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id="toc-Mathematical_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Condensation_clouds" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Condensation_clouds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Condensation clouds</span> </div> </a> <ul id="toc-Condensation_clouds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transonic_flows_in_astronomy_and_astrophysics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transonic_flows_in_astronomy_and_astrophysics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Transonic flows in astronomy and astrophysics</span> </div> </a> <ul id="toc-Transonic_flows_in_astronomy_and_astrophysics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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Available in 18 languages" > <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-18" 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">18 languages</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%AA%D8%B1%D8%A7%D9%86%D8%B3%D9%88%D9%86%D9%8A%D9%83" title="ترانسونيك – Arabic" lang="ar" hreflang="ar" data-title="ترانسونيك" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Transonik_s%C3%BCr%C9%99t" title="Transonik sürət – Azerbaijani" lang="az" hreflang="az" data-title="Transonik sürət" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Transsonische_Str%C3%B6mung" title="Transsonische Strömung – German" lang="de" hreflang="de" data-title="Transsonische Strömung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Velocidad_trans%C3%B3nica" title="Velocidad transónica – Spanish" lang="es" hreflang="es" data-title="Velocidad transónica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B1%D8%A7%D8%B5%D9%88%D8%AA%DB%8C" title="تراصوتی – Persian" lang="fa" hreflang="fa" data-title="تراصوتی" data-language-autonym="فارسی" data-language-local-name="Persian" 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/Transsonique" title="Transsonique – French" lang="fr" hreflang="fr" data-title="Transsonique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%9C%EC%9D%8C%EC%86%8D" title="천음속 – Korean" lang="ko" hreflang="ko" data-title="천음속" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Regime_transonico" title="Regime transonico – Italian" lang="it" hreflang="it" data-title="Regime transonico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%91%D7%A8-%D7%A7%D7%95%D7%9C%D7%99" title="עבר-קולי – Hebrew" lang="he" hreflang="he" data-title="עבר-קולי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Transzszonikus_rep%C3%BCl%C3%A9s" title="Transzszonikus repülés – Hungarian" lang="hu" hreflang="hu" data-title="Transzszonikus repülés" data-language-autonym="Magyar" 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data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Transonic" title="Transonic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Transonic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Transsonisk_fart" title="Transsonisk fart – Swedish" lang="sv" hreflang="sv" data-title="Transsonisk fart" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%92%E0%AE%B2%E0%AE%BF%E0%AE%AF%E0%AF%8A%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%B5%E0%AF%87%E0%AE%95%E0%AE%AE%E0%AF%8D" title="ஒலியொத்தவேகம் – Tamil" lang="ta" hreflang="ta" data-title="ஒலியொத்தவேகம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Transonik_h%C4%B1z" title="Transonik hız – Turkish" lang="tr" hreflang="tr" data-title="Transonik hız" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B7%A8%E9%9F%B3%E9%80%9F" title="跨音速 – Chinese" lang="zh" hreflang="zh" data-title="跨音速" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> 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searchaux" style="display:none">Flight condition in which airflow speeds are concurrently above and below the speed of sound</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the American company, see <a href="/wiki/Transonic_Combustion" title="Transonic Combustion">Transonic Combustion</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1034237262">.mw-parser-output .stack{box-sizing:border-box}.mw-parser-output .stack>div{margin:1px;overflow:hidden}@media all and (min-width:720px){.mw-parser-output .stack-clear-left{float:left;clear:left}.mw-parser-output .stack-clear-right{float:right;clear:right}.mw-parser-output .stack-left{float:left}.mw-parser-output .stack-right{float:right}.mw-parser-output .stack-margin-clear-left{float:left;clear:left;margin-right:1em}.mw-parser-output .stack-margin-clear-right{float:right;clear:right;margin-left:1em}.mw-parser-output .stack-margin-left{float:left;margin-right:1em}.mw-parser-output .stack-margin-right{float:right;margin-left:1em}}</style><div class="stack mw-stack stack-right"><div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:FA-18_Hornet_breaking_sound_barrier_(7_July_1999).