The Doppler Effect(s)
At this stage, the goal of this discussion is to simply highlight a wide range of issues associated with the Doppler effect that may have a bearing on the WSM model without necessarily knowing which issues are directly relevant. We will begin by simply stating that the Doppler effect can be introduced in terms of how a reference point on a wave, e.g. its crest, progresses from source to receiver. For once the wave crest leaves the source, it is no longer affected by the motion of the source, only the wave propagation velocity [c] for that wave medium. However, subsequent wave crests can either be bunched up in front or stretched out behind earlier crests depending on the velocity [v] of the source or receiver or both. As a result, this effect can lead to different perceptions of the wave frequency and wavelength depending on which reference frame the measurements are taken.
For example, the animation above might be seen as the normal Doppler effect in which the source has a velocity [v_{S}]. However, in the wider context of the WSM model, we might consider the wave propagation media as the absolute reference frame against which the velocity of the source [v_{S}] and receiver [v_{R}] are then defined. As such, we might describe the source as transmitting waves into the wave media, which then propagate through the media towards the receiver. However, if the velocity of each reference frame, i.e. source, media and receiver, are different, the ‘observed’ wave will have different frequency [f] and wavelength [λ] attributes:
- Source: velocity [v_{S}], frequency [f_{S}], wavelength [λ_{S}]
- Media : velocity [v_{0}], frequency [f_{0}], wavelength [λ_{0}]
- Receiver: velocity [v_{R}], frequency [f_{R}], wavelength [λ_{R}]
If we assume the media exists as an absolute reference frame, we might also assume it to be stationary [v_{0}], such that wave frequency [f_{0}] and wavelength [λ_{0}] represent some fundamental value. Of course, in practice, any measure of frequency and wavelength has to be linked to one of the reference frames above, which suggests that a number of permutations are possible. However, we will start by characterizing the propagation velocity [c] of a non-dispersive wave in the following terms:
[1]
As indicated, this equation is valid for non-dispersive waves, irrespective of whether we describe them as mechanical or electromagnetic in nature; although the latter might question the need for any propagation media. The generic form of [1] might suggest that it can be applied to any reference frame, where either the frequency [f] or wavelength [λ] can be measured, which then allows the other variable to be calculated, if [c] is assumed to be a constant in all reference frames.
Where does the WSM model fit into this description?
On the assumption that the WSM model requires a physical wave media, i.e. space, it would seem to conform to the basic description of a mechanical wave, although this may be overly simplistic. However, because all matter, e.g. observers and measuring equipment, only exist as a localized wave system within this model, any measurement of the frequency or wavelength has to be associated with some given reference frame. Therefore, the idea of a Doppler effect, of some description, may be key to understanding the wave structure of matter. However, this initial discussion will attempt to outline five ‘viewpoints’ associated with the basic Doppler effect, which may have some bearing on any wave model, i.e.
- The Asymmetric Doppler Effect
- The Normal Doppler Effect
- The Virtual Doppler Effect
- The Lorentz Doppler Effect
- The Relative Doppler Effect
While each of these effects has to be described in terms of a specific configuration of reference frames, i.e. source and receiver velocities, there is also an aspect that may depend on the assumptions of the mathematical model being applied, which may not be proven at this stage. So, as outlined above, mechanical waves are usually ‘seen’ to propagate through a physical media, e.g. sound waves through air, such that the source and receiver velocity can be determined relative to the wave media, which might then be considered as an absolute reference frame. In contrast, special relativity in conjunction with the Michelson-Morley experiment led to a model that appears to require no absolute reference frame, such that the source and receiver velocities are defined purely in terms of their relative velocity to each other. However, before discussing the variant Doppler effects cited above, we might wish to introduce two basic configurations in which waves can be transmitted between source and receiver. In the first case, the source has a relative velocity [v_{S}] with respect to the wave media, while the receiver is stationary.
Based on [1], the source and receiver reference frames might calculate the wave propagation based on a localized determination of frequency [f_{S}, f_{R}] or wavelength [λ_{S},λ_{R}]. However, it is not necessarily clear what frequency and wavelength [f,λ] is actually being propagated through the wave media; although we might suspect that the relative velocity [c±v_{S}] between the source and the wave-front would have some effect. However, let us now consider the second configuration.
While the relative velocity between the source and receiver is still [v], this velocity is now associated with the receiver, not the source. Again, based on [1], the source and receiver reference frames might calculate the wave propagation based on a localized determination of frequency [f_{S}, f_{R}] or wavelength [λ_{S},λ_{R}]. Likewise, we might not be certain as to the frequency and wavelength [f’,λ’] being propagated through the wave media; although given the source is now stationary, we might suspect [f’=f_{S}] and [λ’=λ_{S}].
Note: The use of [f’,λ’] rather than [f_{0},λ_{0}] for the frequency and wavelength of the waves propagating through the media is used to highlight another possible ambiguity. Within the relativistic model, only the relative velocity [v] between the source and receiver is known. While this model allows either of these inertial reference frames to claim to be at rest [v=0], there may still be a velocity with respect to some undetectable media. If so, [f’,λ’] would not correspond to [f_{0},λ_{0}].
The Media Effect?
