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Today, we are going to basically continue
what we have done yesterday, that is Regular
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Water Waves part. First, let me begin writing
some of the expression that we have developed
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yesterday.
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What we had was, in deep water etcetera, so
this is what we had in deep water, what we
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have done was yesterday these expression.
And I as I mentioned to you, if you look at
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these expressions, all parameters are essentially
sinusoidal both with respect to special parameter
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x and with respect to time parameter t.
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So, we will continue today on building upon
these. First of all may be, I will introduce
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you the concept of group velocity and energy
in waves, keeping it to aside. See, one thing
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that we realize that, this theory is based
on the linear theory. Now, linear theory implies
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the the linearity implies that, if I have
for example, a system like a black box, if
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I have an input I and output O.
Let us say, input is I 1 output is O 1 and
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an input is I 2 output O 2, if the system
is linear then, I 1 plus I 2 would give me
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output of I O 1 plus O 2. Or conversely, k
I would give you k O in other words, O and
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I are linearly invited. What it means? If
the box is linear, then I can have solution
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number 1 and 2 and 3 and I can add them all
up and they still remain the solution of the
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system.
Now, if I look at this linear water wave theory;
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I have a wave height eta 1 let us say A 1
cos k 1 x minus omega 1 t. And if I have another
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wave, A 2 eta 2 as A 2 cos k 2 x minus omega
2 t, what it means is that, I can add this
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two up and eta 1 plus eta 2 becomes the sum
of these two. But, the important point is
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that, eta 1 plus eta 2 will also be a wave
possible wave with satisfy the same boundary
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value problem; which means that, linear waves
can be super imposed one other and what I
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will get is also a possible linear wave. So,
there wave impossible that is the most important
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point and you will find out that, it has really
a very high consequence as we go along the
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subject.
So, therefore, the reason we can do that,
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a very simple example. Let us say, this one
it was the solution of phi 1, what was phi
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1; it was satisfying the equation del phi
equal to 0. This was the solution of phi 2
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and I have the equation phi 2 is 0. But now,
this equation is linear, because we know that,
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del square phi 1 plus phi 2 is also 0 simply.
So, therefore, like that we will find out
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that, phi 1 plus phi 2 would also represent
a feasible fluid motion and since phi 1 solution
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is eta 1. Of course, after the boundary condition
phi 2 is zeta 2, we end up getting the eta
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1 plus eta 2 becomes feasible solution.
Now, this is very important because, what
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we will do now is that, we will consider two
waves and add them together and see that,
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if the waves differ by a very small amount
of frequency then, we end up getting a group
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wave. Now, before doing that I will like to
bring back to this eta expression. And I will
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tell you that, it is quite often, when I have
the sinusoidal function easier to write as,
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real part of A e power of i k x minus omega
t as eta, why because. Or I can also write
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this as, real part of A e power i omega t
minus k x; both are same because, you see
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we understand that, e power of i theta is
equal to cos theta plus i sin theta and if
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you make it minus, it becomes minus.
So, real part of both; that means, of e plus
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minus i theta is nothing but, cos theta. So,
this is an expression that is sometimes used
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that because, it becomes easier to do operation
algebraic operation very easily. So, we are
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going to use this expression, because it will
be easier.
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So, now, we consider two waves; One is eta
1, which has which is written as real part
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of A 1 it will we can write it exponential
as e 1 minus i k 1 x plus i omega t. Now,
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I am going to add another wave; eta 2, which
is real part of. Now, if you add this two
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up, we end up getting R we do some manipulation
A 1.
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I will tell you, what is this, this is a simple
algebra. If you did that, simply you know
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what I have done is here you write k 2 as
k 1 plus delta k, omega 2 as omega 1 plus
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delta omega; you simply add, you get. Now,
the most interesting part of this here is
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that, see any function which is any function
which is something into e power of i k x minus
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omega t real part of this can be anything
is actually a waveform.
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It is a waveform travelling in plus direction
with amplitude given by this quantity, that
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is by definition and by this is what it is.
Now, you what is this part, let us look at
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this slowly this means actually this expression
means this into cos k x 1 minus omega t, this
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expression let me take a different pen.
See, this expression implies something here
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this star into cos k x minus omega t. Now,
where to plot this, what is that? It this
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is the sinusoidal form, this is nothing but
a sin form of amplitude given by this star,
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this star is this here. So, that means any
expression with something into e power of
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i k x minus omega t is the waveform with that
something as amplitude.
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Now, look at these expression, here I have
got some something into e power of i k x omega
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t into this full thing, if you take the A
1 here. So, that means here I have got a waveform,
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whose basically amplitude is this. So, I basically
I have a wave, which having amplitude of this
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excuse me.
Now, if you look at this part carefully you
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will see that, this itself is a waveform,
because itself has got e power of minus i
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d k x plus i t omega t; basically it itself
a waveform of length d k and period d omega
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and speed d omega by d k. So, this means that,
the amplitude itself is a oscillatory curve
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and itself is moving.
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In other words, if I were to add that two
together; that means, if I were to add this
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wave and add the other one, which is slightly
different see, maybe here etcetera. If I had
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to add this what I end up getting is, actually
a wave which would look something like, which
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has an envelope; the envelope, which is my
A this envelope which is my A star, itself
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is moving with a wave velocity.
