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The this course, the name of the course as
you have seen is, I just write it once again.
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Actually it is in two modules, I can we can
we can say that, it is in two large modules
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one is Seakeeping, which essentially deals
with how a ship behaves in waves or rather
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one can say motions in waves.
Now, by that what I mean is supposing you
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consider an ocean surface which will have
existing waves and you take a ship which tries
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to move, it undergoes all kinds of motions
up, down etcetera. Now, not only the motions
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as a consequence of the motions there are
many effects such as, the bow of ship might
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come out of water and bang, water may go on
top there can be large acceleration in one
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part and here can be an equipment which undergoes
a large pressure large force, because of that;
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even human beings are subjected to acceleration
and therefore, they tend to fall sick and
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have motion sickness; all these aspects, which
are a consequence of the fact, that the ship
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is moving in waves and therefore, undergoing
motions is the part of seakeeping.
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So, the first module of the course is basically
this part which we will see later on is connected
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to essentially the ship motions in vertical
plane, that is this way this way and about
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the other axis, x axis this way.
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The other part of the course, the second module
which would be approximately half the course,
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manoeuvring is connected to or related to
the behavior of the ship, in the horizontal
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plane, supposing this stage you consider it
to be the ocean surface.
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Now, it wants to turn, make a turn or it want
to take a trajectory which is zigzag, how
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will it do that how would it manoeuvre by
which instrument or equipment part of the
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hull, it will under you know it will induce
this kind of motions, these entire subject
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is manoeuvring, which is what we will do in
the second half of the course.
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Now, what you will get out of the course is,
after the first half for example, what the
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course will lead to you is an ability to assess
the, so called sea keeping qualities, it is
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what sea keeping if I again bring the first
one back the word has come from breaking it
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as how the ship keeps a sea.
Supposing a ship is excessively undergoing
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role or undergoing pitch, people would not
want to be on the boat; supposing you have
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an air craft area and you know it could not
operate or air craft could not land, if the
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deck was undergoing excessive motion. Therefore
one would have to have some kind of a limitation
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this assessment is the ship good can it withstand.
So, much of sea severity of sea, does the
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role exceeds certain amount of degrees, etcetera
all these ability you will be able to do at
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the end of the course or at least have an
idea how to go about.
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So, this is the general instruction, now having
said that, since this course you will see
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here it is motions in waves, motion of what
of ships I can add here, but where in waves,
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so therefore, at the very beginning we require
to understand how do you define a wave. Waves
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becomes the you might say my input, my I have
a surface where I got waves, I must first
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understand this waves which is my environment
in which I put my ship. So, therefore, the
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rest of the today’s class and tomorrow’s
class, we will spent trying to recapitulate
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and summarize, how do we characterize, what
is known as regular waves.
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So, I will now go to this fact of waves, strictly
speaking one should write this with a additional
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you know like qualification, surface gravity
waves, now by definition if I have a free
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surface; free surface meaning an interface
between water and air and if I cause a disturbance
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on the surface, any kind of disturbance under
action of gravity, the surface undergoes an
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undulation and this is known as waves this
is the definition of waves.
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Obviously now, there are many kinds of waves
for example, if you went to an open ocean
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you may purely find a wave surface looking
like that, but to before we get there the
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first part of the study is if I consider a
single periodic oscillation, which gives rise
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to a wave of this nature, that you might have
seen many times depicted many books, many
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many places this is what is called as regular
waves. Now, this regular wave has already
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been studied before this course as a pre requisite
in other courses like main in hydrodynamics,
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but I need to still recapitulate certain parts.
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Now, regular waves we know theoretically that,
regular waves are studied based on potential
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theory, which means that the fluid is assumed
in viscid and the flow is assumed irrotational
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free resolves is also assumed homogeneous
and incompressible that means, actually this
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gives rise to this classical governing equation
called laplace equation as we know, we end
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up getting, where phi is known as the velocity
potential, such that velocity vector q is
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actually good at phi.
This is a very basic definition of irrotational
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motion, fluid flowing irrotational motion
now, for surface gravity waves if you want
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to study turns out, you have to have certain
boundary condition imposed on the fluid.
