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Quick question. I thought that the speed of sound in air was constant, say in the right conditions of pressure and temperature, and humidity... 300 m/s. Now, if I have a sound source that moves towards me at 50 m/s, is the speed of the sound waves 350 m/s when it arrives to me? (this is not about the Doppler effect, which I know about. It's about whether/how to compose the speed of sound and the speed of the source - it seems all the texts I'm seeing assume that the speed is 300 m/s no matter how the source moves)

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The speed of sound is independent of the motion of the source. Sound pressure, which is a scalar, obeys a linear wave equation of the form

$$ \frac{\partial^2 p}{\partial t^2} - c_s^2 \nabla^2 p = S(\vec{x},t) $$

where $c_s$ is the speed of sound and $S$ describes the source. (This equation is an approximation that neglects heating of the air and a few other things, which means it works for ordinary sounds but not for explosive shock waves etc.) The solutions of this equation have the property that disturbances, no matter what the form of the source $S$, spread from the source at the same speed $c_s$.

Physically you can look at it from the reference frame of the car. Passengers in the car would observe the sound being slowed in the front and sped in the back by a strong wind. If they were going near the speed of sound they would observe the sound waves bunching up in the front, forming a single intense sound wave - a sonic boom.

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