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Ok, see in the earlier lectures we have learnt about how to describe a spectrum.
How to describe an irregular sea by means of a spectrum and how to get motions in the
spectrum. So, I was having this for defining… ok, we are going to revisit this part in this
lecture again because when the spectrum that we define earlier, had certain limitation
certain approximation that we did not explicitly set.
What did I do for this spectrum? See I set this record is sum of… ok, what we said
you see, in defining a irregular record we said that it can be composed of adding sine
wave but, what we probably did not emphasize at that time is that all of them, we assumed
to have been travelling in the same direction, which basically means if I add all of this
thing see eta I obtain by making something like a you know i Cos say K i X minus, omega
i t plus beta I, all of them are travelling along plus x direction.
Then what happen if you stand, supposing I was standing here you know I was standing
here and I look my left and right side, what will I see? I will find the crest to be infinitely
long. In other words this wave would have appeared like with the long lines you know
like.
If I were to draw the wave would appear all the crest lines are long, if I stand here because it is infinitely long, because I have
assumed all of them are going to the X direction. So, what happen take any Y value whatever
the Y value is the eta is same, that means if I were to you know like used a X, Y, Z
coordinate this was in an X Z plane, which means whatever Y equal to constant. I cut
a plane my velocities are all in X Z plane which means if I were to see the particle
the way it moves etcetera, the velocity vector you know V vector it is all having only a
dimension of or dependence on X and Z. So, this is essentially a two dimensional fluid
motion there is no motion on the Z direction. So this sea that what we have obtained this
S omega was actually what was called a two dimensional spectrum because what we have
presumed is that the waves are all going in one direction therefore, the motion is two
dimensional, so this is what Y we have got or this is sometime call as long crested sea.
However, now why we are revisiting it is because suppose, you actually go on an ocean and suppose
you actually you know like go on a ship or make an observation you are stranding here
you are not going to see your right hand, left hand side waves to be same. You will
find out a randomness in not only in along X direction also along Y direction, so you
are not going to find this crest line to be same both sides, so obviously somebody may
question that the spectrum that I defined it is good it represents this signal by means
of sin but, what you have said is the waves are long crested whereas, in reality they
are not therefore, we need to now define what is called short crested sea? Now, I we will
come back to this question again see what is happened.
Let us take it this an example somebody went to an ocean there is a boy, there or some
equipment you measure and the data as I say look something like that. Now it was our question
we presumed, we kind of composed that by making them sine waves in one direction what if I
were to add? We have to think that this is not only all the waves coming in one direction
but, also one direction of different length but, also I think that this can be broken
down to waves of all lengths in one direction and all directions.
For example, now if I look see now, I come slowly now, I eta looks like A Cos K x minus
omega t, now this is in X direction supposing the wave was going in X dash direction, what
would be the formula? Formula would be A equal to A Cos K x dash minus omega t. Suppose,
see this is when the ship is going in along X direction and let me call this to be X dash,
if not ship, a wave. This represents a wave travelling in positive X direction suppose,
I want to find out an equation of a wave travelling in positive X dash direction all I have to
do is to write X as X dash. Now, suppose this is theta, then what is X
dash? X dash is it is Y, X Cos theta plus Y sin theta that is a standard transformation,
so I can write this A Cos K. So, this would be this full thing I can put, so this would
be a wave travelling along a direction making theta with positive X axis. So, this one now
what I we will do is that, we can presume this wave to be composed of all waves coming
from not only all frequencies that means omega or K but, also from all thetas.
So, in other words what we are going to think that in this point say, I were to draw a circle
waves here that I measure are not only coming from theta equal to 0 all but, also coming
from theta equal to theta etcetera, etcetera. So, what happen then I can tell eta to be
see earlier I had this, I will write it this way A i omega, let me write it this broad
way K well x, I just write it blank here minus something like that see I did not fill up
the gaps remember what we had earlier if all the waves are assumed to be coming along X
direction I have A i omega cross K x minus omega t plus beta K x minus omega t this is.
