phase speed & group velocity

[skg]

Dear all,
I have some confusion regarding the concepts phase speed (v_p) & group velocity (v_g).

First let me define these terms.

v_p = omega / k

where omega is the angular velocity and k is the wave number.

v_g = d(omega) / d(k)

where d( ) represents the differential of the quantity inside the brackets.

My doubt is whether v_p and v_g are vectors or not.
Please give justification for any answers based on the above equations and physical point of view.
Thankyou in advance for the help.
Regards,
Sandhya.

[aboobacker]

Hi,

You can get some information from the link: http://www.oceanographers.net/forums/showthread.php?t=1071

with regards,

[vinu]

Dear all,
I have some confusion regarding the concepts phase speed (v_p) & group velocity (v_g).

First let me define these terms.

v_p = omega / k

where omega is the angular velocity and k is the wave number.

v_g = d(omega) / d(k)

where d( ) represents the differential of the quantity inside the brackets.

My doubt is whether v_p and v_g are vectors or not.
Please give justification for any answers based on the above equations and physical point of view.
Thankyou in advance for the help.
Regards,
Sandhya.

Let me add these also,

Both the phase velocity and group velocity are vectors. Both has its
own magnitude and direction. The group velocity is smaller than the
phase velocity. However, the direction of group velocity can be in
same or opposite direction of the phase velocity.

Your mathematical definition of the group velocity is correct.
At the same time, it is important that, How we make a conecpt on
these two physical measurements.

I have tried an explanation for the conceptual understanding of
phase velocity and group velocity in this post (the post is bit old), (http://www.oceanographers.net/forums/showthread.php?p=1579)
with some "animation" of wave motion. I think that will help you
increase your confusion further !
(The animation may take a while to download completely).

In that post, the illustration is given for a packet of wave with a form
eta = A(x,y,t).exp.i(kx+ly-w.t). If one want distinctly to see the group
velocity and phase velocity in the real ocean, a simple experiment
can hlep it. Somewhere in the modeling section, a simple
dynamical model is provided. Perturb that model with an oscillating
meridional wind (say; v=V.sin(wt)) in an equatorial beta plane and look at the
evolution of waves. You may end up with Mixed-Rossby-Gravity
waves, emanating from the perturbation, which has westward moving
phase velocity and eastward moving group velocity.

-Thanks

-

[skg]

Hi Vinu and Aboobacker,
Thanks for your reply.
I think the quote by Vinu

" Both the phase velocity and group velocity are vectors. Both has its
own magnitude and direction. The group velocity is smaller than the
phase velocity."

is not correct. The group velocity is a vector and the phase velocity is not a vector. I am sure about this as this was told to me by Shetye (Director, NIO), though he didn't give me an explanation and asked me to think about it :( . I am unable to digest it, as I think dy/dx can not be a vector when x and y are scalars. Thus according to me, c_g can not be a vector from a mathematical point of view because omega and k are scalars.But text books say c_g is a vector! Thus in utter cofusion I posted the doubt here:confused: .
Also group velocity can be smaller than(deep water) or equal to(shallow water) the phase velocity.The post by Aboobacker
http://www.oceanographers.net/forum...read.php?t=1071 (showthread.php?t=1071)
will explain this point.
Anyway thankyou for the discussion.
skg.

[deba]

But, i would definitely like to have clarity on the matter. Could we have some clarifications on this?? I believe they should also be backed by literature references or atleast a detailed mathematical workout. Dr. Vinu, will it be possible to look into that.

[vinu]

Hi,

I think, the statement made by skg is too general and it should
viewed with caution and logic. In what way we able to say a
phase velocity is scalar need a logical view. The "key" is the
dispersion relation. I will try three cases to reach my point.

A classical wave theory yields the dispersion relation
w^2=g.k.tanh.k.H; assuming the notations are well known and
proceeding to the following cases.

Case:1. Deep water waves
Here, tanh.k.H tends to unity. Thus
w^2 = g.k
w/k = g/w
or the phase speed C = g/w; Since "g" and "w" are positive definite
quantities, C = +ve always. In this way, it makes logic to view
the phase speed as a Scalar (i.e. has no specific direction).
What is this "no direction means". Inorder to understand that,
just imagine what happens if a set of wave generated in
the open ocean. It can virtually propagate all the direction in space
and virtually equivalent to "no specific direction". But still you face
the difficulty to find, how you distinguish a wave moving to the
right and a wave moving to the left hand side. For that, there is
no problem for one to assume a postive and negative direction
for the phase speeds, or count the "phase speed" as vectors.

Case:2; Shallow water waves.
Here also, tanh.k.H tends to k, and the dispersion
relation yields w^2 = g.k^2 or
Phase speed C^2 = g.h. Here also, the phase speed is a
positive definite, and you again come to the concept of scalar.
But remember in your mind that, it is our manifestation to
take phase velocity as vectors to distinguish the direction of
wave propagation, i.e. either the wave propagating
to the right and the wave propagating to the left.