So, the envelope itself moves with the speed.
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So, the lots of the student’s lot of people
have difficulty understanding, how the two
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waves which is slightly differing; can actually
cause gives rise to an envelope. Because you
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think that, this is very nice sin curve, this
is another sin curve, when I add the two sin
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curve; which is only differing by a very small
amount in length. How it can actually cause
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a modulation? So, the reason is like this
you know.
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Now suppose, you take this certain length
let us see you know is easier to see in terms
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of length say 10 units. So, every 10 th unit;
10, 20, 30, 40, 50 have a crest. Now, you
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take another one 11 units, what will happen?
11, 22, 33, etcetera in an in 55, this one
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has 55 a that will have 55 a crest you understand,
what I am saying. See, this is you see this
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this way, it is very interesting to illustrate.
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See, if I were to this one two three four
five. So, this 10 say 0, 10, 20, 30, 40, 50,
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60, this is 55. But, the other one we will
have this crest; 11, 22, 33, 44 at 55; it
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will have the top and here this is bottom.
So, what happen if you add this two up, here
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the resulting wave will show actually 0, somewhere
here it will go up, this is exactly, why?
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This modulation will look like etcetera.
You see although its differing slightly, but
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it will have now remember this evolving, so
this entire thing, this front part would also
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appear like moving forward, this this whole
thing will moving look like moving forward.
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So, this is what is called group wave.
And this particular you know this envelope
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and the speed of the envelope is known as
group speed. So, we can also work this group
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speed out very nicely, because we see that,
this particular one in a wave of frequency
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d omega and speed d k. Therefore, the speed
of that envelope is going to be d omega by
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d k, that is by definition.
Now, as soon as d omega d k are not perfectly
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0, you have a group speed. Now, if you make
a wave continuously; it is successive of waves,
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you always have slight frequency difference
as a results, you always end up getting a
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group. Now, how do I get this, now you see
in deep water omega square equal to g k. So,
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d omega by d k equal to or I will say 2 2
omega d omega d k equal to g d k no sorry
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sorry.
Let me write it again 2 omega d omega equal
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to g d k. So, therefore, d omega by d k equals
to g by 2 omega half of g y omega, it is half
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of C; because g by omega was c, we have seen
earlier. So, you see what happen therefore,
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what we end up getting is a group speed, if
I have to call this group speed.
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The speed of that, it becomes half the phase
in deep water excuse me. In the other extreme
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we can show that, when k h tends to 0 that
is very shallow we can actually show that,
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C g tends to C, but that limit is of no interest
to us as I said, for this course sea keeping
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we are mostly concern with deep water cases,
we are not concern with shallow water cases.
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So, now this is very important because, what
happens is that now, the energy of a wave
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always moves with that speed. Let me now come
to now before going to the energy propagation,
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let us now established energy relation itself.
So, groups will be understand.
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Now, let us look at the energy; obviously,
the energy expression if you do kinetic energy
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and potential energy will look to be integration
over volume of see, it is velocity square
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and position. So, it is.
Now, so this is total energy. Actually, if
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I were to do this, if we take a vertical column
of water that is I take a vertical column
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of certain area and look at the energy of
the waves what would happen is that, this
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expression will turn out to be minus h by
eta. In fact, we need not do all these things.
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Now, if you this out, we are not going to
the detail working, but if you work it out,
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it will turn out to be this expression will
turn out to be omega.
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Now, this is how it happens, if you do the
manipulation see this part k eta, what happen
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remember we have said, k eta is small; it
is small amplitude wave theory, eta is height,
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k is lambda inverse of lambda; k eta is proportional
to eta by lambda. So, is now e power of some
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small quantity that to twice e square you
know, if I call k eta to be an epsilon is
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e power of 2 epsilon. So, now this part becomes
very small.
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So, linear theory is consistent to neglect
that, because this is actually second order
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term; because k eta is of order epsilon, 2
k eta is order of 2 epsilon. So, we end up
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getting this relation. Now, this one if I
were to put you all the expression were eta
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you know like eta etcetera, we end of getting
E.
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Again, I am just writing with the derivation
as one-forth rho g A square plus one-forth
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rho g A square cos square k x minus omega
t. In fact, this is my potential energy, this
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is my kinetic energy. And now what happen;
obviously, kinetic energy is you know showing
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this sinusoidal dependence term, because particles
are not constant; they are continuously undergoing
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changes.
Now, what has happened, if I were to integrate
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energy over one period you know like, if I
take a bar here, kind of measure the energy
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from 0 to time, t; total energy in that and
divide by t then, what I will get is, what
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is called average energy over an average surface
area, one unit surface area average energy.
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If you do that, the we end up getting E bar
to be equal to half rho g A square. Actually,
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what you end up getting is that, this becomes
half rho g A square one-forth rho g A square,
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this is one-forth rho g A square. So, the
total is half rho g A square. So, this is
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my average energy remember, this average energy.
Instant energy is different, because of this
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time dependency.
So, average energy over one period; 0 to t
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over a unit area on the surface is this. In
fact, if you see the unit you will find the
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unit of that is energy per unit area, the
unit of that. Now having said that, it turns
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out that if I were to this also fix and attention
of to a vertical fixed observation line, let
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us say.
Some constant line you will find out that,
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energy plus travels to that line, that is
d E by d t turns out to be C g into E bar.