But now what we are going to do, we are going to suppose I presume that look this particular
thing can be broken down to all waves not only from all omegas but, also from all thetas
therefore, I have to put one more this here. So, this let us see this i, this i index is
for omega, so I have to put another index theta here and I can put j here, j here then
this is cos theta plus Y sin theta this becomes of course also i j. So, this basically represents
therefore, as if I am saying that the waves are coming from all direction and all sum,
so it is a, see it is a very interesting thing it is a question of only breaking it down
remember that when you are sitting in a office a satellite has measured ocean height data
it only gives you this data.
t verses eta that is all. It is up to us to think this eta is composed of what kind of
summation of waves, now remember that we could sum all the sine waves because sine waves
summing gives a realistic wave because waves are linear, if I put a black box i add all
the sin then what I get is a sine wave but, here I did not say, I only added only those
sine’s which are along the X axis but, now I want to say that look I can break it down
by adding all the sine’s not only along same X but, also along all X’s.
Obviously, that would give me a simulation of this picture is not going to look like
a long crested anymore because obviously there is a dependence on Y in other words, if I
were to find out eta at sum X and sum Y there are not going to be same earlier there was
no Y, so no matter what Y you to eta was same but, now you are going to be different so
it will have a spread three dimensional picture, so that will be a better model for a 3 D spectrum,
3 D waves. 3 D wave meaning again when I say the word
3 D waves which means the velocity of the particles would have both u v and w earlier
had only u and w that means particles are moving along X axis or Z axis. If you make
a cut Y same but, here it is not going to be same, so this obviously is a three dimensional
wave, so this is what is called short crested sea. You see that means if I were to break
it down to this way and now I plot similarly, you know that process, I will come energy
etcetera because you know the energy is the principle thing.
After all what I have to do is to plot this A, see here what would happening? What was
happening is that, here is that earlier a was a function of only theta and now A is
going to be a function of omega, now A is going to be a function of omega and theta
that is where is this picture did I do that, we can just do it from here also see this
A is going to be a function of remember earlier it was only function of omega in the 2 D case
but, now it is a function of omega as well as theta.
So, having said that so the interest here as I said this is already there here see omega
and theta question is now I need a spectrum obviously, what would happen if I were to
put a spectrum here, in this we will come to this spectrum shape here.
Once again I have to keep drawing this, you know all the time say we will call it this
way and this is theta. So, if I were to now plot all the waves of between this frequency
and this length see now earlier what I did? In a spectrum, I plotted the energy of all
the waves a quantity proportional to a quantity proportional to energy of all the waves coming
between this frequency band. Let us say all the waves of energy between
50 to 60 meter, I plotted one place 60 to 70 meter, I plotted another place like that
I plotted but, now what I have to do is to plot a 2 D plot so it is going to be a bell
shaped curve, it is going to be a shape not going to be a 2 D part. What I have to do
is to plot see in this diagram, so if I call this to be say this is theta, this is omega
then all the waves of length between this d omega that is this omega and angle between
this is what I am going to plot here at the top. So, if I keep plotting like that then
I’ll end up getting a graph like that in fact instead of doing that.
We can also this is a bell shape one I can also do more I will put omega here and I put
theta here. So, what is happening say theta is 0, let just give an example this is alpha
here this is a minus alpha here like that it goes.
So, what I am going to do? I am going to plot here this value, this block which is going
to be a measure of half A squared omega, here I wrote alpha, alpha divided by d omega d
alpha of course, remains this thing. You can also say d alpha, let me also put it this
just for the sake of, so this is what I’m going to put see early I in have this what
I earlier I did I put half a square omega divide by d omega, here I have put this because
remember the overall volume, now the volume under this should become the energy, so this
is what is what we mean by describing short crested sea.
Now, that means it has got a dependence on omega and on theta, now let us lo at some
physical reality part, so this we understand this way of plotting it.
Now, let us say from all which direction the wave comes in now I’m going to plot in this
way theta again because we have to keep on plotting that, so I call this easier to see
this is say omega, see this is omega axis and this angle is the theta.
Now, you see here what I’m saying I have all the waves coming from this, let put it
this way I assume that at this point what I measured, is all the waves coming this side,
then all the waves coming this side, then all the waves coming this side, then all the
waves coming this side. So, what I am saying you see that the measured data consist of
all the waves coming from all sides but, now there is a problem you see here take this
opposite all the waves coming from this side and this side. What would happen? If there
a same frequency, they would cause a standing wave but, however we have a progressive wave.
