The Doppler Effect

While the idea of a Doppler effect is often introduced in terms of the sound or pitch of a train whistle as it approaches and leaves a station, there are many different configurations and interpretations that surround this effect. While the goal of the section of the discussion is not to discuss all in detail, it will attempt to consolidate the basic equations associated with each and draw some comparative conclusions on the effect of each on wavelength.

What, in basic terms, is a Doppler effect?

As per the train example above, it is the train moving with velocity [v] and being the source of the sound waves. We shall assume that all waves propagate through the wave media with a unity velocity [c=1], irrespective of the type of media, but where the wavelength [λ] of the waves propagating through the media becomes a function of the velocity [v] of the wave source.

Note: In the context of the wave models being discussed, we might attempt to minimize the number of media types to those that support matter waves, where the media is space, and sound waves, where the media is air. In this reduced and speculative context, we might assume that both types of waves, i.e. matter and sound, propagate through a 3D wave media as longitudinal waves, where the media is non-dispersive, such that waves of different frequencies all propagate with velocity [c=1].

However, while wavelength is a function of the velocity [v] of the wave source, its measure is also a function of an angle [ϴ] with respect to the axis of motion. If we only consider the source has velocity [v1], then the Doppler effect will be described as the ‘normal doppler effect’ , where the effects of any wavelength change propagates through the reference frame of the wave media. However, there is another type of Doppler effect, which will be described as the ‘virtual Doppler effect’ that is associated with a perceived wavelength change, if the receiver of the waves has a velocity [v2]. At this point, it will simply be highlighted the potential characterization of a Doppler effect may be described in a number of ways.

Normal Doppler Effect

Unless otherwise stated, this first set of discussions about variations of the Doppler effect are restricted to sound waves through the media of air, i.e. acoustic waves. As outlined, this effect is observed in the wave media where the source of the waves is moving with velocity [v1], where we shall initially assume the receiver to be stationary, i.e. [v2=0], with respect to the wave media.

The diagram above is taken from the LaFreniere webpage discussing the Doppler effect and quantifies the change in wavelength [λφ] based on the following equation.

[1]             

However, an examination of the previous diagram shows that the angle [φ] is with respect to the origin and not the current position of the source signified by the black dot to the right of the centre. Further examination of this diagram also shows the wavelength [λφ] only applies to the outer separation of the two off-centre circles, such that we might question the usefulness of this view. However, searching the Internet for references on the Doppler effect generally only focus on the simplified formulation for the forward [λF] and backward [λB] wavelengths with little often said about the angle [φ], let alone the angle [ϴ] with respect to the moving source. Therefore, this discussion will first focus on a simulation, see 94.1.bas, that does not use any equations to plot the normal Doppler effect, as shown below.

So how is this simulation created?

Basically, the simulation is simply drawing a circle every 20 units of time [t], indicated top right. Each circle is centred on the current position of the source at time [t], i.e. the black dot to the right of the centre, which has a velocity [β=0.5] and expands by the speed [c], again as a function of time [t]. However, the simulation is now showing angles [ϴ] with respect to the moving source, i.e. the black forward [0o] and backward [180o] paths along the x-axis, the red vertical [90o] y-axis plus the blue [60o] and green [120o] axes. It is highlighted that the distance between each circle is now the same and represents the change in wavelength [λϴ], not [λφ], associated with the normal Doppler effect. The wavelength factor for the angles [ϴ] described above shown in the simulation bottom-left, which are calculated based on the equation in [2].

[2]             

While the derivation and explanation of the equation in [2] will be detailed in the Ivanov Waves discussion, the following table now shows all values of the change in wavelength determined as a function of angle [ϴ].

B ϴ Rad sin2 cos λ1 ϴ Rad sin2 cos λ1
0.5 0 0.000 0.000 1.000 0.500 90 1.571 1.000 0.000 0.866
0.5 10 0.175 0.030 0.985 0.504 100 1.745 0.970 -0.174 0.957
0.5 20 0.349 0.117 0.940 0.515 110 1.920 0.883 -0.342 1.054
0.5 30 0.524 0.250 0.866 0.535 120 2.094 0.750 -0.500 1.151
0.5 40 0.698 0.413 0.766 0.564 130 2.269 0.587 -0.643 1.245
0.5 50 0.873 0.587 0.643 0.602 140 2.443 0.413 -0.766 1.330
0.5 60 1.047 0.750 0.500 0.651 150 2.618 0.250 -0.866 1.401
0.5 70 1.222 0.883 0.342 0.712 160 2.793 0.117 -0.940 1.455
0.5 80 1.396 0.970 0.174 0.784 170 2.967 0.030 -0.985 1.489
0.5 90 1.571 1.000 0.000 0.866 180 3.142 0.000 -1.000 1.500

Note: Although, LaFreniere labels this to be the relative Doppler effect, it practical terms it is the normal Doppler effect for sound waves, which conforms to the Ivanov wave transforms.

