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3.1.5: Speed of sound

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    The speed of sound in a perfect gas is:

    \[a = \sqrt{\gamma RT},\]

    where \(R\) is the constant of the gas, \(T\) the absolute temperature, and \(\gamma\) the adiabatic coefficient which depends on the gas. In the air \(\gamma = 1.4\) and \(R = 287.05\ [J/KgK]\). Therefore, the speed of sound in the air is 340.3 [m/s] at sea level in regular conditions.

    Mach number

    Mach number is the quotient between the speed of an object moving in the air (or any other fluid substance), typically an aircraft or a fluid particle, and the speed of sound of the air (or substance) for its particular physical conditions, that is:

    \[M = \dfrac{V}{a}.\]

    Depending on the Mach number of an air vehicle (airplane, space vehicle, or missile, for instance), five different regimes can be considered:

    1. Incompressible: \(M< 0.3\), approximately. In this case, the variation of the density with respect to the density at rest can be neglected.
    2. Subsonic (compressible subsonic): \(0.3 \le M < 0.8\), approximately. The variations in density must be included due to compressibility effects. Two different regimes can be distinguished: low subsonic (\(0.3 \le M < 0.6\), approximately) and high subsonic (\(0.6 \le M < 0.8\), approximately). While regional aircraft typically fly in low subsonic regimes, commercial jet aircraft typically fly in high subsonic regimes (trying to be the closest to transonic regimes while avoiding its negative effects in terms of aerodynamic drag).
    3. Transonic: \(0.8 \le M < 1\), approximately. This is complex situation since around the aircraft coexist both subsonic flows and supersonic flows (for instance, in the extrados of the airfoil the flow accelerates and can be supersonic while the flow entering through the leading edge was subsonic).
    4. Supersonic: \(M \ge 1\), and then the flow around the aircraft is also at \(M \ge 1\). Notice that the flow at \(M = 1\) is known as sonic.
    5. Hipersonic: \(M \gg 1\) (in practice, \(M > 5\)). In these cases phenomena such as the kinetic heat or molecules dissociation appears.

    In order to understand the importance of the Mach number it is important to notice that the speed of sound is the velocity at which the pressure waves or perturbations are transmitted in the fluid.

    截屏2022-01-14 下午9.58.51.png
    Figure 3.8: Effect of the speed of sound in airfoils (\(M_a\) corresponds to Mach number).

    Imagine a compressible air flow with no obstacles. In this case, the pressure will be constant along the whole flow, there are no perturbations. If we introduce an airplane moving in the air, immediately appears a perturbation in the field of pressures near the airplane. Moreover, this perturbation will travel in the form of a wave at the speed of sound throughout the whole fluid field. This wave represents some kind of information emitted to the rest of fluid particles, so that the fluid adapts its physical conditions (trajectory, pressure, temperature) to the upcoming object.

    If the airplane flies very slow (\(M = 0.2\)), the waves will travel fast relative to the airplane (\(M = 1\) versus \(M = 0.2\)) in all directions. In this form the particles approaching the airplane are well informed of what is coming and can modify smoothly its conditions. If the velocity is higher, however still below \(M = 1\), the modification of the fluid field is not so smooth. If the airplane flies above the speed of sound (say \(M = 2\)), then in this case the airplane flies twice faster than the perturbation waves, so that waves can not progress forwards to inform the fluid field. The consequence is that the fluid particles must adapt its velocity and position in a sudden way, resulting in a phenomena called shock wave.

    Therefore, when an aircraft exceeds the sound barrier, a large pressure difference is created just in front of the aircraft resulting in a shock wave. The shock wave spreads backwards and outwards from the aircraft in a cone shape (a so-called Mach cone). It is this shock wave that causes the sonic boom heard as a fast moving aircraft travels overhead. At fully supersonic speed, the shock wave starts to take its cone shape and the flow is either completely supersonic, or only a very small subsonic flow area remains between the object’s nose and the shock wave. As the Mach number increases, so does the strength of the shock wave and the Mach cone becomes increasingly narrow. As the fluid flow crosses the shock wave, its speed is reduced and temperature, pressure, and density increase. The stronger the shock, the greater the changes. At high enough Mach numbers the temperature increases so much over the shock that ionization and dissociation of gas molecules behind the shock wave begins.

    3.1.5: Speed of sound is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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