la calle de vórtices de von Kármán</a>. </p><p>A pesar de que no ha sido establecida una teoría científica completa para la generación de ruido por medio de flujos turbulentos, la mayoría de los análisis aeroacústicos están basados en las llamadas <i><a href="/w/index.php?title=Analog%C3%ADa_ac%C3%BAstica&action=edit&redlink=1" class="new" title="Analogía acústica (aún no redactado)">analogías acústicas</a></i>,<sup id="cite_ref-Williams84_1-0" class="reference separada"><a href="#cite_note-Williams84-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> propuestas por <a href="/wiki/James_Lighthill" title="James Lighthill">James Lighthill</a> en los años 1950s durante su estadía en la <a href="/wiki/Universidad_de_M%C3%A1nchester" title="Universidad de Mánchester">Universidad de Mánchester</a>.<sup id="cite_ref-Lighthill52_2-0" class="reference separada"><a href="#cite_note-Lighthill52-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</span></a></sup><sup id="cite_ref-Lighthill54_3-0" class="reference separada"><a href="#cite_note-Lighthill54-3"><span class="corchete-llamada">[</span>3<span class="corchete-llamada">]</span></a></sup> En las <i><a href="/w/index.php?title=Analog%C3%ADa_ac%C3%BAstica&action=edit&redlink=1" class="new" title="Analogía acústica (aún no redactado)">analogías acústicas</a></i>, las ecuaciones que gobiernan el movimiento de fluidos son reagrupadas de tal manera que forman una <a href="/wiki/Ecuaci%C3%B3n_de_onda" title="Ecuación de onda">ecuación de onda</a> en la parte derecha de la ecuación con términos remanentes en la parte izquierda, siendo estos últimos las fuentes de ruido aerodinámico. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aeroac%C3%BAstica&action=edit&section=1" title="Editar sección: Historia"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El desarrollo de la moderna ciencia de la <b>aeroacústica</b> tuvo como motivación el incremento de la emisiones de ruido en aeropuertos, debido a tecnologías emergentes de <a href="/wiki/Avi%C3%B3n_de_reacci%C3%B3n" title="Avión de reacción">aviones a reacción</a> comerciales en la década de 1950. Así pues, el nacimiento de la <b>aeroacústica</b> tuvo lugar con la publicación de los artículos "<i>On sound generated aerodynamically I & II </i>" por James Lighthill<sup id="cite_ref-Lighthill52_2-1" class="reference separada"><a href="#cite_note-Lighthill52-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</span></a></sup><sup id="cite_ref-Lighthill54_3-1" class="reference separada"><a href="#cite_note-Lighthill54-3"><span class="corchete-llamada">[</span>3<span class="corchete-llamada">]</span></a></sup> en los inicios de la década de 1950. Estos artículos son considerados como los referentes de esta ciencia. </p> <div class="mw-heading mw-heading2"><h2 id="Ecuación_de_Lighthill"><span id="Ecuaci.C3.B3n_de_Lighthill"></span>Ecuación de Lighthill</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aeroac%C3%BAstica&action=edit&section=2" title="Editar sección: Ecuación de Lighthill"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Las <a href="/wiki/Ecuaciones_de_Navier-Stokes" title="Ecuaciones de Navier-Stokes">ecuaciones de Navier–Stokes</a>, las cuales gobiernan la <a href="/wiki/Din%C3%A1mica_de_fluidos" class="mw-redirect" title="Dinámica de fluidos">dinámica de fluidos</a> <a href="/wiki/Viscosidad" title="Viscosidad">viscosos</a> compresibles, fueron reagrupadas por Lighthill<sup id="cite_ref-Lighthill52_2-2" class="reference separada"><a href="#cite_note-Lighthill52-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</span></a></sup> en una <a href="/wiki/Ecuaci%C3%B3n_de_onda" title="Ecuación de onda">ecuación de onda</a> no homogénea, así logrando una conexión entre la <a href="/wiki/Mec%C3%A1nica_de_fluidos" title="Mecánica de fluidos">mecánica de fluidos</a> y la <a href="/wiki/Ac%C3%BAstica" title="Acústica">acústica</a>. La expresión resultante es comúnmente llamada "analogía de Lighthill" ya que este modelo representa el campo acústico que no está (de una manera rigurosa) basado en la física del flujo, si no, en una analogía de como este podría representase a través de las ecuaciones fluido-dinámicas para <a href="/wiki/Flujo_compresible" title="Flujo compresible">flujo compresible</a>. </p><p>Considerando la <a href="/wiki/Ecuaci%C3%B3n_de_continuidad" title="Ecuación de continuidad">ecuación de continuidad</a> la cual se expresa como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)={\frac {D\rho }{Dt}}+\rho \nabla \cdot \mathbf {v} =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>ρ</mi> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>ρ</mi> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)={\frac {D\rho }{Dt}}+\rho \nabla \cdot \mathbf {v} =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782373c34fada85b1331bc97cc4f61fc2ec08185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.271ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)={\frac {D\rho }{Dt}}+\rho \nabla \cdot \mathbf {v} =0,}"></span></dd></dl> <p>donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> representan la densidad y velocidad del fluido, con variabilidad espacial y temporal, y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D/Dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>D</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D/Dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55bd3a0737b30e1149ed294a1c36fc09121e1dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.851ex; height:2.843ex;" alt="{\displaystyle D/Dt}"></span> es la <a href="/wiki/Derivada_material" title="Derivada material">derivada material</a>. </p><p>Ahora, considerando la ecuación de conservación de momento escrita como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rho }{\frac {\partial \mathbf {v} }{\partial t}}+{\rho (\mathbf {v} \cdot \nabla )\mathbf {v} }=-\nabla p+\nabla \cdot \sigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>σ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rho }{\frac {\partial \mathbf {v} }{\partial t}}+{\rho (\mathbf {v} \cdot \nabla )\mathbf {v} }=-\nabla p+\nabla \cdot \sigma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f9496d3d91bf2a8850c000401a9d1680e813d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.499ex; height:5.509ex;" alt="{\displaystyle {\rho }{\frac {\partial \mathbf {v} }{\partial t}}+{\rho (\mathbf {v} \cdot \nabla )\mathbf {v} }=-\nabla p+\nabla \cdot \sigma ,}"></span></dd></dl> <p>donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es la <a href="/wiki/Presi%C3%B3n" title="Presión">presión</a> <a href="/wiki/Termodin%C3%A1mica" title="Termodinámica">termodinámica</a>, y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> es el tensor viscoso de las ecuaciones de Navier–Stokes. </p><p>Multiplicando la ecuación de continuidad por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> y sumando la ecuación de conservación de momento, resulta en: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial t}}\left(\rho \mathbf {v} \right)+\nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} )=-\nabla p+\nabla \cdot \sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\left(\rho \mathbf {v} \right)+\nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} )=-\nabla p+\nabla \cdot \sigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3b7b76214143abd052fc63613a87bfe7426cef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.376ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\left(\rho \mathbf {v} \right)+\nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} )=-\nabla p+\nabla \cdot \sigma .}"></span></dd></dl> <p>Notando que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \otimes \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \otimes \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0d40051873b8c7ce4072a75a9a55f34218c828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.662ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} \otimes \mathbf {v} }"></span> es un <a href="/wiki/Tensor" title="Tensor">tensor</a> (véase también <a href="/wiki/Producto_tensorial" title="Producto tensorial">producto tensorial</a>). Tomando la divergencia de la ecuación de momento y sustrayéndole la derivada temporal de la ecuación de continuidad, resulta en: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-\nabla ^{2}p+\nabla \cdot \nabla \cdot \sigma =\nabla \cdot \nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>p</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>σ</mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-\nabla ^{2}p+\nabla \cdot \nabla \cdot \sigma =\nabla \cdot \nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0575fb857ffbf6115d73b8592f7cf294a1e6ac18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.484ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-\nabla ^{2}p+\nabla \cdot \nabla \cdot \sigma =\nabla \cdot \nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} ).