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Again this, we are stating without proving
that there are number of way to established
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that. What it means is most interesting, the
energy travels forward with group speed not
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with phase speed.
So, phase speed has nothing much to do with
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the energy the physical quantity; phase speed
is to do only with the form, this is something
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that we must understand. Whereas, groups speed
to do the physical quantities like energy.
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Particles for example, do not move at all,
they are like a circle, but energy thus move
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forward; obviously, it makes sense because,
if I have calm water here and if I were to
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turn on some wave making device you will find
out that, water is basically moving forward
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and energy will be propagating at group speed.
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So, this part is very interestingly seen in
this way you know like supposing, I tune a
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wave maker at some point of this time you
will find out that, if some time the wave
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is like that; next instants the wave is like
that, so this form would have move at this
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speed, this is actually say, t with time.
So, you know like this is a sometime locational
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front, next instant etcetera.
But, if I were to concentrate on any of the
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phase the form you will find that, this actually
move at twice a speed then, the confusion
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come in peoples mind is that, what happens
to this? Actually what would happen is that,
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as it catches up here, the amplitude becomes
0 and there is no phase speed basically. In
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fact, you simply disappear like these lines
do not exit any further.
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How it happens, why? Now, there is a physical
explanation here, how do I explain this is
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the speed of energy, no it is very simple.
You see, initially this is calm water, if
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I take a line, so at this point I have no
energy at all its calm water, but just next
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point; I have energy because, wave wave is
existing here.
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Now, so naturally next instant this front
has moved here. Obviously, the energy has
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moved up to this point, because just here,
there is no energy. So, this front; obviously,
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carries energy. So, this is exactly why the
wave front is moving with energy. And same
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thing was happen to back, if you switch it;
this back line will also go like that.
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So, this is actually, so classical I keep
telling that it is in fact, on the cover page
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of that book you know that the flap this very
nice actual picture taken from an experimental
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wave you see and photographs. So, this is
extremely important that we have this. Now,
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this is we will find out afterwards, because
we may have a question, why are we studying
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this energy etcetera, where where do they
relate to my ship motion; we will find out
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that, yes there are very very good application
of this eventually, when we talk to ship motion.
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For example, even earlier we have seen in
wave resistance for example, the energy propagation
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is extremely important because, if the energy
was propagating at the same speed as the ship
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then, you end up getting a very large distance,
energy getting trap. Just like in sonic boom
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case of aircraft.
Similar thing would occur in ship motion also
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in spectrum we when we go to that. So, energy
relation becomes very important. Now having
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said, the energy now part, now I want to go
to the one more aspect which becomes very
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important, when we study wave effects on floating
bodies structure, shapes etcetera, that is
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connected to the concept of let me take this
of phase.
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You see here, when I write an equation eta
equal to A cos k x minus omega t well I can
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write this k x minus omega t here, rather
and the graphs looks like this. But now, if
188
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I were to add an angle here phi, what it means
is that? It means that, I actually shift you
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00:24:43,590 --> 00:24:48,330
see what happen rather maybe another one,
I will do better way of doing is eta equal
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00:24:48,330 --> 00:24:55,330
to let me just write this way, A cos k x just
A cos k x let me call initially and phi. What
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00:25:00,080 --> 00:25:07,080
would happen you know that is see here earlier,
when I my phi was 0 at x 0 my eta was A, that
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is when phi is 0; this x here, this A here.
But, when my phi is having some value then,
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00:25:16,650 --> 00:25:22,690
what would happen is that? When x is k x is
equal to minus of that value that is it is
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00:25:22,690 --> 00:25:27,350
somewhere else, this would shift. What it
means is? In fact, I am actually relocating
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the x axis because, I can always write this
as x I can take k out and I can say x plus
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00:25:34,300 --> 00:25:41,300
phi by x etcetera etcetera.
Basically I am changing x to x plus phi by
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k, which means I am changing this origin;
which means that, my my graph would probably
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look like in this case, when x is equal to
minus phi by k then, it is 0 because, this
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00:25:51,300 --> 00:25:58,300
must be 0, so somewhere here. So, it is get
shifted either way this way depends on which
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00:26:00,830 --> 00:26:05,980
sides you take you know.
In fact, I can say that, sin curve and a cos
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00:26:05,980 --> 00:26:11,309
curve is nothing but, phase shifted curve
by 90 degree. So, this shifting of phase is
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very important, why because? What it means
is that, phase tells me the location of its
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00:26:17,860 --> 00:26:23,260
peak with respect to the x coordinate, which
could be of course, in this case physical
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x or time t, depending on what you are looking
at or k x minus omega t.
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00:26:28,350 --> 00:26:33,590
So, when I am writing this expression of eta
earlier, what we wrote that, we say eta equal
206
00:26:33,590 --> 00:26:40,590
to real part of A e power of minus i k x plus
i omega t whichever way you can write it is
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00:26:42,730 --> 00:26:49,730
ok. Now, if I were to add a beta here which
is nothing but, say e power of into e of i
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00:26:54,360 --> 00:27:00,960
beta. Actually this there will be i here that
because entire thing is under I.
