Displacement of waves (Part - 10) - Waves, Class 11 Video Lecture

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till part 9 before going ahead with part
10 now that we have talked about
amplitude and phase as well as
wavelength of a wave let us talk about
displacement in a travelling wave I
introduced travelling wave right so now
we will see how to be determine or how
to we denote the displacement in a
travelling wave because that a
travelling wave is always traveling it
is always moving so it is always
undergoing some displacement in space
with time so how do we get to know that
displacement well amplitude and phase
these are the two parameters which
together describes the displacement of a
wave because amplitude tells us the
maximum displacement of a particular
particle of the medium and face tells us
the state of motion of a particular
particle so together amplitude and phase
will tell us the complete displacement
of the wave now this displacement
function is a periodic function in space
and time now displacement takes place so
here the displacement takes place along
the y axis that is the displacement I am
talking about the displacement of the
particles in the medium so their
displacements takes place along Y axis
therefore we generally denote
displacement as a function of X and T
but we denote it by Y now if you think
of both the waves a transfers as well as
longitudinal as I told the displacement
takes place along Y axis so we denote
displacement as Y which is a function of
X and T what does X ay
is nothing but the propagation of the
wave along the x-axis now when we talk
about a transverse wave here we are
talking about a transverse wave the
displacement of the particle takes place
along x axis I'm sorry the wave
propagates along x axis and the
particles oscillate along y axis so in
that case we say Y as a function of X
and T is denoted by a sine KX minus
Omega T plus 5 so this becomes the
expression for displacement so
displacement is a periodic function in
space as well as time so this can be
represented as a sine function it can
also be represented as a cosine function
because both sine as well as cosine
functions are sinusoidal functions and
it can also be represented as a
combination of sine function and cosine
function I am sure you don't have any
doubts as why do we represent this as a
sine function if you look at this
expression you can see this represents
nothing but the same displacement
equation as used in oscillatory motion
that's because here also all the
particles of the medium exhibit an
oscillatory motion whether we consider
transverse wave or we consider
longitudinal
leiva's even in longitudinal wave what
happens is the particle oscillates right
the only difference is here in this case
the particle oscillates parallel to the
direction of the wave so that is the
only difference otherwise here also the
particle oscillates and we represent it
as in the form of an equation a sine KX
minus Omega T plus Phi
while discussing oscillatory motion I
have already told you that why is it
possible to express displacement as a
combination of sine and cosine functions
so what are the different ways by which
we can represent this we can also
represent it as B cos KX minus Omega T
plus
five as a cosine function this can also
be represented as a combination of sine
and cosine function that is a sign KX
minus Omega T plus Phi plus B cos KX
minus Omega T plus 1 so these are the
various forms in which the displacement
of a traveling wave can be represented
now let us look at the same equation and
derive or reach at an important
conclusion for a travelling wave which
is traveling along the positive x
direction the expression is y is equal
to a sine KX minus Omega T plus Phi
whereas for a travelling wave which is
traveling along negative x-axis
the expression is a sine KX plus Omega T
plus 5 just to observe for along
positive x-axis it is minus Omega T
along negative x-axis it is plus Omega T
now you must be thinking why is it so
when it is moving along positive x-axis
why did we only change the sine of Omega
T let us derive this now this expression
you can also write it in this form this
expression is basically Y is equal to a
sine KX minus Omega T plus Phi Phi is a
phase constant we will study about it
later
so as of now we can ignore the phase
constant and I am sure you have studied
about it in your oscillatory motion as
well and you know that we can consider
the initial phase constant as zero so
let us write this equation in terms of X
so this becomes Y by a is equal to sine
KX minus Omega T or we can write it as
sine inverse Y by a
is equal to KX minus Omega T or this can
be written as K X is equal to sine
inverse Y by a plus Omega T or this can
be written as X is equal to 1 by K sine
inverse Y by a plus Omega T by K so this
is the expression of X in terms of Y and
T now what happens as time T increases
the value of X also increases no matter
whatever the value of this term whether
it is smaller or bigger whatever it is
whether it is positive negative whatever
it is with this expression you can say
that the value of x increases as the
value of T increases that means as time
increases value of x increases so the
value of x increases means the X moves
along positive x-axis you consider this
value as anything and you see that you
as you increase T the value of X also
increases now on the contrary if you
decrease the value of T then the value
of X keeps decreasing that is the that
as X moves towards the negative x axis
right so here we see that as T increases
X also increases so here we conclude
that X keeps increasing with increasing
time whereas in this case what happens y
is equal to a sine KX plus Omega T I am
ignoring Phi so this is again y by a
sine inverse Y by a is equal to KX plus
Omega T or we can write x is equal to 1
by K sine inverse Y by a minus Omega T
by king so in this case what happens as
the time T increases as this value keeps
in as T increases this value keeps
increasing that means a greater amount
keeps getting subtracted from some
amount that means the value of X keeps
decreasing so here in this case X keeps
decreasing with increasing time here X
increases with increasing time so
therefore this denotes that X moves
towards the positive x axis and here the
wave travels along the negative x axis
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