However, if we ignore the implications of special relativity for one moment, the existence of the wave media as an absolute reference frame might suggest that the wave-front has to be receding away from the source or approaching the receiver with a velocity [c±v]. If so, we might be led to a modification of [1] as follows:
[2]
In [2], we see the development of the (1±β) form that will appear in so many of the Doppler equations based on a modification of the basic relationship in [1]. However, in [2], we see the suggestion, based on the physical existence of the propagation media, that questions the constancy of [c] in all reference frames as required by special relativity.
Note: In this context, we are not questioning the wave propagation velocity [c] through the media, only the relative velocity with respect to the source or receiver.
While the validity of the assumptions underpinning [2] need to be clarified and questioned in further discussions, we might consider two outcomes of [2] that might also be queried:
[3]
If we transpose [2] with respect to wavelength [λ], as shown in [3], there is the suggestion that it must be the wavelength [λ] that is either expanded or compressed against some original measure of wavelength [λ’]. However, if this is the case, it might then be seen to imply that the frequency [f] is unaffected. We might consider this effect in terms of a moving source transmitting waves into the media, such that the wavelength is compressed, while the wave frequency is still synchronised to the source - see normal Doppler effect for details. However, we might also get the impression that the following arrangement of [2] can also be valid:
[4]
Now we see the suggestion that it must be the frequency [f] that is either expanded or compressed against some original measure of frequency [f’]. If this is the case, then it might imply that the wavelength [λ] was unaffected. Therefore, we might consider whether this effect has any validity when a receiver is moving towards a stationary source - see virtual Doppler effect for details. However, if we restore the constancy of [c] in all reference frames, we might assume that any change to either wavelength [λ] or frequency [f] has to be reciprocated in the other variable, i.e. [f] or [λ]:
[5]
While we will defer further discussion of [2]..[5] until after reviewing the various Doppler effects, we might clarify that within the WSM model, the normalized velocity [β=v/c] is the relative speed of the source or receiver, as compared to the wave speed [c] through the media. Of course, we may also have to eventually extend the idea of the Doppler effect to include time dilation, although we might still have to question the cause.
[6]
Today, most will recognise the contraction factor [g] within the relativistic factor [γ], as defined by special relativity and based on the Lorentz transformations. However, as indicated, the scope of each type of Doppler effect to be described can depend on the assumptions of an underlying model and the velocity relationship between the source and receiver. So, with some of the issues outlined in mind, the discussion of the Doppler effect(s) will continue throughout the following sub-pages:
- The Asymmetric Doppler Effect
- The Normal Doppler Effect
- The Virtual Doppler Effect
- The Lorentz Doppler Effect
- The Relative Doppler Effect
Summary
Based on the review discussions cited above, it might be suggested that each effect can lead to a different perception of space-time, based on changes to wavelength and frequency. However, what is not clear at this stage is which of the effects discussed might actually be applicable or help explain the validity of the WSM model. While this issue will be tabled for further discussion within the main body of the WSM model review, there may be some value in summarising the following results based on the source or receiver velocity being [β=0.5]. Therefore, the values quoted simply represent the fractional change to wavelength or frequency in comparison to a stationary [v=0] source or receiver:
θ | Normal | Virtual | LaFreniere | Lorentz | φ | Relative | |
0 | 0.500 | 0.667 | 0.577 | 0.577 | 0 | 0.500 | |
30 | 0.567 | 0.698 | 0.629 | 0.629 | 30 | 0.535 | |
60 | 0.750 | 0.800 | 0.775 | 0.775 | 60 | 0.651 | |
90 | 1.000 | 1.000 | 1.000 | 1.000 | 90 | 0.866 | |
120 | 1.250 | 1.333 | 1.291 | 1.291 | 120 | 1.151 | |
150 | 1.433 | 1.764 | 1.590 | 1.590 | 150 | 1.401 | |
180 | 1.500 | 2.000 | 1.732 | 1.732 | 180 | 1.500 | |
210 | 1.433 | 1.764 | 1.590 | 1.590 | 210 | 1.401 | |
240 | 1.250 | 1.333 | 1.291 | 1.291 | 240 | 1.151 | |
270 | 1.000 | 1.000 | 1.000 | 1.000 | 270 | 0.866 | |
300 | 0.750 | 0.800 | 0.775 | 0.775 | 300 | 0.651 | |
330 | 0.567 | 0.698 | 0.629 | 0.629 | 330 | 0.535 | |
360 | 0.500 | 0.667 | 0.577 | 0.577 | 360 | 0.500 |
In this table, we see that results formulated for comparison based on the angle [θ], which we might characterised as existing between the origin of the source at time [t_{0}] and the receiver at [t_{1}]. As such, we can compare the resulting wavelengths for each of the variant Doppler effects discussed.
Note: the column labelled LaFreniere reflects the same results as the Lorentz Doppler effect, but based on a different but equivalent formulation. See Gabriel LaFreniere webpage discussion on the Doppler Effect for more details.
The discussion of the relative Doppler effect also highlights the possibility of another angle [φ], which exists between the source and receiver at time [t_{1}]. However, because there is no direct correlation between [θ] and [φ], no direct comparison of the effects based on these angles can be assumed.