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So, there are, so called boundary conditions
there are two boundary condition that becomes
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one is what we called kinematic boundary condition
and one what is called dynamic boundary condition.
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Now, having gone through all this it turns
out which is what I will simply recapitulate,
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that the surface gravity waves potential turns
out to be equal to without prove we are saying,
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because this is supposed to be in a pre requisite
to a course and you would have known that
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it turns out for a linear surface gravity
wave the phi turns out to be equal to g a
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by omega k. Where well this is a kind of solution
that comes where let me specify this parts,
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h is water depth in this formula, z is positive
upwards, origin is at the mean free surface
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that is z equal to 0 would imply the mean
free surface, a is the amplitude of the wave
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cos h cos h k h.
So, in this case k is known as the wave number
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given by 2 pi by lambda, omega is known as
the wave period 2 pi by t, wave here is actually,
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A having an amplitude
and the wave profile eta is given by A cos
k x minus omega t, actually eta is obtained
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from the fact that is minus 1 by g d phi by
d t at z equal to 0 you can confirm that this
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if I do this operation on this it becomes
like this.
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So, here the wave profile therefore, would
look like a cross curve, if were to call this
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where it can be actually t, it can be also
x, it can be either time axis which means
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you are looking at one point or it can be
at a given time x axis, avail depending on
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that if I were to call this t, then the the
time taken for it evolve is period t rather
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than I want to call this x here, then this
becomes my wave length lambda this is my A
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etcetera, etcetera.
What it means therefore, you see is that according
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to the theory according to the theory in which
we have already made an assumption of smallness
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of the amplitude with respect to length, which
means in evolving this particular getting
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back to that, in evolving this theory we had
made an assumption of linearity. The word
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linearity implies here physically, that we
have said that
its slope that is d eta by d X d eta by d
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Y, that is it horizontal slope
is assumed to be small
this was an assumption inherent based on which
the formula, that the profile is sinusoidal
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has emerged.
So, therefore, what we can say is that the
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sign profile is a theoretical solution of
a regular wave, based on the affect that the
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slope has been assumed to be small, this is
what is called as the number of number of
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terms it is we can call it linear wave theory,
you can call it airy wave theory, you can
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call it first order wave theory or whatever.
So, all these are basically synonymous all
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these means, that you made an approximation
that the slope is small and if that is the
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case then you end of getting a profile given
by sin curve and what is most interesting
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to see here is that, you will find that the
phi expression back here is sinusoidal with
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respect to special axis x, that is with respect
to the special coordinate and with respect
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to also time coordinate t.
So, therefore, if I were to take a gradient
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of y with respect to either time or with space
it would also be sinusoidal and it turns out
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all quantities of physical interest are actually
gradient of phi.
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For example, if I want to find out velocity
say v of the practical it is given by grad
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phi; that means, u is given by d phi by d
x, w vertical it is given by d phi by d z.
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If you want to do u dot that is acceleration
that is d by d t of d phi by d x, if I want
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to do w dot horizontal acceleration it is
d by d t of d phi by d x other important thing
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is pressure if I want to find pressure; the
pressure is given by these expression in here,
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this is my hydrostatic pressure
and this is what is called a linear dynamic
pressure.
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Remember linear dynamic pressure the word
linear because there was another term, which
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was second order which we neglected according
to our linear wave theory, now you find out
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that, this linear dynamic pressure is gradient
of phi in time.
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So, if you were to do this differentiation
of my phi curve obviously, all them is going
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to be sinusoidal in some form or other, so
therefore, a turns out that in wave linear
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wave phenomena, be it velocity, be it acceleration,
be it pressure, everything has a periodicity
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of frequency omega and time period t as you
know omega is 2 pi by t etcetera.
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So, having said that this is only the physical
characteristic, but one most important thing
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of the linear water wave, which we have not
said is the fact that, it follows a relation
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called dispersion relation given by this expression,
this is called dispersion relation.
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Now, this is a very, very important relation
as far as water waves are concerned why, so
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you will find out that, here my left hand
side is omega, now omega which is nothing,
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but to pi by t, so omega T the time period
is related on this side to K. What is K? K
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is nothing but 2 pi by lambda that means,
essentially if for a given water depth h which
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means that if for a given water depth, I have
a wave length lambda it is necessarily of
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certain variate T I cannot have for example,
in a given water depth a wave period of 10
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second and 20 second of same length.