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00:27:00,960 --> 00:27:07,960
This is my phase or this tells me with respect
to x is equal to 0, where the t is equal to
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00:27:09,559 --> 00:27:15,020
0, where the peak occurs. Now, this is very
important because, what happen in water wave
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00:27:15,020 --> 00:27:20,620
mechanics;the floating body response subsequently
things do not occur at the same time, the
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00:27:20,620 --> 00:27:25,540
peak value all.
Say, if something happens highest now, something
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00:27:25,540 --> 00:27:32,370
maybe lagging behind it. See, I want to give
you this example maybe you look at this my
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00:27:32,370 --> 00:27:37,220
hand supposing I twist my hand I said that,
before probably, you will see that, if I were
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00:27:37,220 --> 00:27:42,940
to see my hand; this tip point it goes this
way, but this point may not exactly follow
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00:27:42,940 --> 00:27:47,700
that; that means, if this is having a peak
here, if you observe you will find out that
217
00:27:47,700 --> 00:27:52,620
my this one is actually reaching.
See, I go here as I come back, it is going
218
00:27:52,620 --> 00:27:59,280
up you know, if you do this way as at the
instantly of begin to come at the back; it
219
00:27:59,280 --> 00:28:06,280
is still trying to go up. So, that means that
this tips oscillation is not phase centralize
220
00:28:06,530 --> 00:28:12,020
with my top oscillation. So, that would occur
something like see, it is occurring here that
221
00:28:12,020 --> 00:28:16,350
will occur slight later that, will actually
go like that.
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00:28:16,350 --> 00:28:22,980
So, this blue one would be this point and
red one will be this point, oscillation; this
223
00:28:22,980 --> 00:28:28,960
is phase, this is what exactly what we are
talking of phase you know. And this is extremely
224
00:28:28,960 --> 00:28:35,960
important for us from all points of view because,
later on we will find out that things now
225
00:28:37,330 --> 00:28:42,270
occur simultaneously. When a body is for example,
move the wave is moving up, the body may not
226
00:28:42,270 --> 00:28:47,110
move up, all though it is still sinusoidal.
So, phase is extremely important we need to
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00:28:47,110 --> 00:28:53,620
realize that. Now, what does it means, now
let me look back at this first expression
228
00:28:53,620 --> 00:28:59,270
of that is why I wrote this first page, all
these expression here. I will come back to
229
00:28:59,270 --> 00:29:06,270
this in a minute. See here, what happen you
know you see, eta is a cos curve, but w is
230
00:29:08,380 --> 00:29:15,380
a sin curve. Now, if I were to plot eta and
w now, what would eta look like?
231
00:29:16,260 --> 00:29:23,260
Eta will look like let me take another one
see, eta will look like this the cos curve,
232
00:29:31,090 --> 00:29:37,799
that is eta, but my w is a sin curve. So,
the w will look like is there is a positive
233
00:29:37,799 --> 00:29:41,170
sin curve. So, it is going to look like this
sorry like that.
234
00:29:41,170 --> 00:29:48,040
So, basically you can say that, this is a
phase gap and that is very easily explain,
235
00:29:48,040 --> 00:29:55,040
because a cos curve is nothing but well sin
is nothing but, cos minus 90 degree plus 90
236
00:29:55,960 --> 00:30:02,740
degree yes plus 90 degree. So, what happens
is that, essentially sin and cos differ by
237
00:30:02,740 --> 00:30:09,740
its phase of 90 degree. And I can always say
therefore, that the phase between eta and
238
00:30:10,110 --> 00:30:14,720
omega w sorry is 90 degree.
Now, similarly you will find out there are
239
00:30:14,720 --> 00:30:21,720
all kind of phase, which means the top most
point I have actually not the highest vertical
240
00:30:21,760 --> 00:30:25,929
velocity, which also makes sense, because
the particles are moving this way at this
241
00:30:25,929 --> 00:30:32,530
point; my velocity is most this side and downward
velocity is 0. Whereas, at this point; somewhere
242
00:30:32,530 --> 00:30:38,570
here, my downward is maximum, so that is what
is a phase and that is extremely important.
243
00:30:38,570 --> 00:30:44,340
Now, the one problem that comes is, how do
I measure phase? It is like I have a graph
244
00:30:44,340 --> 00:30:50,610
number of graph; see this one, see another
one, another one here. Essentially, the phase
245
00:30:50,610 --> 00:30:56,840
tells me gap between various graphs; time
gap, but I have to put them in number in some
246
00:30:56,840 --> 00:30:59,780
sense, so what measure phase with respect
to.
247
00:30:59,780 --> 00:31:06,130
So, what normally people do is that, you write
eta to be real part of e i k x minus omega
248
00:31:06,130 --> 00:31:11,950
t or whichever we have write. And measure
everything with respect to this phase which
249
00:31:11,950 --> 00:31:18,950
means mostly phases are measured with respect
to eta; taking eta to be a cos cups with respect
250
00:31:20,830 --> 00:31:27,830
to its x equal to 0, t is equal to 0 means;
if, I were to take an origin here and I consider
251
00:31:27,900 --> 00:31:34,360
eta to be like that that is occurring maximum
here with respect to this signal I am measuring
252
00:31:34,360 --> 00:31:37,380
the phases.
You can you can always debate why this way
253
00:31:37,380 --> 00:31:41,669
I can also measure another way, no problem,
because phase after all information content
254
00:31:41,669 --> 00:31:47,600
is what is the gap between the two; that is
the information content as long as you can
255
00:31:47,600 --> 00:31:48,280
get it.