}"></span></dd></dl> <p>Sustrayendo a ambos lados <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}^{2}\nabla ^{2}\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}^{2}\nabla ^{2}\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffcf0378bc11ca5729c1b39cdc9c21949e8e01b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.253ex; height:3.343ex;" alt="{\displaystyle c_{0}^{2}\nabla ^{2}\rho }"></span>, donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1882ba8f1dc60f0c68a642abb5af093c73910921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}"></span> es la <a href="/wiki/Velocidad_del_sonido" title="Velocidad del sonido">velocidad del sonido</a> en un medio en reposo, esto resulta en: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho =\nabla \cdot \left[\nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} )-\nabla \cdot \sigma +\nabla p-c_{0}^{2}\nabla \rho \right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow> <mo>[</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>σ</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho =\nabla \cdot \left[\nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} )-\nabla \cdot \sigma +\nabla p-c_{0}^{2}\nabla \rho \right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/471371077cd60e5c6ef74c2df8a482bfb7f3f688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:57.274ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho =\nabla \cdot \left[\nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} )-\nabla \cdot \sigma +\nabla p-c_{0}^{2}\nabla \rho \right],}"></span></dd></dl> <p>siendo equivalente a: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho =(\nabla \otimes \nabla ):\left[\rho \mathbf {v} \otimes \mathbf {v} -\sigma +(p-c_{0}^{2}\rho )\mathbb {I} \right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⊗</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mrow> <mo>[</mo> <mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>−</mo> <mi>σ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>ρ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho =(\nabla \otimes \nabla ):\left[\rho \mathbf {v} \otimes \mathbf {v} -\sigma +(p-c_{0}^{2}\rho )\mathbb {I} \right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612e7b49b9ae21a4ad4361a54229226eaeeb7fdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:53.921ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\rho }{\partial t^{2}}}-c_{0}^{2}\nabla ^{2}\rho =(\nabla \otimes \nabla ):\left[\rho \mathbf {v} \otimes \mathbf {v} -\sigma +(p-c_{0}^{2}\rho )\mathbb {I} \right],}"></span></dd></dl> <p>donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8205f06e0d279689ed04a1ac04a3d9c249c637df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle \mathbb {I} }"></span> es el <a href="/wiki/Matriz_identidad" title="Matriz identidad">tensor</a> identidad, y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"></span> representa el doble operador de contracción tensorial. </p><p>La ecuación precedente es la célebre ecuación de Lighthill, una de las principales en la aeroacústica. Esta se trata de una <a href="/wiki/Ecuaci%C3%B3n_de_onda" title="Ecuación de onda">ecuación de onda</a> con un término fuente en el lado derecho de la ecuación, en otras palabras, una ecuación de onda no homogénea. El término de la "doble contracción tensorial" en el lado derecho de la ecuación, es decir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \mathbf {v} \otimes \mathbf {v} -\sigma +(p-c_{0}^{2}\rho )\mathbb {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>−</mo> <mi>σ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>ρ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \mathbf {v} \otimes \mathbf {v} -\sigma +(p-c_{0}^{2}\rho )\mathbb {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc1c057d09b75b7f48bf0ff2100f8f56c9e997ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.862ex; height:3.176ex;" alt="{\displaystyle \rho \mathbf {v} \otimes \mathbf {v} -\sigma +(p-c_{0}^{2}\rho )\mathbb {I} }"></span>, es conocido como el <i><a href="/w/index.php?title=Tensor_de_Lighthill&action=edit&redlink=1" class="new" title="Tensor de Lighthill (aún no redactado)">tensor de Lighthill</a> asociado al campo acústico</i>, comúnmente denotado por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. </p><p>Mediante el uso de la <a href="/wiki/Convenio_de_suma_de_Einstein" title="Convenio de suma de Einstein">notación de Einstein</a>, la ecuación de Lighthill se puede reescribir como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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 (*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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 (*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42759eb09eefdf97045971e5923bf7c0dd64679e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.797ex; height:6.676ex;" alt="{\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 (*)}"></span></dd></dl> <p>donde </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{ij}=\rho v_{i}v_{j}-\sigma _{ij}+(p-c_{0}^{2}\rho )\delta _{ij},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>ρ</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{ij}=\rho v_{i}v_{j}-\sigma _{ij}+(p-c_{0}^{2}\rho )\delta _{ij},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/382ae8bb82e2dee43541875740fde5253339c71a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.824ex; height:3.176ex;" alt="{\displaystyle T_{ij}=\rho v_{i}v_{j}-\sigma _{ij}+(p-c_{0}^{2}\rho )\delta _{ij},}"></span></dd></dl> <p>y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}"></span> es el <a href="/wiki/Delta_de_Kronecker" title="Delta de Kronecker">delta de Kronecker</a>. Cada uno de los términos fuente en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de5760ded748359e36c7fb067c45f5e9642e890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.835ex; height:2.843ex;" alt="{\displaystyle T_{ij}}"></span> juega un rol importante en la generación de ruido. Por ejemplo, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho v_{i}v_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho v_{i}v_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/909899bb99de4c6ad7fe59ba1a207badaf2ae368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.167ex; height:2.343ex;" alt="{\displaystyle \rho v_{i}v_{j}}"></span> describe el ruido debido a las fluctuaciones de flujo (tensor de Reynolds), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43acbf52cc4d4f83f187ceaa49f045114b71772e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.804ex; height:2.343ex;" alt="{\displaystyle \sigma _{ij}}"></span> es el ruido producido por efectos viscosos y de cizalladura (vorticidad) y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p-c_{0}^{2}\rho )\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>ρ</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p-c_{0}^{2}\rho )\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17db87e3552a0865c11173d21c1111f6f829cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.592ex; height:3.176ex;" alt="{\displaystyle (p-c_{0}^{2}\rho )\delta _{ij}}"></span> describe el ruido generado por procesos acústicos no lineales. </p><p>Generalmente es posible despreciar los efectos de <a href="/wiki/Viscosidad" title="Viscosidad">viscosidad</a> en el fluido, es decir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb4831f1e0ca1ba7d007dc6b973e54787e1a4b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma =0}"></span>, esto se debe a que es comúnmente aceptado que los efectos viscosos a grandes <a href="/w/index.php?title=N%C3%BAmero_de_reynolds&action=edit&redlink=1" class="new" title="Número de reynolds (aún no redactado)">número de reynolds</a> están varios órdenes de magnitud por debajo de los otros términos de la ecuación. En el caso de flujos a bajo <a href="/w/index.php?title=N%C3%BAmero_de_reynolds&action=edit&redlink=1" class="new" title="Número de reynolds (aún no redactado)">número de reynolds</a> (efectos viscosos dominantes), el ruido producido es tan bajo que no entra en el umbral percibido por los humanos. Lighthill<sup id="cite_ref-Lighthill52_2-3" class="reference separada"><a href="#cite_note-Lighthill52-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</span></a></sup> presenta una larga y profunda discusión sobre este tema. </p><p>Finalmente, es importante destacar que la ecuación de Lighthill es <b>exacta</b> en el sentido que, ninguna aproximación fue requerida durante la derivación matemática. </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aeroac%C3%BAstica&action=edit&section=3" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Arpa_e%C3%B3lica" title="Arpa eólica">Arpa eólica</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aeroac%C3%BAstica&action=edit&section=4" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notas">Notas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aeroac%C3%BAstica&action=edit&section=5" title="Editar sección: Notas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-Williams84-1"><span class="mw-cite-backlink"><a href="#cite_ref-Williams84_1-0">↑</a></span> <span class="reference-text">(<a href="#CITAREFWilliams_1984">Williams , 1984</a>, pp. 113–124)</span> </li> <li id="cite_note-Lighthill52-2"><span class="mw-cite-backlink">↑ <a href="#cite_ref-Lighthill52_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lighthill52_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lighthill52_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Lighthill52_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">(<a href="#CITAREF_Lighthill_1952">Lighthill , 1952</a>, pp. 564–587)</span> </li> <li id="cite_note-Lighthill54-3"><span class="mw-cite-backlink">↑ <a href="#cite_ref-Lighthill54_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lighthill54_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">(<a href="#CITAREF_Lighthill_1954">Lighthill , 1954</a>, pp. 1–32)</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aeroac%C3%BAstica&action=edit&section=6" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFWilliams1984" class="citation publicación">Williams, J. E. Ffowcs (1984). «The Acoustic Analogy—Thirty Years On». <i>IMA J. Appl. Math.