Speed of sound in air

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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If we say, "The speed of sound under such and such conditions is 300 m/s," we are talking about the speed with which the sound waves propagate through the medium (e.g., through the air). If the source is moving toward or away from you, that does not affect the measurement, but if you are moving through the air, then that does affect your measurement. – james large Aug 27 '15 at 1:18

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