Composite Doppler Effect

As outlined, the normal Doppler effect is associated with the change in wavelength caused by velocity [vS] of the wave source. However, we might attempt to generalise some of the equations that apply to sound waves, starting with the relationship between frequency [f] and wavelength [λ], although if [c] is an attribute of the wave media, then wavelength [λ] becomes a function of [c] and [f] as shown in [3].

[3]             

If all waves propagate with [c=1] through the wave media, the normal Doppler effect leads to a change in the wavelength [λ], while frequency remains synchronised to the source, as represented by [4], where [vs] is the source velocity.

 [4]             

In [4], we are only describing the normal Doppler effect in terms of the forward and backward wavelengths, when the source is moving with velocity [vS] along the axis of motion. However, at this point, we might consider the total path between sender and receiver, where the sender [S] is simply the source of the waves, which propagate through the wave media [M] towards the receiver [R]. As such, we might recognise that the wave really only exists in the media, where the sender and receiver interpret the wave in terms of its perceived attributes of frequency [f] and wavelength [λ], which are a function of velocity [βS, βR]. So, in this context, the sender is only the source of the oscillating frequency [fS], which then propagates outwards into the media with velocity [c=1] with the wave attributes of frequency [fM] and wavelength [λM].

Note: It is highlighted that the frequency [fS] of the wave must remain synchronised at both the sender and receiver, which is why [4] shows the change in wavelength [λM] caused by the velocity [βS] of the sender.

Finally, at the receiver, the wave attributes are interpreted as a function of the receiver velocity [βR], where without knowledge of its velocity with respect to the wave media, as per special relativity, the wave attributes might be calculated on the basis of equation [3]. It might also be highlighted that time dilation, also required by special relativity, would affect the measurement of time [t’] in any moving frame, although this is not an issue for the sound wave example being considered. Having clarified the various reference frames under consideration, we might now extend [4] to account for the velocity of both the sender and receiver.

[5]             

If we consider [5] in terms of a person travelling on the train, where [βSR], we might realise that while the normal Doppler effect exists in the wave media, the velocity of the receiver cancels this effect, i.e. the change in the pitch of the train whistle is only heard on the platform by a stationary observer. However, there are many permutations of sender [βS] and receiver [βR] velocities, both in terms of magnitude and direction that change the scope of the Doppler effect, which are generally summarised in the following table.

λS βS βR λR
1 0 0 1.000
1 0.5 0 0.500
1 -0.5 0 1.500
1 0 0.5 2.000
1 0 -0.5 0.667

If [βSR=0] then there is no perceived Doppler effect. If [βS=±0.5] then the previous outlined of the normal Doppler effect apply, which propagates through the wave media towards the receiver. However, the last two options suggest a change in the received wavelength [λR] that are dependent on the receiver velocity [βR], which will now be described in terms of the virtual Doppler effect.

Virtual Doppler effect

If we simply consider the forward and backward virtual wavelengths, when the source is stationary, then equation [5] reduces to the following form:

[6]             

Based on the default sign of [βR] in [6], the positive value of [βR=0.5] indicates the receiver is moving away from the wave source at half the wave velocity [c=1] propagating through the media toward the receiver with velocity [c-v=0.5]. Likewise, when the value of [βR=-0.5] is negative, the receiver is moving towards the wave source with velocity [c+v=1.5].

λS βS βR λR
1 0 0.5 2.000
1 0 -0.5 0.667

Lorentz Doppler Effect

In the previous sections of this discussion, the examples have been constrained to sound waves, such that there are no relativistic effects, specifically time dilation, associated with the velocity of either the source or the receiver. In this context, the normal Doppler effect for sound waves was associated with the Ivanov transforms – see Ivanov Waves for details. However, consideration of relativistic effects, which might apply to LaFreniere’s electron model, suggests that the Ivanov transforms have to be modified to account for time dilation, although we may need to later question whether there is any causal mechanism to justify this change. However, we will start by simply comparing the Ivanov and Lorentz transforms as shown in [7].