</i> (32).</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AAeroac%C3%BAstica&rft.atitle=The+Acoustic+Analogy%E2%80%94Thirty+Years+On&rft.au=Williams%2C+J.+E.+Ffowcs&rft.aufirst=J.+E.+Ffowcs&rft.aulast=Williams&rft.date=1984&rft.genre=article&rft.issue=32&rft.jtitle=IMA+J.+Appl.+Math.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFLighthill1952" class="citation publicación">Lighthill, M.J. (1952). «On Sound Generated Aerodynamically. I. General Theory». <i>Proc. R. Soc. Lond. A</i> (211).</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AAeroac%C3%BAstica&rft.atitle=On+Sound+Generated+Aerodynamically.+I.+General+Theory&rft.au=Lighthill%2C+M.J.&rft.aufirst=M.J.&rft.aulast=Lighthill&rft.date=1952&rft.genre=article&rft.issue=211&rft.jtitle=Proc.+R.+Soc.+Lond.+A&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span> <a rel="nofollow" class="external text" href="http://links.jstor.org/sici?sici=0080-4630(19520320)211%3A1107%3C564%3AOSGAIG%3E2.0.CO%3B2-7">Disponible</a> en <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>.</li> <li><span id="CITAREFLighthill1954" class="citation publicación">Lighthill, M.J. (1954). «On Sound Generated Aerodynamically II. Turbulence as a Source of Sound». <i>Proc. R. Soc. Lond. A</i> (222).</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AAeroac%C3%BAstica&rft.atitle=On+Sound+Generated+Aerodynamically+II.+Turbulence+as+a+Source+of+Sound&rft.au=Lighthill%2C+M.J.&rft.aufirst=M.J.&rft.aulast=Lighthill&rft.date=1954&rft.genre=article&rft.issue=222&rft.jtitle=Proc.+R.+Soc.+Lond.+A&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span> <a rel="nofollow" class="external text" href="http://links.jstor.org/sici?sici=0080-4630(19540223)222%3A1148%3C1%3AOSGAIT%3E2.0.CO%3B2-2">Disponible</a> en <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>.</li> <li>L. D. Landau and E. M. Lifshitz, <i>Fluid Mechanics</i> 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75. <a href="/wiki/Especial:FuentesDeLibros/0750627670" class="internal mw-magiclink-isbn">ISBN 0-7506-2767-0</a>, <a rel="nofollow" class="external text" href="https://www.amazon.com/gp/reader/0750627670/">ver en amazon</a>.</li> <li>K. Naugolnykh and L. Ostrovsky, <i>Nonlinear Wave Processes in Acoustics</i>, Cambridge Texts in Applied Mathematics vol. 9, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> (1998) chap. 1. <a href="/wiki/Especial:FuentesDeLibros/052139984X" class="internal mw-magiclink-isbn">ISBN 0-521-39984-X</a>, <a rel="nofollow" class="external text" href="http://books.google.com/books?vid=ISBN052139984X">Pre-visualizar en Google</a>.</li> <li>M. F. Hamilton and C. L. Morfey, "Model Equations," <i>Nonlinear Acoustics</i>, eds. M. F. Hamilton and D. T. Blackstock, <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a> (1998) chap. 3. <a href="/wiki/Especial:FuentesDeLibros/0123218608" class="internal mw-magiclink-isbn">ISBN 0-12-321860-8</a>, <a rel="nofollow" class="external text" href="http://books.google.com/books?vid=ISBN0123218608">Pre-visualizar en Google</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aeroac%C3%BAstica&action=edit&section=7" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.olemiss.edu/depts/ncpa/aeroacoustics/">Aeroacústica en la Universidad de Misisipi</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.mech.kuleuven.be/mod/aeroacoustics/">Aeroacústica en la Universidad de Lovaina</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.multi-science.co.uk/aeroacou.htm">International Journal of Aeroacoustics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051030160132/http://www.multi-science.co.uk/aeroacou.htm">Archivado</a> el 30 de octubre de 2005 en <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>..</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304114401/http://www.grc.nasa.gov/WWW/microbus/cese/aeroex.html">Ejemplos en Aeroacústica de la </a><a href="/wiki/NASA" title="NASA">NASA</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.aeroacoustics.info/">Aeroacoustics.info</a>.</li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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