Now, this you can contrast for example, with
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electromagnetic radiation, where you can have
different wave length, different frequency,
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but there is a speed is constant, we will
find out here that, now what is my speed of
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a wave see, when I have this wave here next
instant it goes like that, next instant it
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goes like that, the form of the wave is traveling.
In fact, I can show that if I were to write
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this eta to be A cos k x minus omega t this
can be written as A cos k x minus omega by
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k t and you will see that if you have to travel
at the speed omega by k, with respect to this
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wave to you it will look stationery, which
essentially mean the form is traveling with
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the speed omega by k.
So, omega by k becomes my phase speed or some
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people call it celerity and it is interesting,
because omega by k is nothing, but lambda
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by T if you see because omega is 2 pi by t
k is 2 pi by lambda and obviously, it make
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sense, because what is happening is that,
this is lambda and the time for it to the
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form to actually go down one cycle and come
up which would imply as if this has gone here
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is T.
So, therefore, you would think this form has
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travelled from here to here in time T, so
this is what is phases speed, remember it
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is the form that is traveling not the particle
itself, we will see later on that the particle
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here, they move in a circular fashion we will
come to that in a minute.
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Now, having said that this omega square etcetera,
now I will actually come to one simplification,
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what is happening here you see that in this
expression this is all for water depth h,
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now in if for example, h was very deep in
other words k h is you know like 2 pi by lambda
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into h is very high, tan h k h will tend to
actually 1 like this relation this g k, this
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tan h k h this will tend to actually 1 and
even this relation will all simplify.
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In such cases what happens my relation this
relation becomes omega is equal to g k and
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in fact, this relations, in fact or become
exponential k z e power of k z minus k z.
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So, in deep water when I assume h tends to
infinity then k h; obviously, tends to also
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infinity because you know k h is 2 pi by lambda
into h.
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Then tan h k h tends to 1, because tan h k
h actually if you if you write x here tan
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h graph looks something like that equals to
1 that is tan h x and there are similar relation
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for sin h and cos h x for large values of
x.
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So, if I put those thing it turns out in deep
water well, the word deep water implies for
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h tending to infinity for now I am saying
this, now we will now find a practical name
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of what is h in a minute it turns out phi
reduces to g A by omega e power of well actually
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this will become depending on which side the
z is the z is plus opposite. So, therefore,
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it is plus k z and of course, eta remains
same and my dispersion becomes omega square
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equal to g k.
Now, this is another thing what I will like
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to tell you here, let us look at this this
relation
you will find here two things, one is that
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phi is a function of x sinusoidal is a function
of z exponentially and function of t sinusoidal.
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So, therefore, what is happening when I differentiate
this number 1 is that, it always remains sinusoidal,
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because when I differentiae it supposing I
differentiate it with respect to x it is sinusoidal,
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if I differentiate with respect to z also
it is sinusoidal, because this term get differentiation
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if I do it against t also sinusoidal.
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But other thing is that its z dependence is
exponential; that means, that if I were to
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see a property of a water wave along z, every
property is reducing sinusoidal and every
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property is harmonic in the horizontal plane,
whether it is with respect to space or with
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respect to time. This is very important because,
we will find out afterwards then when I put
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a ship here; obviously, it is it would also
respond to this harmonic excitation you know
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something is being pushed every 10 second,
so it will also have that every 10 second
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oscillation.
So, this is extremely important from water
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wave that things come down exponentially,
how does it come down, with e power of k z,
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what is k z, e power of two pi by lambda into
z. Now it turns out that normally if lambda
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or rather I will tell you if z is more than
equal to lambda by 2 then e power of k z that
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is becomes e power of minus well, minus z
I will say because z is plus opposite minus
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pi, this value itself is something like 0.04.
And normally just like, when we say you know
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the sin theta is theta, if theta is less than
4 degree similarly we will say that normally,
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when e power of the exponential part has become
more than e power of minus pi it is almost
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0.
