# Zero Group Velocity: What Does it Mean?

• Zarquon
In summary, an infinitely long mass-spring transmission line has a dispersion relation with a group velocity of zero. This means that the current in a pure standing wave is constant at any given time, with only a variation in magnitude. This is different from a traveling wave, where the phase of the current varies with position, and a zero phase velocity would imply faster-than-light energy propagation.
Zarquon
An infinitely long "mass-spring transmission line", consisting of masses (m) connected by springs (spring constant s) obeys the following dispersion relation:

ω = \sqrt{4s/m} sin(kd/2).

The group velocity is

dω/dk = d/2 \sqrt{4s/m} cos(kd/2).

What does zero group velocity "mean" for this system?

This is a question from an exam i took recently. The answer was that you'd get a standing wave, which doensn't make sense to me. My answer was that you'd get stationary packets of traveling waves (like this: http://www.falstad.com/dispersion/groupzero.html ) (whereas you can get standing waves at any frequency. I still think I'm right, so if someone here can shed some light I'll know whether I should start arguing with my professor.

To see why a pure standing wave has a zero phase velocity, let's look at the equation for the current, I, in a standing wave.

I(x,t)=Imax[sin(kx)][cos(wt)] ;where w is omega=2*pi*freq and 0 < kx < 180 degrees

At any time, t, the phase of the current is a constant, i.e. the phase of the current does not vary with position x as it does for a traveling wave. In a pure standing wave, the position, x, only affects the magnitude of the envelope of the current. Since the phase of the current is a constant over any 0-180 degrees, the difference in any two phase measurements at different locations along that 0-180 degree path is zero. Note that a phase delay of zero for a pure traveling wave would imply faster-than-light energy propagation.

## 1. What is zero group velocity?

Zero group velocity refers to the case where the group velocity of a wave or particle is equal to zero. Group velocity is the speed at which the overall shape or envelope of a wave or particle travels, and zero group velocity means that this shape is not moving.

## 2. How is zero group velocity different from zero phase velocity?

Zero group velocity and zero phase velocity are two different concepts. Zero phase velocity refers to the case where the phase velocity of a wave or particle is equal to zero, meaning that the wave or particle is not moving at all. In contrast, zero group velocity means that while the wave or particle is still moving, the overall shape of the wave is not changing.

## 3. What are some examples of phenomena with zero group velocity?

One example of a phenomenon with zero group velocity is a standing wave, where the overall shape of the wave appears to be stationary even though the individual particles are still moving. Other examples include certain quantum systems and materials with special properties, such as photonic crystals.

## 4. What does zero group velocity imply about the behavior of waves or particles?

Zero group velocity is indicative of a situation where the wave or particle is in a state of resonance, where energy is being stored and exchanged between different modes of the system. It can also suggest that the system is at a critical point, where small changes in external conditions can lead to significant changes in the behavior of the system.

## 5. How is zero group velocity relevant in practical applications?

Zero group velocity has important implications in various fields, such as quantum mechanics, optics, and acoustics. It is used in technologies such as photonic devices and sensors, and is also being studied for potential applications in areas like quantum computing and information processing.

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