Case:3. Planetary waves.
When you come to the planetary wave regime (i.e. waves
which feels the rotation effects),
the dispersion relation always have rotation component in it, more
generally it is in the form (f^2/k^2). So the point is, as long as the wave
feels the rotation, it leaves a phase speed with
dependency on "wave numbers". Since the wave number plane
is a space plane, it has the direction. Here you will get
phase velocities with "specific directions" according to
the choice of "wave number" and you can derive both
positive and negative phase speed, accordingly. For
example in the equatorial beta-plane, you have
strict constraints on phase speed (including
the direction) depending on wave number.


The interesting thing is, even if the shallow water wave speed
C^2 = g.H is direction independent, the rotation effects makes
the wave disperse under rotation, and give "direction specific"
waves. But this is only for those, who deal with waves of time scale
compared to the rotation of the frame (earth). For example
the internal gravity waves are direction specific and depends on
the stratification and rotation.


I think, the "key" is also on the "definition of vector". If you spend
some time on "What the way you want define a vector", you may
get a good view on this problem, I think so.




-Thanks


.

[bhatt.vihang]

hello,

As magnitude of vector is a scalar quantity and mostly we deal with numbers, we follow phase and group velocity are scalars. However, velocity has magnitude and direction both. It is also evident from the equation of phase velocity (wave vector dependence). The wave vector defines the direction of the wave. In shallow water waves, we approximate the equation to derive the magnitude of waves. However, one has to incorporate details of wave number when directions are to be determined. The dispersion relation for shallow water waves just says "waves are nearly non dispersive".

[skg]

Thank you Vinu for the detailed reply.
When I referred to waves, I was having wind generated surface gravity waves (wind waves and swells) in mind, not planetary waves. Sorry for not mentioning it earlier. And my confusion is only from the mathematical point of view. In mathematics
scalar/scalar =scalar
vector/scalar=vector
d(scalar)/d(scalar) = scalar....
When I think in terms of Physics behind it, I also feel that both c_g and c_p should have direction. But as I already mentioned, I was told that it is not the case.
I will welcome further discussions/literature on this.
Regards,
skg.

[vinu]

Hi,

I think Bhatt has given a concise answer for skg's question.

Also, it is noteworthy that, evenif the phase speed has a wave
vector, the usual vector decomposition rules does not
satisfy for the phase vector. That means, for a two dimensional wave
in xy-plane, the phase speed in x-direction is, Cx = w/k and in y-direction,
is Cy=w/l and the resultant phase speed is, C = w/sqrt(k^2+l^2)
If we decompose "C" into C.cos(theta) and C.sin(theta), we can not
reach Cx or Cy. The physical reason is, Cx referes to the
speed with which the intersection of phase line of the wave advances along
X-direction and NOT the speed of the projection of Phase vector
along X-direction. Similarly, Cy referes to the speed with wich the
intersection of phase line of the wave advances along Y-direction.

A detailed pictorial description may find from the GFD book
by Pedlosky (1982).

-Vinu

-

[skg]

Thanks Vinu and Bhatt.
Vinu, can you make the definitions for Cx and Cy clearer?
In Cx=w/k, k is wave number, not wave vector right? I want to make it clear as w is scalar, wave number is scalar and wave vector is vector &
scalar/vector = Not defined.
In Cy= w/l , what is l? Can you make your point below :

"If we decompose "C" into C.cos(theta) and C.sin(theta), we can not
reach Cx or Cy. The physical reason is, Cx referes to the
speed with which the intersection of phase line of the wave advances along
X-direction and NOT the speed of the projection of Phase vector
along X-direction. Similarly, Cy referes to the speed with wich the
intersection of phase line of the wave advances along Y-direction."

a bit more clear? I don't understand what is meant by phase line. I know it in the case of an autonomous differential equation. But can you tell me in this context?
I don't have access to the book by Pedlosky. Can anybody give me a soft copy or post it here?
Thanks and regards,
skg.

[vinu]

Hi again,

Evenif wave number is, by definition, number of waves in unit distance (space),
we may account that direction of space when we want to specify the direction
of wave propagation.

The 'k' and 'l' are wave numbers in X and Y directions.
For a 2D wave, if we measure the wave characterestics along two axes
(i.e. X and Y) we get two wave speed. They may or may not be equal,
according to the geometry of the axis.

Phase line of a wave is a "line through constant wave
proporties", for example a line through constant height. It is equivalent
to the "wave front".

A figure is attached. It is partially redrawn from Pedlosky book. In that
pciture, three items are shown. (1) A set of waves radiating from a
source at the center and its phase line (b) A measure of wave number
in XY plane at one part of the wave (c) A Physical view of wave propagation
and "lines of constant height" (called phase lines).

The phase speed along X-direction of the reference frame is the
speed at which the "intersection of phase lines" at X-axis
advances in time. Similiarly for the phase speed along Y-axis also.
Now try to project the mean Wave Vector "K" onto X-axis and
try to find its speed. That will be different from the former calculation.

Key parameters,

k = 2.pi./lamda_x
l = 2.pi./lamda_y
K = sqrt (k^2+l^2)
C = w/K
Cx = w/k = C.K/k (where BIG K=mean wave vector, BIG C=mean phase speed)
Cy = w/l = C.K/l


http://www.oceanographers.net/forums/attachment.php?attachmentid=93



Thanks

-

[skg]

HI Vinu,
Thank you very much for the nice explanation and clarifications. I will be getting a copy of Pedlovsky's book soon and then I can explore it further.
Regards,
Sandhya.