[Q]Do you know about exact form of Group velocity and meaning of Group Delay?

In summary, the question is whether the exact form of Group velocity is \frac{dw}{dk} or \frac{dw}{dk}|_{\bar{k}} and whether Group velocity is independent of K, the propagation number, for a linear function of k.The second question is about the meaning of Group delay and the formula for it, \frac{d\varphi }{dw}. Group delay refers to the time it takes for a wave packet to travel through a medium. The formula for group delay is the derivative of the phase with respect to frequency, which represents the change in phase as the frequency changes. In other words, it measures the rate at which the phase of a wave changes with respect to the frequency.
  • #1
good_phy
45
0
Hi, It is long time to come here sine i graduate University.

Anyway, My question is whether exact form of Group velocity is [tex]\frac{dw}{dk}[/tex]

or [tex]\frac{dw}{dk}|_{\bar{k}}[/tex]

I want to know whether Group velocity is independent of K, propagation number

Becasue Group velocity is 'proper' speed of generally complex wave in comparision with

phase velocity So These exists only one Group velocity of certain wave which should not be

dependent of any K


Second Question is what does Group delay means? I have to drive Formula of Group delay

,[tex]\frac{d\varphi }{dw}[/tex] but i don't know what does it means.


Please Solve my question.
 
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  • #2
good_phy said:
Hi, It is long time to come here sine i graduate University.
Anyway, My question is whether exact form of Group velocity is

[tex]\frac{dw}{dk}[/tex]

or

[tex]\frac{dw}{dk}|_{\bar{k}}[/tex]

I want to know whether Group velocity is independent of K, propagation number

If [itex]\omega[/itex] is a linear function of k, then [itex]d\omega/dk[/itex] is a constant and the group velocity is well-defined.

If [itex]\omega[/itex] is not a linear function of k, then [itex]d\omega/dk[/itex] varies with k and one cannot speak of "the" group velocity, strictly speaking. Nevertheless, if a wave packet includes only a small range of k's, one can speak approximately of a group velocity by evaluating [itex]d\omega/dk[/itex] at [itex]\bar k[/itex], for a certain period of time. After a "long" period of time, the wave packet "falls apart."
 
  • #3


Hello! Thank you for your question. Group velocity is defined as the velocity at which the envelope of a wave packet propagates. It is given by the formula \frac{dw}{dk}, where w represents the angular frequency and k represents the wave number. This formula is independent of k, as you mentioned. This means that the group velocity is the same at all points along the wave, regardless of the propagation number.

Group delay, on the other hand, refers to the time it takes for the peak of the wave packet to propagate through a certain distance. It is given by the formula \frac{d\varphi}{dw}, where \varphi represents the phase of the wave. Group delay is important in signal processing and communication systems, as it affects the timing and accuracy of data transmission.

I hope this helps to clarify the concepts of group velocity and group delay for you. If you have any further questions, please don't hesitate to ask.
 

1. What is the exact form of Group velocity?

The exact form of Group velocity is given by the expression vg = dω/dk, where ω is the angular frequency and k is the wave vector. It represents the velocity at which the overall shape or envelope of a wave packet propagates through a medium.

2. How is Group velocity different from Phase velocity?

Group velocity and Phase velocity are two different ways of measuring the speed of a wave. While Group velocity represents the speed at which the overall shape of a wave packet propagates, Phase velocity represents the speed at which the phase of the wave propagates. In other words, Phase velocity is the speed at which the peaks and troughs of a wave move, while Group velocity is the speed at which the entire wave packet moves.

3. What is the meaning of Group Delay?

Group Delay is a measure of the time it takes for a wave packet to travel through a medium. It is defined as the negative derivative of the Group delay with respect to the angular frequency: τg = -dω/dvg. It represents the amount of time a wave packet is delayed compared to a reference wave packet with a different frequency.

4. How is Group Delay related to Group velocity?

Group Delay and Group velocity are closely related. They are both measures of how a wave packet propagates through a medium. Group Delay is directly proportional to Group velocity, so an increase in Group velocity will result in a decrease in Group Delay and vice versa.

5. What are the applications of understanding Group velocity and Group Delay?

Understanding Group velocity and Group Delay is important in many fields, including optics, acoustics, and signal processing. It is used to analyze and manipulate wave propagation in various mediums, and is also important in the development of technologies such as radar, telecommunications, and medical imaging.

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