256
00:31:48,280 --> 00:31:55,040
Now, you know that supposing I say, that like
I I I let me give another example. I say that,
257
00:31:55,040 --> 00:32:02,040
eta equal to A cos k x minus omega t plus
45 degree and then, some other signal say
258
00:32:02,679 --> 00:32:09,679
R is A cos k x minus omega t plus A 89 degree.
What would happen is that, you can always
259
00:32:12,360 --> 00:32:18,450
say with respect to some difference eta is
having a phase 45, R is having a phase 89,
260
00:32:18,450 --> 00:32:24,380
but I can also make this 45 as 0 and make
this actually as 44 degree right that makes
261
00:32:24,380 --> 00:32:29,540
more sense generally in our study, because
we will find out in sea keeping study.
262
00:32:29,540 --> 00:32:34,110
What we are trying to do is that, measuring
everything with respect to eta, because eta
263
00:32:34,110 --> 00:32:38,870
is my primary input. Remember what we are
trying on to do, what we want to do is? I
264
00:32:38,870 --> 00:32:45,870
have a wave, my ultimate aim is what does
the wave do to my ship, how dose my ship move,
265
00:32:47,120 --> 00:32:51,940
how does my body move. So, eta or the wave
is my primary input.
266
00:32:51,940 --> 00:32:55,799
So, I want to basically measure everything
with respect to eta, but it is not mandatory,
267
00:32:55,799 --> 00:33:02,030
it is upto you as long as, you do not make
a mistake what my point is essentially this.
268
00:33:02,030 --> 00:33:08,980
So, given another example that, the phase
versions are very very very well established
269
00:33:08,980 --> 00:33:15,980
with respect to for example, various signals.
See, if if wave profile eta goes like this,
270
00:33:18,720 --> 00:33:25,720
it turns out u will having the same phase,
this is my eta, this is my u then, my w goes
271
00:33:31,340 --> 00:33:38,340
like that then, my no w goes no no z component
acceleration x component of acceleration will
272
00:33:43,080 --> 00:33:50,080
go like this, that is u dot and this will
go like that w dot.
273
00:33:51,179 --> 00:33:55,830
So, you know what I am trying to say that,
these are all the the as far as the acceleration
274
00:33:55,830 --> 00:34:00,720
velocity are concerned, the relation is very
well established. This, of course, everybody
275
00:34:00,720 --> 00:34:07,720
knows very well that in a sinusoidal signal;
the displacement that is eta or phi, the velocity
276
00:34:12,020 --> 00:34:18,839
that is phi or you can say, phi dot if I want
to tell. And the next integration all differ
277
00:34:18,839 --> 00:34:25,829
by 90 degree 90 degree, that is why displacement
and velocity 90 degree, velocity acceleration
278
00:34:25,829 --> 00:34:30,509
90 degree. Displacement acceleration opposite
that everybody knows it in any signal.
279
00:34:30,509 --> 00:34:36,729
There is also very easy to see, because if
you write anything as you know like X equal
280
00:34:36,729 --> 00:34:43,729
to A cos something say, omega t then, X dot
is going to be you know minus omega A sin
281
00:34:46,159 --> 00:34:53,159
omega t, X double dot is going to be minus
omega square A. So, we can see this and this
282
00:34:55,220 --> 00:34:59,279
opposite 180 degree, this and this is 90 degree
etcetera.
283
00:34:59,279 --> 00:35:05,470
So, this is very well established, but what
we will but what what will happen to our subject
284
00:35:05,470 --> 00:35:10,259
subsequently is that, I have this wave here.
Now, I know everything of that, but I have
285
00:35:10,259 --> 00:35:17,259
going to have a body here, this body’s motion
are response or force or pressure whatever,
286
00:35:17,440 --> 00:35:21,970
I have to figure it out; it will also be sinusoidal,
which I will come later on, but it will not
287
00:35:21,970 --> 00:35:27,190
occur exactly in the same phase. It will occur
some time later, some time before phase is
288
00:35:27,190 --> 00:35:31,029
very important.
And also it has the very practical meaning
289
00:35:31,029 --> 00:35:36,190
because, for example, take pressure if all
the pressures were going to occur at the same
290
00:35:36,190 --> 00:35:42,339
phase then, I would have a very large load;
when there was a you know like crest coming
291
00:35:42,339 --> 00:35:48,059
and just the opposite, when trough coming;
it does not happen that way. So, this is very
292
00:35:48,059 --> 00:35:51,509
important that we understand about the phase
relation.
293
00:35:51,509 --> 00:35:58,509
Having said that now, Let us try to work out
the some simple problems, I know rest of the
294
00:35:59,529 --> 00:36:06,529
time let me try to, so we will now spend the
you know last few minutes of this class on
295
00:36:09,249 --> 00:36:13,450
trying to do some kind of problem with also
involve maybe phase.
296
00:36:13,450 --> 00:36:20,450
Now, suppose there is tank the problem says;
there is a towing tank 150 long, 5 meter wide,
297
00:36:27,170 --> 00:36:34,170
there is a wave maker in one end and this
is what we, this pen is not so sharp, there
298
00:36:45,769 --> 00:36:52,769
is another one. Let us say, the problem says
that let us make we are making a regular and
299
00:37:17,200 --> 00:37:24,200
T equal to well.
