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mysearch
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On first reading, the description of ‘group velocity [vg]’ appears to be quite straightforward. However, I also found a number of speculative explanations as to ‘how’ and ‘why’ the group velocity may exceed the ‘phase velocity [vp]’. Therefore, in order to get a better intuitive understanding of the issues, I am ‘attempting’ to simulate 4 different types of beat wave configurations:
Each simulation uses the same basic approach in which the amplitude of two individual phase waves is added, and calculated, for all values of [x], which is then displayed as a single frame in the animation. The process is then repeated for incrementing values of time [t] to create the next frame within the animation. While the first simulation in the list above produced ‘sensible’ results, i.e. [vg=vp], all the subsequent permutations, based on the same algorithms, lead to ‘unexplained’ results. For example, the second simulation changes only the direction of one of the phase waves, but appears to suggest a group velocity that is 10x the phase velocity, although the actual value calculated was [vg=1]. My initial lines of thoughts:
I have attached ‘simulations.pdf’ in order to give some more details of the simulation results for anybody who might be interested in this topic. I have also attempted to attach two gif files to give ‘examples’ of the animations, although the upload size required 500 frames to be reduced to just 25 and so are they very jerky.
Anyway, I would appreciate any help on offer as I am sure somebody must have already resolve these issues. Thanks
- 1-way, non-dispersive phase waves
- 2-way, non-dispersive phase waves – see 2ND.gif attachment
- 1-way, dispersive phase waves
- 2-way, dispersive phase waves – see 2D.gif attachment
Each simulation uses the same basic approach in which the amplitude of two individual phase waves is added, and calculated, for all values of [x], which is then displayed as a single frame in the animation. The process is then repeated for incrementing values of time [t] to create the next frame within the animation. While the first simulation in the list above produced ‘sensible’ results, i.e. [vg=vp], all the subsequent permutations, based on the same algorithms, lead to ‘unexplained’ results. For example, the second simulation changes only the direction of one of the phase waves, but appears to suggest a group velocity that is 10x the phase velocity, although the actual value calculated was [vg=1]. My initial lines of thoughts:
- There is an mistake in the simulation. If so, it is not obvious to me, but see next bullets.
- The equation used to calculate [vg=dw/dk] might have to accommodate the direction of the phase wave velocity [vp]. For example, given that [vp=w/k], such that [k=w/vp], then treating [vp] as a vector having magnitude and direction might suggest that the sign of [k] must also be direction dependent. However, it is unclear whether this would explain the anomalous result associated with the 1-way simulations.
- The equation [vg=dw/dk] would be greater than [vp], when [dw>dk], which appears possible in dispersive media – see equation [1] in attached pdf for details. However, it is not clear this would explain the non-dispersive cases.
- The simulations are only showing an 'pattern shift'. For example, when the phase waves propagate with velocity [c] in opposite directions, the phase relationship between these waves constantly changes with time, i.e. on every frame. So while the ‘shift’ in the beat waveform might suggest a propagation velocity greater than [c], it has nothing to do with anything propagating through the media, rather it simply reflects the rate of change in phase shift between the phase waves.
I have attached ‘simulations.pdf’ in order to give some more details of the simulation results for anybody who might be interested in this topic. I have also attempted to attach two gif files to give ‘examples’ of the animations, although the upload size required 500 frames to be reduced to just 25 and so are they very jerky.
Anyway, I would appreciate any help on offer as I am sure somebody must have already resolve these issues. Thanks