Sound and apparent wind -- Any Doppler effect?

[Kairos]

If two cars are driving side by side at the same speed, their relative speed is zero but do they nonetheless perceive a sound Doppler effect from the other car's siren because of the apparent headwind generated by their speed?

 

[hmmm27]

If they're running parallel at a constant velocity, then no.

 

[Kairos]

but I do not understand why because the wavelength is expected to be shortened at the level of the cars, since the original wavelength is pushed behind the vertical bar of this scheme :

by the apparent headwind

 

[hmmm27]

but I do not understand why

Would you have understood if I had said "yes" ?

because the wavelength is expected to be shortened at the level of the cars,

shortened as compared to what ?

since the original wavelength is pushed behind the vertical bar of this scheme :View attachment 290889 by the apparent headwind

I have no idea what you mean by that.

I can see the circles - I assume they represent the wavefront, the closest circle to the driving vehicle would be the most recent.

If both vehicles were parked, and there was a steady wind, would you expect a Doppler effect ?

 

[jbriggs444]

Consider two consecutive of wave fronts from one car to the other. Does the transit time for the first differ from the transit time for the second?

but I do not understand why because the wavelength is expected to be shortened at the level of the cars, since the original wavelength is pushed behind the vertical bar of this scheme :View attachment 290889 by the apparent headwind

The wavelength is not pushed. The wave front is pushed.

This is similar to a boat crossing a river problem. The tiny portion of the wave front that is aimed just right to go from one car to the other will have its forward velocity reduced by the cross-wind.

What effect would you expect this to have on the wavelength along the path from bottom car to top?
What effect would you expect this to have on the frequency of wave fronts passing the mid-point on that path?

Consider another argument...

If you decide that there is an apparent headwind for the bottom car and a resulting doppler shift at the sending end, what about the doppler shift at the receiving end?

 

[A.T.]

the wavelength is expected to be shortened at the level of the cars

Doppler shift is about frequencies not wavelengths. You have to consider the propagation speed along the line connecting the cars.

 

[Kairos]

Thank you for all these interesting arguments!
As the frequency is related to the wavelength through $f=c/\lambda$, if I understand well the frequency is unchanged because the speed of the sound wave $c$ is decreased by the wind in exactly the same ratio as the wavelength in the direction orthogonal to the road? (and to the wind)

 

[jbriggs444]

Thank you for all these interesting arguments!
As the frequency is related to the wavelength through $f=c/\lambda$, if I understand well the frequency is unchanged because the speed of the sound wave $c$ is decreased by the wind in exactly the same ratio as the wavelength in the direction orthogonal to the road? (and to the wind)

Yes.

I would consider frequency to be the invariant and wavelength to be the thing that varies based on velocity, but the result is the same regardless.

 

[vanhees71]

What's invariant is the "phase" of the wave, i.e., $\omega t-\vec{k} \cdot \vec{x}$.

 

[Kat-hi]

If two cars are driving side by side at the same speed, their relative speed is zero but do they nonetheless perceive a sound Doppler effect from the other car's siren because of the apparent headwind generated by their speed?

Does it matter if they are the same size?

 

[hmmm27]

Does it matter if they are the same size?

If what are the same size ? The posed question is taken as two point sources moving at the same velocity through still air, ie: no aero effects.

 

[Kairos]

What's invariant is the "phase" of the wave, i.e., $\omega t-\vec{k} \cdot \vec{x}$.

The invariance of frequency would have seemed in agreement with the conservation of energy (E=hf). It is not true?
Why is phase invariance more correct?

 

[vanhees71]

It's a very general concept leading to a consistent derivation of all kinds of Doppler effects. One must not forget that for the acoustic Doppler effect you have a preferred frame of reference, which is the rest frame of the medium (usually air in everyday life). Then you can have all kinds of situations, depending on whether the observer or the source is moving wrt. this rest frame or both. The concept of "phase invariance" let's you derive all these cases by using appropriate Galilei boosts (non-relativistic case) or Lorentz boosts (relativistic case). There's a nice AJP paper for the non-relativistic case, using this principle to derive all kinds of situations (including the most general case for both moving source and detector in arbitrary directions):

A. H. Spees, Acoustic Doppler Effect and Phase Invariance, American Journal of Physics 24, 7 (1956)
https://doi.org/10.1119/1.1934120

 

[Sparrow_1827]

As source is moving at a speed say v m/s wrt medium towards receiver and
receiver is moving at speed v m/s wrt medium away from the source then by the formula :

f = [(c - v)/(c - v)]*f_o = f_o

**f_o = original frequency ; c = speed of sound ; f = resultant frequency to the reciever ; v<c