What is Wave?
A Wave, in simple words, can be described as a mere disturbance along any plane surface. For instance, consider an ocean, where the Waves move along the ocean surface from one place to another. The crest of a Wave moves at intervals and therefore travels a specific distance over a specific time period. In this chapter, you are dealing with Phase and Group Velocity, which are properties of Waves. When a Wave moves in Groups, the speed or Velocity at which it moves is known as the Group Velocity.
Furthermore, you should also be able to differentiate and establish the relation between Phase and Group Velocity in physics. This will help you in clearing your concept while preparing you for advanced understanding.
What is Group Velocity?
The Group Velocity of a Wave is defined as the Velocity at which an entire envelope of Waves moves through a medium. A most common Example, in this case, can be that of throwing stones in a water body which causes multiple Waves on the surface of water.
On throwing the stone, a ripple is created around the point where the stone drops. The ripple is formed of small Wavelets which propagate away from the dropping point in multiple directions. Here, a Wavelet having the shortest Wavelength propagates faster than others.
However, to understand what Group Velocity is, you should also have an idea of simple harmonic motion too. This will help you in understanding the concepts better and in a more natural way.
What is Phase Velocity?
Considering the fact that a Wave consists of two significant parts crest and trough, its Phase Velocity is also dependent on the same. Students should have prior knowledge of it to understand what Phase Velocity is. It is the Velocity at which a specific component of a Wave, say crest, propagates in space. This feature or Velocity is directly dependent on the time period and Wavelength. Alongside, you should also be clear about the relation between Phase Velocity and Group Velocity. For such understanding, knowing the mathematical formula or representation is highly beneficial.
What is the Relation Between Phase Velocity and Group Velocity?
To dig deeper into the concept of the relation between Group Velocity and Phase Velocity, take a look at the mathematical expression explained below.
The expression for Phase Velocity is presented below -
\[V_{p} = \frac{\lambda}{T}\]
Here, \[V_{p}\] is the Phase Velocity, \[\lambda\] (read lambda) is the Wavelength, and T is the time period.
The expression for Group Velocity is -
\[V_{g}= \frac{{\delta}w}{{\delta}k}\]
Here, \[V_{g}\]is the Group Velocity, \[{\delta}\]w is the angular frequency of the Wave, and \[{\delta}\]k is the angular Wavenumber.
Therefore, students should first define Group Velocity and Phase Velocity in their respective mathematical formats. Consequently, they can relate both these velocities in the following manner -
\[V_{g}\] = \[V_{p}\] + \[k\frac{dV_{p}}{dk}\]
Among the valid points to note here while you define Phase Velocity and Group Velocity, most crucial is that both are directly proportional to each other.
To know more about Phase Velocity and Group Velocity, you can look into our online learning programs. There you will get access to high quality study notes drafted by our eminent faculties. You can also get an in-depth learning experience to prepare well for your Exam. You can also download our Vedantu app which provides easy access to Phase Velocity and Group Velocity pdf and others, allowing you to access study material at any time.
Relation Between Group Velocity and Phase Velocity
Students must have got an insight into the Group Velocity and Phase Velocity from the above-given text. You might also have understood the relationship between the Group and Phase Velocity.
It might have helped you to gain confidence after enhancing your understanding of the topic. However, a few of you might still be struggling to crack the sure-shot trick to understanding chapters.
Let's get to know some of the tricks to unlock the best strategy.
Concepts, not Words
Just like there is a secret ingredient in a dish that enhances its taste and makes it 10 times better, similarly, there is a secret ingredient in theory cum practical subjects like Physics, which says that theory and numerical both should be aced by visualizing and understanding its key concepts. Understanding the concepts and not cramming them helps you a lot because it prepares you even for the twisted questions. Another very important fact that you should always keep in mind is that the human mind loves stories, so you shall try and make an interesting narrative of things which you are writing and this way, you will be able to retain things for a longer time.