The very simple problem find out omega which
300
00:37:28,599 --> 00:37:35,599
of course, you know absolutely we do by this
2 pi by T. So, this is turn out to be 2 by
301
00:37:38,890 --> 00:37:45,890
2 into pi equal to 3.14 radian per second.
This is very elementary you just to know about
302
00:37:48,200 --> 00:37:52,619
how to get omega, k etcetera. We will come
to the phase problem in a second k of course,
303
00:37:52,619 --> 00:37:59,619
is 2 pi by lambda.
Now, how do I do this 2 pi by lambda now you
304
00:38:01,829 --> 00:38:07,170
tell me. So, you have to use the dispersion
relation first, so dispersion relation tells
305
00:38:07,170 --> 00:38:12,269
me how to get actually this thing, but we
can also without that we can do see omega
306
00:38:12,269 --> 00:38:18,729
square is equal to g k. So, k is simply omega
square g, this this also can be omega square
307
00:38:18,729 --> 00:38:24,279
by g. We can straight forward use that, which
is that which is the same thing you just using
308
00:38:24,279 --> 00:38:27,729
the same thing differently.
So, this will turn out to be something like
309
00:38:27,729 --> 00:38:34,729
omega square is in this case, pi square by
a approximately come to be about 1 1 meter
310
00:38:38,989 --> 00:38:45,989
power 1 and lambda therefore, come to be approximately
6.25 meter, 2 pi; basically 2 pi length. Now
311
00:38:50,640 --> 00:38:57,640
it says that, find out maximum towing particles
velocity in the tank; let us say tell me this,
312
00:39:03,839 --> 00:39:10,839
now you see here we will have to go back to
our expression for what we have done before
313
00:39:12,579 --> 00:39:19,519
of u and.
Now, you see here, what is the maximum u max
314
00:39:19,519 --> 00:39:26,519
this is this because, you know it is remember
that, maximum value of e k z is only 1 because,
315
00:39:29,359 --> 00:39:36,029
z cannot be positive; z is 0 to negative.
So, this amplitude is maximum value, similar
316
00:39:36,029 --> 00:39:41,470
this is maximum value. So, you but then, remember
that that total maximum is not going to be
317
00:39:41,470 --> 00:39:46,989
simply some, because the maximum there is
another point. u square plus omega square
318
00:39:46,989 --> 00:39:50,989
you cannot do by this two addition because,
when this happens this is 0 remember, when
319
00:39:50,989 --> 00:39:55,799
this happens; this is 0.
So, the maximum would occur actually this
320
00:39:55,799 --> 00:40:01,499
only, the modulus do not use u square plus
omega square with this value and this is square
321
00:40:01,499 --> 00:40:06,269
as you did. So, we end up getting this maximum
particle velocity to be something like, if
322
00:40:06,269 --> 00:40:13,269
you work it out about 0.60 meter. Now, what
is celerity c is going to be omega by k. How
323
00:40:22,859 --> 00:40:29,859
much it comes to be? It comes to 3.12 meter
per second. And now the question is the the
324
00:40:34,380 --> 00:40:41,380
interesting question when a phase is there,
that this question says like this, no no no
325
00:40:44,109 --> 00:40:51,109
it is phase speed not group speed.
We are we are talking of phase speed. See,
326
00:40:51,349 --> 00:40:57,239
phase speed means this form speed, group speed
is the front front speed, which I will come
327
00:40:57,239 --> 00:41:01,150
the next problem is on the group speed, but
here this this particular question is now
328
00:41:01,150 --> 00:41:08,150
on the phase, what is the phase?
This you see now, so observer is standing
329
00:41:40,440 --> 00:41:47,440
here, the question is what the phase of the
wave 2.5 meter here is. So, you see now, this
330
00:41:55,630 --> 00:42:02,630
is how do you, how do you work it out. Now
you see, this is observer standing here and
331
00:42:03,059 --> 00:42:06,890
what the question is what would be the phase,
if I were to go to 2.5 meter towards the wave
332
00:42:06,890 --> 00:42:13,890
maker say, this is my wave maker.
So, now see with respect to this of I were
333
00:42:15,969 --> 00:42:22,969
to call this to be x, what is my eta? See,
suppose this is my x I start from here then,
334
00:42:23,279 --> 00:42:30,279
it becomes eta equal to A cos k x minus omega
t. But, now the question is that, I need to
335
00:42:32,309 --> 00:42:37,519
call this measure with respect to this or
rather I have to find out this with respect
336
00:42:37,519 --> 00:42:44,519
to x equal to here. So, now, this is 2.5 meters,
so what is happening that, I have to measure
337
00:42:45,519 --> 00:42:52,519
this with respect to a parameter when x has
been made see by this x plus 2.5 meter. Because
338
00:42:54,700 --> 00:42:59,259
see, if I were to measure something any point
here is x, but if I measure from here; it
339
00:42:59,259 --> 00:43:04,940
is x plus 2.5.
So, I that the expression for that with respect
340
00:43:04,940 --> 00:43:11,940
to this point is going to be, A cos k x plus
2.5 minus omega t, so this if you work it
341
00:43:14,569 --> 00:43:21,569
out, it becomes k x minus omega t plus k into
2.5. Now, k into 2.5 is therefore, the phase.
342
00:43:24,069 --> 00:43:31,069
What is k 2.5? 2.5 into 2 pi by lambda what
we found out earlier is.