So, what happen is that, that is very important we have to say that all the waves that is
coming are only over one half sector because the other half sector is exactly opposing
and if I consider them, I end up getting a standing wave or if it is a different frequency
see if I use this as A 1 and this as A 2, I end up getting a standing wave or a part
progressing in one side. So, what happens therefore, from physical point you can easily
understand from physical point of view, I did not have to take presume this integration
this that means if I were to well if I were to call eta to be an integration over d theta,
over d omega of a omega theta you know Cos K X Cos theta, plus Y sin theta minus omega
t plus beta something like that. If the graph know you understand, this is
only a integral form of the equation instead of writing there I said summation is integrate
from a omega theta this but, over d omega d theta and here it was 0 to infinity and
here it was 0 to 2 pi but, now we see that this should be actually one half. So, one
half means 0 to pi or it is easier to write in terms of minus pi by 2 to plus pi by 2
because that gives you a dominant direction and spread both sides.
See zero to 2 pi is a question of opinion from which you call zero, zero to 2 pi means
I called this we have all the way this much, now here I am saying the waves will be only
over this one of the half sector. So, I can call this see if I call zero to 2 pi from
here I am calling but, if I call this way then I have got basically minus pi by 2 to
plus pi by 2. The question what I am saying is that whether you call zero to pie or pie
to 2 pie or you know like minus pie by 2 to plus pie by 2 is the same thing because you
are talking of waves coming over in a half single. So, it is a mere convention to use
physically or few more I will tell you more easier to we should use minus pie by 2 to
plus pie by 2 why? Because remember winds waves are created because of winds primarily
these are wind driven waves, no wind generated waves.
So, wind is when to go in to blow in some direction wind and you would expect most waves
to be in that direction but, we are saying that this wave can be actually from ninety
degree plus minus, means one eighty degree. Obviously this will be the primary and you
will think waves coming also from somewhere this direction and also up to this direction,
so if I take this as base then the waves become minus pi by 2 to plus pi by 2 rather I would
have a most dominant wave, very high kind of wave at the direction the wind blows there
will be other waves coming both sides 90 degree spread that is more logical after all I only
looking at the integration. So, if I do for example, zero to minus pi
by 2 and zero to plus pi by 2 with zero is that direction of the wind they need makes
more physical sense and of course, it satisfies my property also, so we typically this is
done. So, we therefore what we are saying, we say that I can model once again coming
back to this irregular wave I can model this irregular wave by summing or taking waves
from all direction but, now I find out that the directions I can take, I need to take
only over one half of the full circle that I can write more easily from minus pi by 2
to plus pi by 2 in integral sense. Sir but, this is assumption that the waves
coming from different direction might of the same omega for standing waves.
No even otherwise, see even otherwise the point is that if no same omega because after
all what is happening that when you are considering integral for each omega there is an omega,
there is an omega for each omega. So, if you take each omega and each omega they will be
a standing wave so therefore, I cannot have because we are looking at a propagating wave
therefore, what we are doing is that, we do not have that or if you want if you want you
can say you have that its amplitude zero. So, the dominant waves are all coming from
basically within this half section that is what we are saying that makes more sense in
the modeling part. Now, comes the question of spectrum well we
have seen that obviously the spectrum you have to plot this part, we need to plot, so
called what I just now I showed in a diagram we showed that we are plotting this height
right d omega, d theta etcetera but, how do I know model it? See now I have a modeling
question, I really do not know how it has spread after all when I get a signal like
this you know I can make a Fourier analysis and get Fourier analysis allows me only my
frequency distribution but, I cannot have the spreading distribution, theta distribution
I cannot get that information see if I get us, give you a signal I tell you to f f t,
I can find a omega verses omega but, here I am making that there is also theta there
that I cannot not get from f f t signal processing you must, we must understand this, so then
what happen I have to make a model of it?
Now, comes the question of this model, now you see same wave it goes like that again
we get back to the same wave now person a made it a 2 D model, so here the formula for
S omega 2 D verses omega here this person A has made this and has a formula for that
it looks like this, omega verses S omega now same signal is analyzed by person B and he
wants to write this in terms of a kind of a you know like a omega theta omega, he wants
to write it as 3 D omega theta verses omega and theta. You know if you want to plot you
will obviously end up plotting that now he the same signal remember.