Going Beyond Books is Also Important
Always try to look for resources other than your textbooks. It may sound rubbish but it isn't. You should explore more online videos, study material, tests, graphs, flowcharts etc. The Internet is considered to be a distracting thing but sometimes it may help you try to use it for your benefit. It offers you a number of learning sites which gives you a number of options to choose from. Use the internet as a weapon in preparation time. It provides everything that you could ever need. If you like to prepare while someone is explaining it to you, you may try video explanation, if you get a better understanding by reading, you may go through the book and then the study material available online and whenever you feel like your preparation is complete, you may try attempting online quizzes which are easily available.
Group Study
The Group might help you achieve milestones especially when you consider the theoretical subjects. You just need to understand what is the best for you that can help you understand things better. And if you are a person who gets things when you are discussing, Group study is the best option for you. But, make sure that before opting for the Group study, you have at least studied the text once, individually. Later, you may discuss things with each other, even the doubts and topics which you or your friend is finding difficult.
Repeated Recalling of the Concepts
Recalling all the concepts time and again is important as it gives you better conceptual clarity which is very essential for the subjects which include theory. Giving all the concepts equal time and heed is very important. Theory may be a little boring but when you make a story out of it using Examples, it makes things much easier and interesting for you. Memorizing all the concepts with the help of Examples and mnemonics is also an efficient way. Always try to complete the learning of concepts on time and also with sincerity. Make an efficient study plan and keep in mind the targets that you've set for yourself.
Best Way to Remember- Write it Down
The best way to remember something is to learn and write it down. It helps you to remember the concepts for a longer period of time and that too, easily. You must have observed this from your parents and grandparents that they always write whenever they want to remember something and its benefits are not unknown to any. Writing helps you to memorize and understand the concept. Make it a habit to write all the points and that too, in your own words. Hence, you shall always learn, write, check and then plan your next course of action.
FAQs on Relation Between Group Velocity And Phase Velocity
1. What is the basic definition of phase velocity?
Phase velocity is the speed at which a specific point of constant phase on a wave, such as a crest or a trough, travels through a medium. It essentially describes how fast the shape of a single-frequency wave propagates. The formula for phase velocity (v_p) is the ratio of angular frequency (ω) to the wave number (k): v_p = ω/k.
2. What is group velocity and how does it differ from phase velocity?
Group velocity is the speed at which the overall shape or envelope of a wave packet moves. A wave packet is a collection of waves with slightly different frequencies. The key difference is that while phase velocity tracks a single point on one wave, group velocity describes the speed of the entire group, which is what carries energy and information.
3. What is the fundamental mathematical relation between group velocity and phase velocity?
The group velocity (v_g) is mathematically defined as the derivative of the angular frequency (ω) with respect to the wave number (k). The formula is: v_g = dω/dk. This relationship shows that the group velocity depends on how the frequency of the wave changes as its wave number changes, a property known as dispersion.
4. Can you provide a real-world example to explain the difference?
Think of ripples spreading out after you throw a stone into a pond. You might see small, individual ripples moving quickly; this is the phase velocity. However, the main disturbance, or the 'group' of the largest ripples, moves outwards more slowly. The speed of this main group of waves is the group velocity.
5. Under what conditions are group velocity and phase velocity equal?
Group velocity and phase velocity are equal only when waves travel through a non-dispersive medium. In such a medium, the phase velocity is the same for all frequencies. Since all the waves in the packet travel at the same speed, the group as a whole moves at that same speed. An example is light travelling in a vacuum.
6. What happens to these velocities in a dispersive medium?
In a dispersive medium, the phase velocity depends on the frequency of the wave. This causes different waves in a packet to travel at different speeds, making the group velocity different from the phase velocity. This is why a prism splits white light into different colours—glass is a dispersive medium where the speed of light depends on its frequency (colour).
7. Why is group velocity often considered more physically important than phase velocity?
Group velocity is considered more physically significant because it represents the speed at which energy and information are actually transmitted by a wave. The phase velocity can, in some special cases, be greater than the speed of light in a vacuum (c), but this does not violate the theory of relativity because no signal or energy is being transferred at that speed. The group velocity, which carries the signal, can never exceed c.