So, the rule of thumb is that, that is how
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this word came that is water depth is more
than half the wave length, then anything that
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is happening below that is, so small almost
0 then you can ignore it. So, we therefore,
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we say that if water depth was more than half
the wave length, we consider the water depth
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to be deep water or it is a deep water case.
On the other hand if the water wave is less
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of course, that would not hold, in fact, we
call that too intermediate water, now why
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I am saying is because in our course of seakeeping
normally, we always deal with deep water cases
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why because a typical wavelength would be
about 100 meter, 200 meter, 300 meter, may
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be 400 meter, but ocean depth were ships operate
runs in 1000 of meter.
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So, mostly the operation of ship would be
in deep water, whatever the water is therefore,
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rest of the course when we talk we will only
consider deep water cases, because ship operating
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deep water, when we are looking at ship, you
can always refer to the exact equation with
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h, it is an unnecessary complication; there
is no point of doing it, because when you
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actually put numbers it will turn out that
e power of k h you know tan h k h will become
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almost 1, 0.99999 something.
So, having said that this is my property of
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water wave, now I will try to tell you about
the practical you know velocity etcetera etcetera
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well before that let us now spend little more
time on dispersion relation still.
191
00:26:15,680 --> 00:26:22,680
So, I have this omega square equal to g k,
now speed c what is c, c is omega by k, so
192
00:26:25,100 --> 00:26:32,100
that is g by omega, now g by omega is what
g by, now you see it is g by omega means speed
193
00:26:32,880 --> 00:26:39,880
depends on frequency inversely, so if I what
to write omega T it will become 2 pi by T.
194
00:26:41,540 --> 00:26:47,580
So, therefore, speed depends on period, but
if I want to write in terms of the length
195
00:26:47,580 --> 00:26:54,580
g by c omega, I can represent as root over
of g k, because omega is root over of g k,
196
00:26:55,710 --> 00:27:02,710
so this will turn out to be root over of g
by k it is root over of g by lambda by 2 pi,
197
00:27:04,100 --> 00:27:11,100
so what do I find c with root lambda, c with
1 by omega, c with T.
198
00:27:13,770 --> 00:27:20,770
So, this tells us that celerity of water wave
or phase beat depends on length and of course,
199
00:27:21,670 --> 00:27:27,280
length and frequency are connected therefore,
it depends on length or period or frequency,
200
00:27:27,280 --> 00:27:34,280
which means in deep water if I were to tell
omega or t or lambda or c all of them, in
201
00:27:37,050 --> 00:27:42,610
deep water and also in shallow water for a
given h would immediately relate to each other,
202
00:27:42,610 --> 00:27:46,020
there can be only one value for this and this
and this and this.
203
00:27:46,020 --> 00:27:51,900
So, you cannot have for example, two different
wave lengths travelling at two different speed,
204
00:27:51,900 --> 00:27:58,220
this is where I was earlier mentioning water
waves differ from electromagnetic radiation,
205
00:27:58,220 --> 00:28:03,580
because in electromagnetic radiation or in
sound sound waves for example, sea is constant
206
00:28:03,580 --> 00:28:09,770
for example, in sound waves if sea was not
constant you probably would not hear me, because
207
00:28:09,770 --> 00:28:13,150
the ear’s frequencies would have reached
your ears differently.
208
00:28:13,150 --> 00:28:19,470
If the electromagnetic radiation was not of
same speed, you would not see the picture
209
00:28:19,470 --> 00:28:24,260
white, because all colors would be different,
but in water waves it is, so so this is why
210
00:28:24,260 --> 00:28:29,900
water waves are called dispersive, because
if 10 water waves start of 10 different length
211
00:28:29,900 --> 00:28:34,740
at 1 point, then eventually the longer one
would have travelled this much shorter one,
212
00:28:34,740 --> 00:28:40,450
longer one would have travelled further.
So, you know it would have separated out this
213
00:28:40,450 --> 00:28:46,700
is what we called dispersion relation very,
very important relation and what happen, now
214
00:28:46,700 --> 00:28:53,700
coming back to this this part, this pi we
are writing this c g A by omega, some people
215
00:28:54,670 --> 00:28:59,910
will write in terms of some other parameter,
because I can now write omega in terms of
216
00:28:59,910 --> 00:29:06,910
t in terms of lambda in terms of k.