343
00:43:32,249 --> 00:43:37,109
k is 1 sir
k is 1 absolutely. So, it is 2.5 radiant straight
344
00:43:37,109 --> 00:43:41,749
forward. So, it is like this you end up getting
therefore, that 2.5 radiant, which will be
345
00:43:41,749 --> 00:43:48,749
about 144 degree. This is the idea for the
phase. Now, we will see one more case is where
346
00:43:50,380 --> 00:43:57,119
the question was asked regarding this group
speed, that we will we will work out.
347
00:43:57,119 --> 00:44:04,119
Now, here it says, another again a wave maker
problem; it is making wave of frequency omega,
348
00:44:20,319 --> 00:44:27,319
period well actually in fact, omega period;
period is given as 2 second and amplitude
349
00:44:27,359 --> 00:44:34,359
A is given as 0.25 meter, tank length; length
of the tank is.
350
00:44:34,910 --> 00:44:41,910
Now here, the first question approximately,
how long will it take a wave front to propagate
351
00:44:42,640 --> 00:44:49,640
from the wave maker to the tank. First question
is, how much time will it take front to reach
352
00:44:49,880 --> 00:44:56,880
the other end of the tank. Now, the question
here what the question was asked, this time
353
00:44:58,819 --> 00:45:05,819
taken for wave front this is going to be L
by C g, because the front will travel.
354
00:45:22,749 --> 00:45:26,829
So, what is C g here, we can first find out.
Now, in this particular problem assume deep
355
00:45:26,829 --> 00:45:33,829
water, so my T is given. Now, how much is
C first of all, you have to find out your
356
00:45:34,729 --> 00:45:41,729
C? C is we can always C is omega by k.
And let us see, how we can write in terms
357
00:45:46,079 --> 00:45:53,079
of you know like T. See, omega square being
g k omega by k equal to g by omega equal g
358
00:45:56,609 --> 00:46:03,609
by to 2 pi lambda. So, C is this thing, g
into lambda by 2 pi. Now, of course, lambda
359
00:46:04,920 --> 00:46:09,960
we have to again work out let say no actually
no sorry sorry g by omega we can write simply
360
00:46:09,960 --> 00:46:16,960
g by not lambda T, this is T making a mistake.
So, this is like this, g T by 2 pi how much
361
00:46:17,499 --> 00:46:24,499
this comes? T is 2 second. So, it is g by
2 pi into 2. So, g by pi, there is approximately
362
00:46:25,650 --> 00:46:32,650
let us say, 3 3.2 3.2 meter per second.
So, therefore, C g is 1.6 meter per second.
363
00:46:38,660 --> 00:46:45,660
So, therefore, time taken is going to be 100
by 1.6 second, approximately is equal to 100
364
00:46:50,519 --> 00:46:57,519
by 1.6 would be about how much? 62.5 second.
So, these are the group speed. Now, I will
365
00:47:06,450 --> 00:47:11,819
I am going to do one part, which I will now
leave it for you to work out, because you
366
00:47:11,819 --> 00:47:16,349
know it is with respect to this thing, phase
again.
367
00:47:16,349 --> 00:47:23,349
Now, see here I I will I will write it down,
consider an observer situated along the tank
368
00:47:45,579 --> 00:47:52,579
side, time. This next part of course this
of course is very simple this part, because
369
00:48:38,479 --> 00:48:43,450
the answer is that, it will be the phase speed
you know like after you have the wave front
370
00:48:43,450 --> 00:48:48,039
has passed you are standing and you are trying
to now see the the crest passing.
371
00:48:48,039 --> 00:48:51,869
Now you see, what would happen is that, the
crest would pass much faster. So, you will
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00:48:51,869 --> 00:48:58,869
actually see it, but it important point is
next one, what is the this also we have done
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00:49:24,579 --> 00:49:31,579
last time. See here, this is what we have
done before, what is the phase of the wave
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00:49:56,670 --> 00:50:02,200
elevation 1.5 meter closer to the wave maker
relative to the wave elevation of the position
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00:50:02,200 --> 00:50:06,319
of the observer. Important point is this,
see this phase is being measured relative
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00:50:06,319 --> 00:50:11,109
to the wave elevation, you have to measure
phase always with related to something, you
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00:50:11,109 --> 00:50:14,509
just cannot say phase is so and so, it make
no sense.
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00:50:14,509 --> 00:50:20,380
The word phase necessarily connected to a
measurement. But, now there is a second question
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00:50:20,380 --> 00:50:27,380
is that, what is this this phase this this
part, what if the observer is travelling moving
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00:50:39,569 --> 00:50:46,569
with a velocity 1 meter per second either
towards the wave maker or the other way around
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00:50:55,289 --> 00:51:02,289
the wave maker.
I will I will explain this picture in a minute,
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00:51:02,819 --> 00:51:09,819
this I will leave it to you for thought. What
happen here now, this is a tank here, wave
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00:51:10,849 --> 00:51:17,029
maker is here and waves are going passed it,
now you are standing here let us say, standing
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00:51:17,029 --> 00:51:23,099
here. Initially of course, you are saying
that with respect to that what is the phase?