So, what is happening one interesting point is obviously, what is happening is that see
energy under this wave is S 2 D omega d omega, this is my energy of the wave zero to infinity
and energy under this wave, this signal, this model is going to be but, this two must be
same because they represent the same energy, total energy after all energy cannot change
it is a question of viewing it the same waves, I am thinking person A thought well I am going
to model it by saying it is coming all from direction one, person B says no I am going
to think it is coming from direction one, direction two, direction three etcetera, etcetera.
Energy remains same area under that it will remain same at volume under this.
So, I have this fundamental relation that is area under S 2 D, if I will to write it
again that is if I were to write this in S 2 D omega d omega must be same as. So, here
comes the question of modeling, you know what is done is that people model this S 3 D, it
is this in us this is the very important assumption, this is assumed as first of all it is assumed
that it can be broken down to two part f 1 this thing and f 2 theta, as if see this is
an assumption you presume that I can break it down to this a part purely function of
omega and a part purely function of theta and now this part which is taken to be S 2
D omega and we call it f theta. So, it is presumed therefore see this is very
important that a 3 D spectrum can be written down as a 2 D spectrum multiplied by a function
which is function of theta alone, this is known as spreading. You can easily know why
it is spreading function because what we have done we have taken S 2 D, see if I were to
now lo at this side this is my omega i have then S 2 D part here and I simply spread it
this side, this is theta side by a function multiplied that with the function and go down,
paper it down that is gives me spreading see for example, if f theta was one then it will
be equally spread means there is a equal chance of all waves coming from all directions.
But we know from physics that wind is blowing one side and you expect most waves in that
side and only a small waves on other side as you go ninety degree of the wind, you expect
very less wind waves coming out. So, wind is blowing in suppose wind blows this side
you expect most waves are this side and very waves coming this side very less wave, I mean
mostly seen I eighty percent here may be little less on coming from 45 degree, even less from
30 degree practically nothing from 90 degree 45, 60, 90 like that.
So, this obviously it is always spread, it will it is suppose to have this theta is suppose
to have this if I were to plot zero to pi by 2 or here minus pi by 2 you expect this
spreading to have a bell shape that’s make sense isn’t it? Because this tells me the
strength of the wave coming in direction zero, wind direction and obviously the strength
will reduce the percentage of waves coming from direction away from zero would be less
as you go further from zero it is spread.
This idea we must get in your mind, once again I will tell you the spreading function concept,
so if I were to put theta here, if I were to put f theta you would expect it to first
of all f theta will not be, will be zero beyond minus pi by 2 or plus pi by 2. So, if I call
this plus pi by 2 minus pi by 2 you will expect it to be maximum here, zero here and it will
be something like something like that, that is your that is a realistic model.
Of course, you can all say why not that well we can always do that people have also analyze
this spreading function somewhat by taking the wave slope you know actually although
I say you cannot measure but, you could measure nowadays with sophisticated instrumentation
from a random signal of slope also, after all it is all going X direction I have a slope
in one direction, if it is coming from this direction I have slope in different direction,
so if I measure the slope I get a feel the content of content of directional you know
waves, so people have been doing obviously it is found most waves are in the direction
of the wind so this is what we do. Now, comes the question of how do I model
it so we look that to this equation again and we go back to this now I find out this
let us write in another paper.
So, I have S 3 D omega theta is equal to S 2 D omega f theta, the other next come the
question, well I am going to put it here now integration over that d omega d theta is same
as integration about that I mean the integration about that this.
Now, this see carefully follow area under that is S 3 D this d omega d theta but, S
3 D is written as S 2 D omega and f theta d theta, f theta, so when you integrate that
it will be between the omega and between this theta this is just from identity from the
approximation we used because we said this is, this into this.
Now, however understand that this side is also same as S 2 D omega d omega, this side
same as this because we found the area is same, so if this is same as this what do I
get? I say I get that minus pi by 2 to pi by 2 must be equal to one, what we end up
getting is that we say f theta is a model, I can use any formula but, do not you must
use a formula for f theta, so that this condition is satisfied this is very important for us
to understand. We must have this condition satisfied f theta d theta must be equal to
one over this, so this is a spreading function and we must have this relation.
Now, what can be possible spreading function well as you understand since it is not fully
clear people have model the spreading function many ways.