So, there this part can be rewritten depending
217
00:29:07,340 --> 00:29:13,480
on what parameter you want to use as independent
parameter it can be something like g A by
218
00:29:13,480 --> 00:29:20,480
well root over of g k for example, I can write
g A by omega, as g A by well not in this case,
219
00:29:21,910 --> 00:29:28,710
in the deep water case g k, so root over of
g k and I can relate that root over of g 2
220
00:29:28,710 --> 00:29:35,320
pi by lambda a etcetera. So, this is a question
of only convenience of how we proceed, it
221
00:29:35,320 --> 00:29:39,870
really depends on the way we want to look
at it.
222
00:29:39,870 --> 00:29:46,870
Now let us, look at some of these numbers
therefore, you know like relation of this
223
00:29:47,200 --> 00:29:54,200
number, now typically let us say, let us take
a wave period of T before before that let
224
00:29:59,290 --> 00:30:06,020
me also see this other way round, now lambda
it turns out therefore, equal to 2 pi g by
225
00:30:06,020 --> 00:30:10,980
omega square, this is from the dispersion
relation omega square is equal to g k straight
226
00:30:10,980 --> 00:30:14,450
forward.
So, this we are trying to relate, now a relationship
227
00:30:14,450 --> 00:30:21,450
this turns out to be if you are to put a s
I unit that lambda in meter g is you know
228
00:30:22,660 --> 00:30:29,660
like meter per second square and period T
in terms of second, actually this I can write
229
00:30:30,240 --> 00:30:37,240
this way g by 2 pi square by T square, so
this turns out to be g T square by 2 pi.
230
00:30:38,920 --> 00:30:45,920
So, this in the this thing becomes almost
1.56 T square in SI unit, so that tells me
231
00:30:48,330 --> 00:30:55,330
that if T was 8 second, lambda would have
been 8 square into 1.5 around 100 meter approximately
232
00:31:02,390 --> 00:31:09,390
if T was to be 10 second, lambda would be
around 150 meter, say T was about 12 second,
233
00:31:12,070 --> 00:31:18,090
lambda would be approximately 144 into 1 and
half about 200 as, so let say and it is all
234
00:31:18,090 --> 00:31:21,130
approximate number.
So, this gives me a feel about the relation
235
00:31:21,130 --> 00:31:26,070
between these number and this let us look
at the speed value, say for this what is the
236
00:31:26,070 --> 00:31:31,360
phase speed very simple 100 by 8 because lambda
by T, it would have been something like 12
237
00:31:31,360 --> 00:31:37,400
meter per second here, you will see 200 by
12 it would be approximately 16 meter per
238
00:31:37,400 --> 00:31:41,370
second.
So, you see speed is increasing this is of
239
00:31:41,370 --> 00:31:47,630
course, a reason why a long wave like tsunami
you have travel speed, so high phase speed
240
00:31:47,630 --> 00:31:52,600
of travel, so high you know it can be in in
many kilometers per second per hour.
241
00:31:52,600 --> 00:31:59,600
So, this is exactly the you know like part
connected to dispersion relation, now I will
242
00:32:02,840 --> 00:32:09,840
just want to also tell about the particle
path, that it takes it turns out if we were
243
00:32:10,460 --> 00:32:17,460
to do a study in deep water this particles
actually travel in a circular way, this is
244
00:32:29,230 --> 00:32:36,140
my circular way. And obviously, since I mention
earlier that everything diminishes with this
245
00:32:36,140 --> 00:32:40,950
thing, in here the particle will move this
way in here the particle will move this way
246
00:32:40,950 --> 00:32:44,290
etcetera.
So, the particle motion is basically circular
247
00:32:44,290 --> 00:32:48,760
path that means, if I were to put a particle
here, it actually stays in same place going
248
00:32:48,760 --> 00:32:55,520
like that, it does not move forward according
to linear theory, so there is no net mass
249
00:32:55,520 --> 00:33:01,970
moving forward as per linear theory it is
only the form that is moving forward.