[7]             

While a more detailed analysis of the Lorentz transforms will be deferred to a later discussion, we might initially compared the two variant simulations, see 94.2,bas,  showing both the Ivanov and Lorentz Doppler effects, when the source velocity [βS=0.5]. 

In the Ivanov simulation left, we see the source frequency [f1=0.02], which based on [λ=c/f1] translates into a wavelength [λ=50] being propagated into the wave media, although the actual wavelength is a function of the source velocity [β=0.5] and angle [ϴ]. If we now assume that the wave source in the Lorentz simulation, right, is subject to time dilation, then the frequency [f1=0.02] is reduced to [f2=0.017], which in-turn effects the wavelength [λ=57.73] propagated into the wave media, although this is again a function of the source velocity [β=0.5] and angle [ϴ].

Note: The black and white divisions in the Ivanov and Lorentz simulations reflect the half-wavelength [l1=50/2] and [l2=57.73/2]. In the Ivanov model left, we see the calculated results of wavelength [l1’] are expressed as an absolute figure and as a ratio of [l1]. As we might expect, this ratio conforms to the Ivanov transforms. While, in the Lorentz model right, the calculated results of the wavelength [l2’] are expressed as an absolute figure, but the ratio figures are with respect to [l1], not [l2]. In fact, if the ratio were calculated with respect to [l2], the ratio figures would align to the Lorentz transforms. Why?

Let us try to consider the ‘why’ question raised in the note above. First, we might illustrate how frequency [f2] and wavelength [λ2] are calculated from the Lorentz factor [g], based on [8], which is normally associated with special relativity.

 [8]             

Based on [8], we see that time dilation affects the rate of time in the frame moving relative to the wave media. Why this might be so will be deferred to later discussions. For now, we might simply accept the assumption of time dilation such that the source frequency changes from [f1] to [f2], which in-turn leads to the change in wavelength from [λ1] to [λ2], as per [8]. However, if we now subject these figures to the Ivanov transforms, as per the animation blow, we get the same figures as the Lorentz Doppler effects, shown right in the previous animation above. 

If this is the case, it appears that the Lorentz transforms are only showing the change in wavelength relative to the conceptual wavelength at the source, i.e. [l1=c/f1=50] and not the actual wavelength [l2=c/gf1=57.73] propagating through the wave media.

Note: While possibly too early to reach any definitive conclusions, we appear to have only a Doppler effect, which we might sub-divide in terms of a normal and virtual effect based on the relative velocity of the source and/or receiver of the waves. The normal Doppler effect is explained by the Ivanov transforms, which if first subject to time dilation, yields the same results as the Lorentz transforms.

While the previous results need to be reviewed further in the subsequent discussion entitled Ivanov Waves and Lorentz Transforms, we might table some issues for consideration. Special relativity rests on the assumptions underpinning the Lorentz transforms, which require length contraction and time dilation. In the context of the following discussion of Ivanov Waves, the idea of standing wave compression might provide the basis of a possible causal mechanism for length contraction, although the Ivanov transforms in [7] make no reference to time dilation. However, while the Lorentz transforms, also shown in [7], allude to time dilation, Lorentz never provided any causal explanation of how this might occur. In fact, Lorentz only interpreted the requirement for time dilation as a mathematical requirement, such that time dilation in the LaFreniere electron model is still an open issue. In addition, the previous simulations possibly suggest that the Ivanov transforms may be the more fundamental set of equations as they appear to reflect the change in wavelength with velocity [β] and angle [ϴ] with to [l2=57.73], which corresponds to the time dilated frequency [f2=0.017].

Note: While this discussion will not pursue these issues any further at this point, it might highlight one further issue that needs to be considered. The simulations presented correspond to the absolute frame of reference with respect to the wave media. Of course, special relativity proceeds on the assumption that this reference frame does not exist, see Michelson-Morley experiment for more details, such that any observer comoving with the source will be subject to time dilation, although they will not perceive the effect. In this context, the Lorentz transforms are the translation between any two arbitrary inertial reference frames, where one assumes itself to be the stationary frame.