Most usual we have modeling is to use a Cos theta graph, why Cos theta graph you see if
you use Cos theta, a Cos theta here it is pi by 2 here is zero Cos theta basically goes
like that Cos theta has this look know that it is zero maximum at zero and at pi by 2
maximum value at theta equal to zero and at pi theta pi by 2 it is zero, so it has got
this kind of a spread. So, essentially f theta it would obviously
you cannot use sin theta as a model because sin theta would be exactly opposite, you would
want to have cost eta f theta remember I wanted to have f theta which is maximum here and
as you go down, lowering down this is one possible way but, people do not model that
there is a problem of using cos theta alone why because you see here you know it is like
this the graph or maybe it is like this but, what you want is actually like this because,
so I comes down papering and this can be done by using cos square theta or power of cos
theta. So, typically you will find out now that 2
by pi is a possible function that will actually look like this and it also has an interesting
property that the slope becomes zero here, the function maybe we should lo pertinent
it becomes like that it goes to zero, if I use this forget this just cos square theta
you take, if I take cos square theta it will go to actually maximum here go down to zero
here as well as the slope is zero here but, then of course,, if we use cos square theta
the integration is not one. So, therefore you have to constant coming 2 pi by, now from
this analogy we can find out that there can be f theta of the form which is which will
lo like cos.
Well you can have define m you can find out of course, you will find out that larger the
m, more no peaky it is A as m goes up, see as what I mean if you use cos square theta
that of course, there is a constant 2 by pi. If we use cos four theta, it will be some
other constant because obviously the area under that has to be one as I said it may
be, in fact it may be something like eight by three pi that can be worked out but, my
point of view of saying is that as you use more power then you have got peakier. What
does a peakier means? Obviously, it means this less and less spread.
So, you know if you use now here is the question of you know modeling let us say that you went
to north Atlantic you find that well there is a three dimensional wave all right but,
majority of the wave is much more in one direction much less in spread. So, you would use a function,
choose a function from literature which is the higher part of cos more focused in fact
larger the air, we call more focused the waves because it more in one direction smaller the
m more spread cos square the lowest one that is used is typically 2 by pi cos square normally
you we do not use anything lower than that. Once again the spread is something like spread
you know like how much the directional content is there and it is very illogical to think
that waves from all direction are of same probability. Cos say it again? No, you do
not have to see minus pi by 2 to plus pi by 2 we are taking because we have actually chosen
see what we are doing is, I will tell you what I we will write this way it is this I
will answer your question for minus pi by 2 equal to zero else.
Now, the question is actually why we to minus it is not question of functions value of theta
what we chose is that we say from physical ground I can use from any zero to 2 pi as
long as I use what you say is correct as long as I use from zero to pi sum one you need
it will become same thing question is what to take? So, it satisfies us or easier to
visualize I if I to minus pi by 2 to plus pi by 2 because what would happen then I can
think that zero is my dominant wave direction and I am spreading.
If I take 0 to pi you know like pi then, I have to think my dominant wave direction is
pi by 2 I can always take from in fact alpha minus pi by 2 to alpha plus pi by 2 and make
alpha anything you want 360 degree, then you take 182 you know like 4 whatever but, we
do not want to do it because one thing you know that we should not unnecessarily complicated
things, why we have chose always X wave direction because it is easier to think that way, so
that is the reason why you can choose anywhere does not matter.
What yours question is only with the reference point, if I choose the reference point is
ninety degree dominant then I have zero to 90 to 180 degree, if I use 270 degree of dominant
well I will have 180 to 360 degree. So, should you use 270 or 180 as dominant direction,
rather it is better to use a zero degree of dominant direction that is what we are doing
so this is how it is? So, this how the spreading goes here and now
what is happening, is that if I were to draw this you know like spectrum there several
people drawn in separate ways.
If I were to draw in a theta direction rather it is difficult for me to plot this say, so
this angle is theta so what would happen it is something like your this graph you are
kind of rotating this graph rotating, this graph both sides and pulling it down that
is what you are doing right because theta is this see this is theta.
So, we are essentially this graph you know this graph you are rotating it this and pulling
it down see if we, if I were to look at this way rotating it and pulling it down both sides
as you go ninety become zero. So, this amount it is coming down is f theta, so this is one
we have. So, if you look at that you are going to get some picture you will find picture
will looking like that you know a kind of a shape will come like that not really it
does not look like a conical shape because question of the way you are looking and if
you were to draw in terms of this as theta, this as omega you will find out this thing
coming down to zero as you go less.