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/FA-18_Hornet_breaking_sound_barrier_%287_July_1999%29.jpg/220px-FA-18_Hornet_breaking_sound_barrier_%287_July_1999%29.jpg" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/FA-18_Hornet_breaking_sound_barrier_%287_July_1999%29.jpg/330px-FA-18_Hornet_breaking_sound_barrier_%287_July_1999%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/FA-18_Hornet_breaking_sound_barrier_%287_July_1999%29.jpg/440px-FA-18_Hornet_breaking_sound_barrier_%287_July_1999%29.jpg 2x" data-file-width="2100" data-file-height="1500" /></a><figcaption>Aerodynamic condensation evidences of <a href="/wiki/Prandtl%E2%80%93Meyer_expansion_fan" title="Prandtl–Meyer expansion fan">supersonic expansion fans</a> around a transonic <a href="/wiki/F/A-18_Hornet" class="mw-redirect" title="F/A-18 Hornet">F/A-18</a></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sears-Haack.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Sears-Haack.png/220px-Sears-Haack.png" decoding="async" width="220" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Sears-Haack.png/330px-Sears-Haack.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Sears-Haack.png/440px-Sears-Haack.png 2x" data-file-width="1089" data-file-height="716" /></a><figcaption>The <a href="/wiki/Sears%E2%80%93Haack_body" title="Sears–Haack body">Sears–Haack body</a> presents a cross-sectional area variation that minimises <a href="/wiki/Wave_drag" title="Wave drag">wave drag</a>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Shock_wave_above_airliner_wing_(7).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Shock_wave_above_airliner_wing_%287%29.jpg/220px-Shock_wave_above_airliner_wing_%287%29.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Shock_wave_above_airliner_wing_%287%29.jpg/330px-Shock_wave_above_airliner_wing_%287%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Shock_wave_above_airliner_wing_%287%29.jpg/440px-Shock_wave_above_airliner_wing_%287%29.jpg 2x" data-file-width="3648" data-file-height="2432" /></a><figcaption>Shock waves may appear as weak optical disturbances above airliners with <a href="/wiki/Supercritical_airfoil" title="Supercritical airfoil">supercritical wings</a></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Transonic_flow_patterns.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Transonic_flow_patterns.svg/220px-Transonic_flow_patterns.svg.png" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Transonic_flow_patterns.svg/330px-Transonic_flow_patterns.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Transonic_flow_patterns.svg/440px-Transonic_flow_patterns.svg.png 2x" data-file-width="286" data-file-height="234" /></a><figcaption>Transonic flow patterns on an <a href="/wiki/Airfoil" title="Airfoil">airfoil</a> showing flow patterns at and above <a href="/wiki/Critical_Mach_number" title="Critical Mach number">critical Mach number</a></figcaption></figure></div></div> <p><b>Transonic</b> (or <b>transsonic</b>) flow is air flowing around an object at a speed that generates regions of both subsonic and <a href="/wiki/Supersonic_speed" title="Supersonic speed">supersonic</a> airflow around that object.<sup id="cite_ref-:2_1-0" class="reference"><a href="#cite_note-:2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The exact range of speeds depends on the object's <a href="/wiki/Critical_Mach_number" title="Critical Mach number">critical Mach number</a>, but transonic flow is seen at flight speeds close to the <a href="/wiki/Speed_of_sound" title="Speed of sound">speed of sound</a> (343 m/s at sea level), typically between <a href="/wiki/Mach_number" title="Mach number">Mach</a> 0.8 and 1.2.<sup id="cite_ref-:2_1-1" class="reference"><a href="#cite_note-:2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The issue of transonic speed (or transonic region) first appeared during World War II.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Pilots found as they approached the sound barrier the airflow caused aircraft to become unsteady.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Experts found that <a href="/wiki/Shock_wave" title="Shock wave">shock waves</a> can cause large-scale <a href="/wiki/Flow_separation" title="Flow separation">separation</a> downstream, increasing drag, adding asymmetry and unsteadiness to the flow around the vehicle.<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Research has been done into weakening shock waves in transonic flight through the use of <a href="/wiki/Anti-shock_body" title="Anti-shock body">anti-shock bodies</a> and <a href="/wiki/Supercritical_airfoil" title="Supercritical airfoil">supercritical airfoils</a>.<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Most modern <a href="/wiki/Jet_engine" title="Jet engine">jet</a> powered aircraft are engineered to operate at transonic air speeds.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Transonic airspeeds see a rapid increase in drag from about Mach 0.8, and it is the fuel costs of the drag that typically limits the airspeed. Attempts to reduce wave drag can be seen on all high-speed aircraft. Most notable is the use of <a href="/wiki/Swept_wing" title="Swept wing">swept wings</a>, but another common form is a wasp-waist fuselage as a side effect of the <a href="/wiki/Whitcomb_area_rule" class="mw-redirect" title="Whitcomb area rule">Whitcomb area rule</a>. </p><p>Transonic speeds can also occur at the tips of <a href="/wiki/Rotorcraft" title="Rotorcraft">rotor</a> blades of helicopters and aircraft. This puts severe, unequal stresses on the rotor blade and may lead to accidents if it occurs. It is one of the limiting factors of the size of rotors and the forward speeds of helicopters (as this speed is added to the forward-sweeping [leading] side of the rotor, possibly causing localized transonics). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Discovering_transonic_airflow">Discovering transonic airflow</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=2" title="Edit section: Discovering transonic airflow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Issues with aircraft flight relating to speed first appeared during the <a href="/wiki/Supersonic_speed" title="Supersonic speed">supersonic</a> era in 1941.<sup id="cite_ref-:35_5-0" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Ralph Virden, a test pilot, crashed in a fatal plane accident.<sup id="cite_ref-:23_6-0" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> He lost control of the plane when a shock wave caused by supersonic airflow developed over the wing, causing it to stall.<sup id="cite_ref-:23_6-1" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Virden flew well below the speed of sound at Mach 0.675, which brought forth the idea of different airflows forming around the plane.<sup id="cite_ref-:35_5-1" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In the 40s, <a href="/wiki/Kelly_Johnson_(engineer)" title="Kelly Johnson (engineer)">Kelly Johnson</a> became one of the first engineers to investigate the effect of compressibility on aircraft.<sup id="cite_ref-:35_5-2" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, contemporary <a href="/wiki/Wind_tunnel" title="Wind tunnel">wind tunnels</a> did not have the capability to create wind speeds close to Mach 1 to test the effects of transonic speeds.<sup id="cite_ref-:23_6-2" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Not long after, the term "transonic" was defined to mean "across the speed of sound" and was invented by <a href="/wiki/National_Advisory_Committee_for_Aeronautics" title="National Advisory Committee for Aeronautics"> NACA</a> director <a href="/wiki/Hugh_Latimer_Dryden" title="Hugh Latimer Dryden">Hugh Dryden</a> and <a href="/wiki/Theodore_von_K%C3%A1rm%C3%A1n" title="Theodore von Kármán">Theodore von Kármán</a> of the California Institute of Technology.<sup id="cite_ref-:35_5-3" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Changes_in_aircraft">Changes in aircraft</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=3" title="Edit section: Changes in aircraft"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Initially, <a href="/wiki/National_Advisory_Committee_for_Aeronautics" title="National Advisory Committee for Aeronautics">NACA</a> designed "dive flaps" to help stabilize the plane when reaching transonic flight.<sup id="cite_ref-:35_5-4" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This small flap on the underside of the plane slowed the plane to prevent shock waves, but this design only delayed finding a solution to aircraft flying at supersonic speed.<sup id="cite_ref-:35_5-5" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Newer wind tunnels were designed, so researchers could test newer wing designs without risking test pilots' lives.<sup id="cite_ref-:4_7-0" class="reference"><a href="#cite_note-:4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The slotted-wall transonic tunnel was designed by NASA and allowed researchers to test wings and different <a href="/wiki/Airfoil" title="Airfoil">airfoils</a> in transonic airflow to find the best wingtip shape for sonic speeds.<sup id="cite_ref-:4_7-1" class="reference"><a href="#cite_note-:4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>After <a href="/wiki/World_War_II" title="World War II">World War II</a>, major changes in aircraft design were seen to improve transonic flight.<sup id="cite_ref-:23_6-3" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The main way to stabilize an aircraft was to reduce the speed of the airflow around the wings by changing the <a href="/wiki/Chord_(aeronautics)" title="Chord (aeronautics)">chord</a> of the plane wings, and one solution to prevent transonic waves was swept wings.<sup id="cite_ref-:35_5-6" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Since the airflow would hit the wings at an angle, this would decrease the wing thickness and chord ratio.<sup id="cite_ref-:35_5-7" class="reference"><a href="#cite_note-:35-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Airfoils wing shapes were designed flatter at the top to prevent shock waves and reduce the distance of airflow over the wing.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Later on, Richard Whitcomb designed the first <a href="/wiki/Supercritical_airfoil" title="Supercritical airfoil">supercritical airfoil</a> using similar principles.