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00:51:23,099 --> 00:51:26,799
Simple problem we have done last time, but
now the question is more interesting. Now,
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00:51:26,799 --> 00:51:30,099
I want to define the wave with respect to
moving frame of reference.
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00:51:30,099 --> 00:51:33,989
See earlier, what happen if I want to find
it out all I have to do is, to measure this
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00:51:33,989 --> 00:51:40,479
as my reference frame. If I have my cos k
x minus omega t, I simply define x with respect
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00:51:40,479 --> 00:51:46,039
to that, but now my x is very because what
is happening is that, I am actually walking
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00:51:46,039 --> 00:51:50,200
towards this or walking towards.
Now, remember this supposing I was with the
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00:51:50,200 --> 00:51:56,430
crest see the and I was travelling with the
phase speed, what will be the phase gap? 0,
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00:51:56,430 --> 00:52:00,979
because I will always see a crest in front
of me, it was going slower; my crest passed
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00:52:00,979 --> 00:52:06,950
by, if I other way around means something.
So, what would happen? There will be a phase
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00:52:06,950 --> 00:52:10,219
gap, but the phase gap is not going to be
constant because, constant what would happen
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00:52:10,219 --> 00:52:15,599
in this case; phase gap would keep on evolving
with time; that means, my phi that is k x
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00:52:15,599 --> 00:52:19,519
minus omega t plus phi would become a function
of time t.
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00:52:19,519 --> 00:52:24,719
So, therefore, what happens is that, in this
problem you end up finding out a evolving
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00:52:24,719 --> 00:52:29,289
phase because, there is not going to be constant
and that is what may be you could work out
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00:52:29,289 --> 00:52:35,630
for the next class and you would kind of see
it is. So, I am going to more or less yeah
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00:52:35,630 --> 00:52:41,779
end it here, today on the wave part.
This gives us a kind of a basic understanding
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00:52:41,779 --> 00:52:48,779
of linear water wave that we are going to
use continuously, when we study sea keeping.
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00:52:48,969 --> 00:52:54,279
I will close it by saying that, water waves
you know are sometime little bit difficult
403
00:52:54,279 --> 00:53:00,140
to understand because, by wave we do not we
always mean the form, where the particle never
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00:53:00,140 --> 00:53:05,029
follow the form; particle have a different
motion. So, the dynamics involve is not as
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00:53:05,029 --> 00:53:07,469
severe as it might think, but they also be
more.
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00:53:07,469 --> 00:53:11,839
What I am trying to say is that, what you
think intuitively may not always work. It
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00:53:11,839 --> 00:53:16,799
just does not work because, the very fact
that two waves can interact is an interesting
408
00:53:16,799 --> 00:53:21,299
example, I can look at calm water and I can
tell you theoretically it is equal to two
409
00:53:21,299 --> 00:53:28,299
waves of exactly opposite phase. It is like
saying, 4 is equal to you know 2 plus 2 minus
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00:53:29,369 --> 00:53:36,339
4 or equal to plus 4 minus 4 equal to or 0
rather other way round, 0 is equal to 2 minus
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00:53:36,339 --> 00:53:40,920
2 or minus 2 plus 2 or whatever So, you know
the waves are like that.
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00:53:40,920 --> 00:53:47,920
So, this there is some kind of conceptual
understanding required what we talked of only
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00:53:49,369 --> 00:53:53,259
one wave, when we add the will start because,
when we add we will find out that, I can always
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00:53:53,259 --> 00:53:59,690
say that calm water as I say some of n numbers
of waves. And now we will find out in a wave
415
00:53:59,690 --> 00:54:03,920
maker I will just end it by saying, supposing
I will say that and I say that, I suppose
416
00:54:03,920 --> 00:54:08,150
you have a wave maker and you want to make
a wave by pushing this way and we find out
417
00:54:08,150 --> 00:54:13,059
that, if I actually have two waves of out
of phase; I have no wave at all.
418
00:54:13,059 --> 00:54:17,940
Now, the question is, how do I make it to
move, make a you know like wave maker move,
419
00:54:17,940 --> 00:54:22,670
so there is a calm water. The interesting
part is that, if supposing person one wants
420
00:54:22,670 --> 00:54:27,700
to make a wave of phase a, so he wants to
push it and if person b wants to make a wave
421
00:54:27,700 --> 00:54:32,049
of phase minus say at that instant, he has
to pull it. So, both will pull and push and
422
00:54:32,049 --> 00:54:36,259
the plate will remain steady and therefore,
it is like force is plus 1 minus 1 is zero,
423
00:54:36,259 --> 00:54:40,359
water also remains 0.
So, you know it get explain by that. So, you
424
00:54:40,359 --> 00:54:46,349
cannot make, you cannot move a plate to make
a 0 wave, because the movement is going to
425
00:54:46,349 --> 00:54:52,219
be 0, is like doing nothing. So, this is what,
but then we will always tell that, yes to
426
00:54:52,219 --> 00:54:58,430
calm water is 1 minus 1 with that, I am going
to end it. And we will go tomorrow next class
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00:54:58,430 --> 00:55:03,400
onwards to the you knows the actual we will
begin to put the ship in waves in regular
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00:55:03,400 --> 00:55:04,569
waves, thank you.