In fact lot of plots have been made in actual literature of directional content they will
look something like this you know, I will tell you the picture will look something like
you know there is some bar charts there it is difficult to plot here but, it will, it
is going to with lot of spikes over this domain, certain domain because not all waves of all
comes certain omega and certain theta if you were to measure it comes out like a cluster
like a like a bell shape. People have done it measure as I said nowadays
it is becoming fairly sophisticated to also be able to determine directional content from
what is known as wave slope. We have say ocean satellites they are collecting whole lot of
ocean data, they collect besides wave heights. Nowadays they are also collecting slopes similarly,
the data voice are they are they keep collecting slopes, so long back it was only wave height
but, now you also have slopes coming in and from there directional content has been worked
out many times so they look like that, so this is what is a 3 D spectrum. Now of course,
having said 3 D spectrum now suppose I have going to spectrum now so you see opposite
now you see how do I reconstruct the signal?
So, I have this spectrum S 3 D omega theta, so I want to find out what is eta t so there
the like some people want to find out this eta t what we can do of course, is that this
is given by straightforward a let me put it K here i, omega i this will be a random value.
So, what you have to do is that you will obtain this by i equal to something one to n, so
what is happening, you can reconstruct this way and where a omega i theta j will be given
by 2 S 3 D. See earlier what is happening, if you remember
the this is why I am, why I say it this is that here the question is like that I have
this formula given after all this formula is known to me now. So, I have this formula
given I can find out this reconstruct a realistic realization of the ocean wave, reconstruct
what should be the eta t means reverse problem I have a spectrum here see in a 2 D case,
I have the spectrum here find out what would be go from here to there where do we use it?
What is the use of this kind of information very useful because think of this.
I have a ship; I have to find out what is this realistic like realization. Now, somebody
tells me use c spectrum by this formula and I go to use the c spectrum, he says take c
state four, so I know this formula so I have this given to me this is given to me but,
I want to know this means physically what kind of waves let us talk of tank wave tank
I want to make a model study I cannot fit this, what do I fit? I have to fit this, after
all in a tank I have to create waves which will be like this, what is this? Is this should
be such that it is statistically equivalent to this, so I must have to create a real time
signal remember this is a real time signal in real time. So, if I want a 2 D spectrum
I have to create real time spectrum, let’s say you are doing a model test for a ship
or for an off shore structure in sea state 3, so I have to send the waves which is same
as sea state 3.
What do I send? Well I have a tank here, I have a wave maker here which oscillates this
way as it oscillates it creates waves so this is S and this is eta, so I have actually eta
by S defined you know typically to push it to centimeter and this frequency you are going
to get this height etcetera. So, the question is that I want to create
this, this is my place where I want to put my structure, so I want to create a realistic
wave I want to test this structure in sea state three, so I go to formula, I end up
getting this spectrum but, this spectrum I cannot fit I have to break it down to this
signal and then only get this signal fit and of course, I get back a signal out similar
which I have to reconstruct back to this after all reality is in real time there is no imaginary,
no frequency domain etcetera. So, what is happening I require to create
this? So, I said for 2 D spectrum I created that by simply saying eta t was equal to two
S 2 D, well let me put it the other way round A omega i, omega i Cos K i x minus omega i
t plus say some beta or theta, there is a beta we call it beta i. So, you see here an
A the omega i was given as square root of 2 S 2 D omega i d omega this how we have got
it.
