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Sound waves travel so fast. We apply the rules of adiabatic process (quasi static) to derive at the equation for speed of sound. My doubt is that how can sound be considered a quasi static process since it travels at high speed?

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Because the wave is going so fast, the little parcels of air that participate in the compression/rarefaction process of sound wave travel have almost no time available in which to exchange compression heat or rarefaction cooling between one another as the wave passes through them. In addition, the absolute magnitude of the pressure differences contained in the wave are very small, making the magnitude of the heating/cooling process tiny. This means the overall process can be considered quasistatic.

niels nielsen
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There is an answer on stack exchange, How is sound wave quasistatic?