<sup id="cite_ref-:4_7-2" class="reference"><a href="#cite_note-:4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Mathematical_analysis">Mathematical analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=4" title="Edit section: Mathematical analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Streamline_Patterns_for_Flow_Regimes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Streamline_Patterns_for_Flow_Regimes.png/220px-Streamline_Patterns_for_Flow_Regimes.png" decoding="async" width="220" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Streamline_Patterns_for_Flow_Regimes.png/330px-Streamline_Patterns_for_Flow_Regimes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Streamline_Patterns_for_Flow_Regimes.png/440px-Streamline_Patterns_for_Flow_Regimes.png 2x" data-file-width="1202" data-file-height="1362" /></a><figcaption>Streamlines for three airflow regimes (black lines) around a nondescript blunt body (blue).<sup id="cite_ref-:13_9-0" class="reference"><a href="#cite_note-:13-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Prior to the advent of powerful computers, even the simplest forms of the <a href="/wiki/Compressible_flow" title="Compressible flow">compressible flow equations</a> were difficult to solve due to their <a href="/wiki/Nonlinear_system" title="Nonlinear system">nonlinearity</a>.<sup id="cite_ref-:23_6-4" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A common assumption used to circumvent this nonlinearity is that disturbances within the flow are relatively small, which allows mathematicians and engineers to <a href="/wiki/Linearization" title="Linearization">linearize</a> the compressible flow equations into a relatively easily solvable set of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> for either wholly subsonic or supersonic flows.<sup id="cite_ref-:23_6-5" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This assumption is fundamentally untrue for transonic flows because the disturbance caused by an object is much larger than in subsonic or supersonic flows; a flow speed close to or at Mach 1 does not allow the <a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">streamtubes</a> (3D flow paths) to contract enough around the object to minimize the disturbance, and thus the disturbance propagates.<sup id="cite_ref-:13_9-1" class="reference"><a href="#cite_note-:13-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Aerodynamicists struggled during the earlier studies of transonic flow because the then-current theory implied that these disturbances– and thus drag– approached infinity as local Mach number approached 1, an obviously unrealistic result which could not be remedied using known methods.<sup id="cite_ref-:23_6-6" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>One of the first methods used to circumvent the nonlinearity of transonic flow models was the <a href="/wiki/Hodograph" title="Hodograph">hodograph</a> transformation.<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This concept was originally explored in 1923 by an Italian mathematician named <a href="/wiki/Francesco_Tricomi" title="Francesco Tricomi">Francesco Tricomi</a>, who used the transformation to simplify the compressible flow equations and prove that they were solvable.<sup id="cite_ref-:0_2-3" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The hodograph transformation itself was also explored by both <a href="/wiki/Ludwig_Prandtl" title="Ludwig Prandtl">Ludwig Prandtl</a> and O.G. Tietjen's textbooks in 1929 and by <a href="/wiki/Adolf_Busemann" title="Adolf Busemann">Adolf Busemann</a> in 1937, though neither applied this method specifically to transonic flow.<sup id="cite_ref-:0_2-4" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Gottfried Guderley, a German mathematician and engineer at <a href="/wiki/Technical_University_of_Braunschweig" title="Technical University of Braunschweig">Braunschweig</a>, discovered Tricomi's work in the process of applying the hodograph method to transonic flow near the end of World War II.<sup id="cite_ref-:0_2-5" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> He focused on the nonlinear thin-airfoil compressible flow equations, the same as what Tricomi derived, though his goal of using these equations to solve flow over an airfoil presented unique challenges.<sup id="cite_ref-:0_2-6" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:23_6-7" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Guderley and Hideo Yoshihara, along with some input from Busemann, later used a singular solution of Tricomi's equations to analytically solve the behavior of transonic flow over a <a href="/wiki/Supersonic_airfoils" title="Supersonic airfoils">double wedge airfoil</a>, the first to do so with only the assumptions of thin-airfoil theory.<sup id="cite_ref-:0_2-7" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:23_6-8" class="reference"><a href="#cite_note-:23-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Although successful, Guderley's work was still focused on the theoretical, and only resulted in a single solution for a double wedge airfoil at Mach 1.