Understanding the 4-Stage Doppler Model: Exploring Sound Waves in Air

[mysearch]

Hi, I was hoping that somebody might be able to help me better understand/verify a Doppler model from the perspective of both a stationary and moving reference frame. For the current purpose, this Doppler model might be generally described in terms of sound waves in air, where air is assumed to be a non-dispersive medium and the waves propagate with constant velocity [c] through this media. In this context, it is assumed that there are no relativistic effects, only Doppler, i.e. ($1±β$]. While I have attached a diagram, see 'configs.jpg', the model probably needs a bit of explaining:

There are 2 basic configurations, where the central reflector is either stationary [$β=0$] or moving [$β=0.5c$]. In the [$β=0$] configuration (top), waves propagate from the left [$S_L$] and right [$S_R$] sources towards the central reflector and return to the left [$R_L$] and right [$R_R$] receivers having undergone a [$180^o$] phase shift at the central reflector. In this case, the frequency [f] and wavelength [$λ$] on each interface is assumed to be [$f_0$] and [$λ_0$]. However, I also assume this configuration would produce a standing wave, basically conforming to [1] below, which oscillates in time as the waves propagates towards and away from the reflector as the IN-OUT waves change phase with respect to each other.

[1] $A_0 cos \left( \omega t - kx \right) + A_0 \left( \omega t + kx + \phi \right) = 2A_0 \left[sin(kx)\right] \left[ sin ( \omega t) \right]$

As far as I can see there is no obvious ambiguity in the stationary configuration as there are no Doppler effects and all frames of reference are equal. However, when the reflector is moving to the right with velocity [$β=0.5c$], i.e. the bottom configuration, things appear to become more complicated as it would seem that each interface ends up with a different wavelength [$λ$] plus the perception of these values will presumably differ depending on the frame of reference in which measurements are taken; hence the following cases:

Case-1: Stationary Observer & Reflector
The attachment 'case1.gif' provides a basic animation of this configuration, which tries to mimic the top diagram in configs.jpg and the description above. However, now it is possible to see the dynamics of the standing waves produced, i.e. red curve in bottom trace. What is possibly interesting is the ‘grey’ outline which reflects the instantaneous standing wave aggregated over time.

Case-2: Stationary Observer & Moving Reflector
In the [$β=0.5c$] case, the left [$S_L$] and right [$S_R$] sources and the observer are still stationary and, as such, there is no Doppler effected waves propagating through the media towards the reflector. However, waves from the left [$S_L$] source are ‘chasing after’ the moving reflector, while waves from the right [$S_R$] source are moving towards the moving reflector. The results assumed in this case are shown in the animation 'case-2.gif'. As suggested, the source waves, left and right, propagating through the media are unchanged, but the reflected waves are affected by the rate of arrival at the reflector, which causes the change of wavelength being suggested. Interesting, the animation also suggests that the standing wave is maintained, but undergoes an expansion or compression on either side of the reflector, which the stationary observer could presumably prove by measuring the amplitude of the superposition wave along the x-axis? Again, the grey trace is the time aggregated effect of the standing wave, which highlights where the standing wave nodes will exist in this case, which differ from case-1.

Case-3: Observer comoving with Reflector
In essence, this case is physically identical to the previous case, except the perception of the waves is now determined by an observer collocated with the reflector. However, it not necessarily clear to me how to model this configuration, although animation 'case-3.gif' proposes a possible interpretation. While there is no change to the incoming waves, as per case-2, the comoving observer would measure and presumably interpret the inbound waves to have a different wavelength based on the Doppler effect ($c±v$] caused by the velocity [v], such that frequency would be determined based on [$f=c/λ$].

But would the reflected waves also be subject to another ($1±β$] effect on the return path?

The animation suggests – yes, because if the comoving observer could measure the amplitude of the superposition ahead and behind its current position, it is assumed that it would detect an additional expansion and compression of the standing waves as shown. While I understand that few may be interested in all this detail, I would appreciate any comments, insights or corrections regarding the animation and outline results presented. Thanks

 

[AndyW]

Hello, thank you for reaching out for clarification on this Doppler model. I can help you understand and verify the model from both a stationary and moving reference frame.

Firstly, in the stationary configuration (β=0), the waves propagate towards the central reflector and return with a 180° phase shift at the central reflector, creating a standing wave. This standing wave can be described by the equation you provided, where A0 is the amplitude, ω is the angular frequency, t is time, k is the wave number, x is the position, and φ is the phase shift.

In the moving configuration (β=0.5c), things become more complicated as the reflector is now moving. In this case, the wavelength of the waves on each interface will be different due to the Doppler effect. The perceived wavelength will also differ depending on the frame of reference in which measurements are taken. This is because the observer's motion affects the perceived frequency and wavelength of the waves. However, the standing wave will still exist in this configuration.

In the case where the observer is stationary and the reflector is moving, the observer will perceive the inbound waves to have a different wavelength due to the Doppler effect. However, the reflected waves will also be affected by the rate of arrival at the reflector, causing a change in wavelength. This can be seen in the animation you provided, where the standing wave undergoes expansion and compression on either side of the reflector.

In the case where the observer is comoving with the reflector, the perceived wavelength of the inbound waves will be affected by the Doppler effect (c±v). However, the reflected waves will also be subject to another (1±β) effect on the return path. This means that the standing wave will undergo additional expansion and compression, as shown in the animation.

Overall, the animations and results you have presented seem to be accurate and align with the principles of the Doppler effect. I hope this helps you better understand and verify the Doppler model in both stationary and moving reference frames. Let me know if you have any further questions or concerns.