Group Velocity

 

 

A signal that transmits information normally have a small frequency interval (lateral bands) around the high frequency carrier. This  signal is a “frequency” group and forms a wave packet. The group velocity is the propagation velocity of this wave packet (or group of frequencies).

In the following image we can observe the group velocity (ug) and phase velocity (up) in a electric field wave:

group-and-phase-velocity

As we did in the post about the phase velocity, let’s solve some exercises.

Exercise 1:

Two waves travel through a long and straight wire. Their wave functions are y1 = 0,005 . cos (6,7x-550t)  and y2 = 0,005 . cos (7,0x-580t) being x and y in meters and t, in seconds. Write the resultant wave function. What is the group velocity?

The resultant expression is:

group-velocity

so we can calculate the values of y and obtain:

group-velocity-parameters

Therefore, y will be:

group-velocity-steps

The group velocity will be:

 

group-velocity-result

Exercise 2:

The group velocity concept appears when considering the superposition of harmonic waves with different frequencies in a dispersive medium; thus, in those media where the frequencies are propagated with different wave velocities. Their value is given by the following formula:

group-velocity-derivative

where w=w(k) is the so-named wave packet dispersion. If the phase velocity is v(v=w/k), the following relation is met:

group-velocity-disperssion

and from the expression above, we obtain:

group-velocity-simplication

Therefore, as a monochromatic wave has, by definition, a unique angular frequency, w, we will have:

group-velocity-zero

Comparison between Phase Velocity and Group Velocity

In the previous exercise, we have seen a case where the group and phase velocities are equal. In order to go deeper, let’s recall the definition of group velocity and let’s give it a more interesting meaning by relating the concept with the transmission of information.

Therefore, we can state that the movement of a group of narrow-band waves can be considered as a wave train of sinusoidal waves (blue line of points, in the picture below) with a central frequency and waveform number, while the envelope (continuous red line in the picture below) slowly changes in the space and time.

group-and-phase-velocities

The speed of the envelope is the group velocity. This velocity corresponds with the velocity of the wave packets. From the point of view of the dynamic, the group velocity has a physical relevance due to the rate of the energy transportation, which make this concept being more important than the phase velocity.

Now, let’s have a look at the phase and group velocities relationship:

When the group velocity is equal to the phase velocity (red point in the image below):

phase-velocity

As we saw in exercise 2, this happens in the monochromatic light.

When the group velocity (red line) is smaller than the phase velocity (black point):

group-velocity-point

As we can observe, this means that the phase of a wave changes faster (the wave travels faster) than the envelope of the waves group.

Finally, we have the case when the group velocity (black point) is greater than the phase velocity (red point):

both-velocities

 

Therefore, the envelope (red point) travels faster than the phase velocity (black point).

 

There are some online applets that allow us to practice this concept. For example, on this one, you can make several simulations (it’s in Spanish, but it’s quite straight forward 🙂 ):

group-velocity-app

 

Conclusions

In telecommunications engineering, one of the most important topics that we study is the transmission lines. In this study, we analyze the parameters of the lines but also the behavior of the waves that travel through them.

In this post and the previous one, we have study one of the most relevant aspects of that behavior. In fact, the phase velocity is defined as the velocity the wave propagates through a line. The group velocity is the velocity of a wave packet (for instance, an impulse) propagates through the transmission line.

Any questions or comments will be very welcomed 😀

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