<sup id="cite_ref-:0_2-8" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Walter_G._Vincenti" title="Walter G. Vincenti">Walter Vincenti</a>, an American engineer at <a href="/wiki/Ames_Research_Center" title="Ames Research Center">Ames Laboratory</a>, aimed to supplement Guderley's Mach 1 work with numerical solutions that would cover the range of transonic speeds between Mach 1 and wholly supersonic flow.<sup id="cite_ref-:0_2-9" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Vincenti and his assistants drew upon the work of <a href="/wiki/Howard_Wilson_Emmons" title="Howard Wilson Emmons">Howard Emmons</a> as well as Tricomi's original equations to complete a set of four numerical solutions for the drag over a double wedge airfoil in transonic flow above Mach 1.<sup id="cite_ref-:0_2-10" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The gap between subsonic and Mach 1 flow was later covered by both <a href="/wiki/Julian_Cole" title="Julian Cole">Julian Cole</a> and <a href="/wiki/Leon_Trilling" title="Leon Trilling">Leon Trilling</a>, completing the transonic behavior of the airfoil by the early 1950s.<sup id="cite_ref-:0_2-11" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Condensation_clouds">Condensation clouds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=5" title="Edit section: Condensation clouds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At transonic speeds <a href="/wiki/Prandtl%E2%80%93Meyer_expansion_fan" title="Prandtl–Meyer expansion fan">supersonic expansion fans</a> form intense low-pressure, low-temperature areas at various points around an aircraft. If the temperature drops below the <a href="/wiki/Dew_point" title="Dew point">dew point</a> a visible cloud will form. These clouds remain with the aircraft as it travels. It is not necessary for the aircraft as a whole to reach <a href="/wiki/Supersonic" class="mw-redirect" title="Supersonic">supersonic</a> speeds for these clouds to form. Typically, the tail of the aircraft will reach supersonic flight while the nose of the aircraft is still in subsonic flight. A bubble of supersonic expansion fans terminating by a wake shockwave surround the tail. As the aircraft continues to accelerate, the supersonic expansion fans will intensify and the wake shockwave will grow in size until infinity is reached, at which point the bow shockwave forms. This is Mach 1 and the <a href="/wiki/Prandtl%E2%80%93Glauert_singularity" title="Prandtl–Glauert singularity">Prandtl–Glauert singularity</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Transonic_flows_in_astronomy_and_astrophysics">Transonic flows in astronomy and astrophysics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=6" title="Edit section: Transonic flows in astronomy and astrophysics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In astrophysics, wherever there is evidence of shocks (standing, propagating or oscillating), the flow close by must be transonic, as only supersonic flows form shocks. All black hole <a href="/wiki/Accretion_(astrophysics)" title="Accretion (astrophysics)">accretions</a> are transonic.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Many such flows also have shocks very close to the black holes. </p><p>The outflows or jets from young stellar objects or disks around black holes can also be transonic since they start subsonically and at a far distance they are invariably supersonic. Supernovae explosions are accompanied by supersonic flows and shock waves. Bow shocks formed in <a href="/wiki/Solar_wind" title="Solar wind">solar winds</a> are a direct result of transonic winds from a star. It had been long thought that a bow shock was present around the heliosphere of our solar system, but this was found not to be the case according to <a href="/wiki/Interstellar_Boundary_Explorer" title="Interstellar Boundary Explorer">IBEX</a> data published in 2012.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/transonic" class="extiw" title="wiktionary:Special:Search/transonic">transonic</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a href="/wiki/Anti-shock_body" title="Anti-shock body">Anti-shock body</a></li> <li><a href="/wiki/Subsonic_flight" class="mw-redirect" title="Subsonic flight">Subsonic</a> flows</li> <li><a href="/wiki/Supersonic" class="mw-redirect" title="Supersonic">Supersonic</a> flows</li> <li><a href="/wiki/Hypersonic" class="mw-redirect" title="Hypersonic">Hypersonic</a> flows</li> <li><a href="/wiki/Prandtl%E2%80%93Meyer_expansion_fan" title="Prandtl–Meyer expansion fan">Supersonic expansion fans</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transonic&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:2-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAnderson2017" class="citation book cs1">Anderson, John D. Jr. (2017). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/927104254"><i>Fundamentals of aerodynamics</i></a> (Sixth ed.). New York, NY. pp. 756–758. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-259-12991-9" title="Special:BookSources/978-1-259-12991-9"><bdi>978-1-259-12991-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/927104254">927104254</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+aerodynamics&rft.place=New+York%2C+NY&rft.pages=756-758&rft.edition=Sixth&rft.date=2017&rft_id=info%3Aoclcnum%2F927104254&rft.isbn=978-1-259-12991-9&rft.aulast=Anderson&rft.aufirst=John+D.