Phase Velocity & Group Velocity Video Lecture - Civil Engineering (CE)

senior plasma physics lecture 7 when you
disturb a plasma in some way it will
produce waves so here we look at the
simplest possible plasma oscillating
mode in doing that we'll introduce
concepts such as the phase and group
velocities then we look at basic plasma
oscillations and the associated
frequency of that oscillation so how do
we mathematically represent waves in
plasma physics there are two types of
waves electromagnetic and electrostatic
or plasma waves can be represented by
some kind of sinusoidal function are
given in this form where K is the wave
number which is 2 pi on the wavelength
and Omega is the angular frequency where
F is the frequency
notice that the periodic function that
we've just given can be expanded in this
form as you can see this is a periodic
function but in complex number notation
we keep it in the exponential form for
mathematical convenience as will become
apparent in future lectures when we
start to use this function the amplitude
e in this function can be either
electric or magnetic field depending on
the situation
let's define the phase velocity of a
wave to do that recall the previous
expression for the wave function we can
rewrite this function in terms of the
angle theta for simplicity now imagine a
graphical representation of this wave
function and it is moving to the right
with velocity V theta now imagine that
we are standing on the top of the crest
of that wave and moving along in the
same direction with the same velocity V
theta that means from our perspective
the phase angle theta of the wave
doesn't change so the rate of change of
theta from our perspective is zero now
substitute back the expression for theta
and carry out this derivative it
simplifies to this rearranging this
expression gives us the velocity of the
wave in terms of Omega and K this
velocity is known as the phase velocity
whenever there is a superposition of
many waves with various frequencies and
velocities such as this which is really
just a superposition of two ways we have
sine a velocity - this bundle of waves
known as the group velocity in this case
we can look at it as the velocity of the
envelope VG but recall from the last
slide that there is also a phase
velocity of a wave so the velocity of
the individual crests in that envelope
is called the phase velocity and the
group and phase velocities do not
necessarily have to equal it's possible
to derive the group velocity of a
superposition of waves but the steps
that need to be taken are a bit more
involved than the derivation of the
phase velocity the end result is that
the group velocity is defined as the
Omega decay
now let's look at one of the most
fundamental oscillation modes of the
plasma imagine you had a group of ions
and a group of electrons here the AI is
represented by red and the electrons
represented by blue if they are free
charges as they usually are in the
plasma then they'll move towards each
other but the ions are so much more
massive than the electrons that it's the
electrons that end up doing most of the
moving as the electrons are drawn
through most of the ions the chances of
all those electrons being captured by
the ions is very small so most of the
electrons will swing through to the
other side the electrons will eventually
stop and be attracted to the ions again
and the whole process repeats this
oscillation frequency is known as the
plasma frequency quite often it's
written in its angular form in many
textbooks it's referred to as the plasma
frequency but the oscillation frequency
is not really Omega will refer to it as
the plasma angular frequency where M
naught is the plasma density and M here
is the electron mass the derivation of
the plasma frequency uses the fluid
equations and the Poisson equation which
is given by this expression we also
assume that we're dealing with cold
electrons so their thermal energy KT is
zero because the ions are so much more
massive than the electrons we assume
that they are fixed in space