If you have a doubt what we have done is well here what we have done is something like this
you see this one what is this S 2 D omega, this omega i let us say d omega equal to half
A omega i square, A well we can call it A omega as A i equal to square root of two S
2 D d omega at this omega you K therefore, I end up getting this eta by summing this
all. So, what I mean for each one, I find out what is for? I break it down to several
parts for each one, I find out what is the amplitude of the wave of this frequency add
them up with a random phase. Here what I have to do in a 3 D case I have
to do the same thing but, I have to do the larger breaking, I have to break not only
over the I axis but, also over the j axis. So, what would what I’m saying therefore,
is that see it is, this is you will understand this, so if I take the same 2 D spectrum of
say height let us say height equal to five meter I will end up getting a may be wave
like that. Now, if I use a 3 D spectrum what I will probably
find out is a slightly more randomness in the sense that this might have much larger
number of waves of one particular frequency like typical or rather if I were to use a
extreme example a 2 D might give something like say more like this, in a very special
case 3 D is going to give me slightly more spread some somewhere small somewhere bigger
etcetera. In other words this will have much more focus
omegas because all are in one direction this one will be a little more spread you see,
so anyhow this is only a qualitative assessment the question therefore, you see here that
when I talk about experiment now I have this interesting point, I have this tank I have
to create this wave, so what we do you know we break it down to this signal then I break
it down to the feed I have to give to this wave maker and I give that see
Now, I have to get this I have to find out what is the corresponding this S value and
that I give it to here I push it that way. So, now see if I want to make a 2 D spectrum
I simply give this signal if I want to give a 3 D spectrum I will give this signal so
as well as the tank is concerned I can make actually of course, in the 3 D spectrum in
a tank I cannot really make in a long tank, that is not correct because in a long because
the point is that there, we are making all the wave in a one direction in fact that is
I have to correct that. Well we can create this way the spectrum time
signal but, however I really cannot create waves in a long tank use a 3 D spectrum because
in this case I am actually sending waves in all one direction actually for this purpose
only I need a so called you know like wave basin, which looks like that I will just spend
this little time.
You you’ll have a wave like IIT madras has got wave basin like that we are, we have got
this wave maker this side earlier days people use to send wave this side and wave this side
and then combining you could get nowadays very interesting phenomena is there you can
actually have multiple segment of wave maker, if you move them like a snake type if you
move this one here, this one here, this one here etcetera you will find out you can make
directional wave how you know it is the I also tell you this suppose I moved this mode
the crest would have come here, this is slightly the crest have come here, this is slightly
as the crest, so you end up getting a wave like that.
So, what happen it is very complicated theory but, you can move this wave maker like a snake
type like when it is here next one is here, next one is here, like that and then you can
make directional wave very complicated but, you can break but, having said that, so what
we understood I want to just spend this time on this class on 3 D spectrum only.
See what I understood is that I have 2 D spectrum, I have 3 D spectrum. 3 D seems to be more
realistic after all when you go to ocean you state 3 D why the hell do we use 2 D spectrum?
What is the use of 2 D spectrum then? Somebody may ask the question why I spend so many times
how to find out the response in 2 D spectrum now there is a good reason for it.
You see as I said in 2 D spectrum, I’m presuming all the waves coming from one direction therefore,
you would expect it is much more all are focused in one direction and suppose I have a ship
which responds much heavily for wave coming in one direction then it is going to go up
and down much more, now same wave if I same ship for same wave height if I assume come
from all direction this is going to be slightly mellowed in this direction see then what will
happen extreme example single wave suppose this full thing is modeled by one wave of
5 meter height. So, in 2 D case there is a wave coming of 5 meter height this side, 3
D case I will say there is a wave of 4 meter height this side one meter height from other
side. Now, this body has seems to respond mostly
for wave in this direction so in the case one the 2 D case, I end up getting usually
a larger response larger heave for example, or larger in the case of being larger role
therefore, if a ship is designed which can survive 2 D spectrum, it would normally be
lesser in 3 D spectrum therefore, if you could actually after all you are going to design
the ship for 2 D spectrum coming from all direction.
So, if I could make the ship survive a 2 D spectrum from all direction then I would more
or less think that that would be more than sufficient for it to survive a 3 D spectrum,
this is one of the reason why 2 D spectrum despite its assumption is still valid, is
still used for all my you know actual calculating otherwise we would not have done it we would
not have used at all 2 D spectrum but, even now it is used.
Now, tomorrow’s class we will find out the response in 3 D spectrum which is you are
going to find out it is nothing but, one more level of algebra see here I have added all
the omegas and responses in omega direction there I have to repeat that in theta direction.
So, here I have one piece of paper with omega and this thing here, I have to do each piece
of paper for one theta ten pieces and add them together it is just one level of algebra,
so there is nothing thinking involving that it is only a level of algebra involved but,
still having said 2 D is useful because 2 D gives me usually a large usually but, not
always a larger response a more conservative kind of a design comes out of that.