+Jr.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F927104254&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATransonic" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:0_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:0_2-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-:0_2-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-:0_2-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-:0_2-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-:0_2-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-:0_2-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-:0_2-11"><sup><i><b>l</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVincentiBloor2003" class="citation journal cs1">Vincenti, Walter G.; Bloor, David (August 2003). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1177/0306312703334001">"Boundaries, Contingencies and Rigor"</a>. <i>Social Studies of Science</i>. <b>33</b> (4): 469–507. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F0306312703334001">10.1177/0306312703334001</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0306-3127">0306-3127</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13011496">13011496</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Social+Studies+of+Science&rft.atitle=Boundaries%2C+Contingencies+and+Rigor&rft.volume=33&rft.issue=4&rft.pages=469-507&rft.date=2003-08&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13011496%23id-name%3DS2CID&rft.issn=0306-3127&rft_id=info%3Adoi%2F10.1177%2F0306312703334001&rft.aulast=Vincenti&rft.aufirst=Walter+G.&rft.au=Bloor%2C+David&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1177%2F0306312703334001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATransonic" class="Z3988"></span></span> </li> <li id="cite_note-:1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTakahashi2017" class="citation book cs1">Takahashi, Timothy (15 December 2017). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/1162468861"><i>Aircraft performance and sizing. fundamentals of aircraft performance</i></a>. Momentum Press. p. 107. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60650-684-4" title="Special:BookSources/978-1-60650-684-4"><bdi>978-1-60650-684-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1162468861">1162468861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aircraft+performance+and+sizing.+fundamentals+of+aircraft+performance&rft.pages=107&rft.pub=Momentum+Press&rft.date=2017-12-15&rft_id=info%3Aoclcnum%2F1162468861&rft.isbn=978-1-60650-684-4&rft.aulast=Takahashi&rft.aufirst=Timothy&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F1162468861&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATransonic" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTakahashi2016" class="citation book cs1">Takahashi, Timothy (2016). <i>Aircraft Performance and Sizing, Volume I</i>. 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Warrendale, PA: SAE International. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4271%2F760477">10.4271/760477</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2060%2F19760009938">2060/19760009938</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118185921">118185921</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SAE+Technical+Paper+Series&rft.atitle=Airfoil+Section+Drag+Reduction+at+Transonic+Speeds+by+Numerical+Optimization&rft.volume=1&rft.date=1976-02-01&rft_id=info%3Ahdl%2F2060%2F19760009938&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118185921%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4271%2F760477&rft.aulast=Hicks&rft.aufirst=Raymond+M.&rft.au=Vanderplaats%2C+Garret+N.&rft.au=Murman%2C+Earll+M.&rft.au=King%2C+Rosa+R.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.4271%2F760477&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATransonic" class="Z3988"></span></span> </li> <li id="cite_note-:13-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-:13_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:13_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamm1990" class="citation book cs1">Ramm, Heinrich J. (1990). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/228117297"><i>Fluid dynamics for the study of transonic flow</i></a>. New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-60129-748-3" title="Special:BookSources/1-60129-748-3"><bdi>1-60129-748-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/228117297">228117297</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+dynamics+for+the+study+of+transonic+flow&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=1990&rft_id=info%3Aoclcnum%2F228117297&rft.isbn=1-60129-748-3&rft.aulast=Ramm&rft.aufirst=Heinrich+J.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F228117297&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATransonic" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChakrabarti1990" class="citation book cs1">Chakrabarti, Sandip (1990). <i>Theory of Transonic Astrophysical Flows</i>. 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