Frequency shift in case of wind and no relative motion between observer and source

Question

I've learned the following things about Doppler effect in the case of sound waves

1. The Doppler effect formula is to be used in the air reference frame (where $v_{sound}=343 m/s$, at room temperature)
2. Doppler effect is present only if there is relative motion

That means that if observer and source are both steady in the ground reference frame, but there is wind blowing from one to the other no frequency shift occur because there is no relative motion between source and observer in the air (moving) frame.

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The thing that I do not understand is the following. Consider the situation illustrated before (source and observer steady, wind blowing): the speed of sound perceived by the observer is not $v_{sound}=343 m/s$ anymore but it should be $v_{sound}=343m/s+v_{wind}$ if the wind blows towards the observer.

Now if the source emits a sound wave of frequency $f$ and wavelenght $\lambda$ the following holds: $$v_{sound}=f \lambda$$

But $v_{sound}$ changes in the case of wind even if there is no relative motion between observer and source (hence no Doppler effect) so $f$ (or $\lambda$) should change also in this case. And this in contrast with what I stated at the beginning about Doppler effect. How can that be?

I'm probably missing something in this, because it seems a paradox, I appreciate any help.

Answer 1

With respect to the air the wavelength will be $\lambda = 343/f$ everywhere. With respect to the ground the wavelength will be $\lambda = v_{sound}/f$ but $v_{sound}$ will depend on position with respect to the source - picture your classic Doppler shift diagram with a set of not quite concentric circles.

Downwind from the source the wavelength, measured with respect to the ground, will get longer proportional to the speed, according to $v_{sound}=f\lambda$ but the frequency will be the same as it was at the source.