250
00:33:01,970 --> 00:33:08,600
So, the particle are just moving like this,
you can also get a feel about the particle
251
00:33:08,600 --> 00:33:15,600
velocities for the simple reason that see,
this is A this circle distance is two pi A
252
00:33:16,130 --> 00:33:22,470
and a particle is traveling is 2 pi A distance
on average in period t, so therefore, you
253
00:33:22,470 --> 00:33:26,940
know 2 pi A by t would be a kind of you know
like speed.
254
00:33:26,940 --> 00:33:31,890
Now, one thing in dispersion relation that
is very important is that, we come back to
255
00:33:31,890 --> 00:33:38,890
dispersion relation is because see, this vertical
axis which is my A this has no connection
256
00:33:39,970 --> 00:33:45,080
with the dispersion relation I could have
another wave this much or another wave very
257
00:33:45,080 --> 00:33:51,480
small all of them will have same frequency
omega say length is same.
258
00:33:51,480 --> 00:33:57,450
So, this two are connected, but for the same
omega lambda combination I cannot different
259
00:33:57,450 --> 00:34:04,450
A’s, so in other words the dispersion relation
is connecting my quantities in the horizontal
260
00:34:05,570 --> 00:34:10,760
axis, time axis and x axis, but nothing in
the vertical axis.
261
00:34:10,760 --> 00:34:17,760
In other words if I have an 8 second wave,
it is 100 meter what is the height, well 100
262
00:34:18,440 --> 00:34:23,240
meter wave can have a height of 1 meter, can
have 2 meter, can have 3 meter this is not
263
00:34:23,240 --> 00:34:30,220
specified why it is so, linear theory because
remember that I have made an assumption that
264
00:34:30,220 --> 00:34:37,220
well eta x eta y is small.
What is this approximately, they are equal
265
00:34:38,720 --> 00:34:45,720
to A by lambda in in that order, so what is
happening A by lambda has been taken as small,
266
00:34:46,530 --> 00:34:53,530
but no specifically of A what is small one
can be small 0.02 is also small therefore,
267
00:34:53,590 --> 00:34:58,050
if I take 0.01 for example, my A would have
been certain number, 0.02 would be twice the
268
00:34:58,050 --> 00:35:03,460
number. So, A cannot be specified, so the
theory applies for small a as long as a by
269
00:35:03,460 --> 00:35:10,460
lambda a small, but not for in specifically
of a, this is very important to understand.
270
00:35:12,550 --> 00:35:19,550
So, now the next point very most important
point is actually to talk about is pressure,
271
00:35:22,310 --> 00:35:29,310
the other part of water wave would be pressure
p, now what did we see before, t is minus
272
00:35:32,920 --> 00:35:39,920
rho g z minus rho d phi by d t. Now, let me
write the phi expression again, because this
273
00:35:41,780 --> 00:35:48,780
is important for us to realize this it was
g A by omega e power of k z sin k x minus
274
00:35:52,540 --> 00:35:59,540
omega t now much is rho d phi by d t.
Let us work it out see, minus rho d phi by
275
00:36:02,480 --> 00:36:09,480
d t this linear dynamic pressure, if I do
minus here, so this 1 minus let me just write
276
00:36:09,830 --> 00:36:16,830
it down, this becomes minus omega comes in
g A by omega, that is right, so that means,
277
00:36:29,350 --> 00:36:36,350
it is minus g A a power of k z, no cos x minus
omega k etcetera.
278
00:36:41,520 --> 00:36:46,820
Now, what is happening obviously, you can
understand this that again, now if I see this
279
00:36:46,820 --> 00:36:53,820
eta it is something like A cos k x minus omega
t, so naturally there is a highest dynamic
280
00:36:56,490 --> 00:37:02,580
pressure, either positive or negative occurs
under crest of trough for this, but remember
281
00:37:02,580 --> 00:37:08,480
in all this formula my z is 0 here.
282
00:37:08,480 --> 00:37:15,460
Now, let us look at the two part of the pressure
by a plot, the hydrostatic pressure and the
283
00:37:15,460 --> 00:37:22,460
liner dynamic pressure, now let us look under
trough, so I have got here say now, under
284
00:37:23,270 --> 00:37:30,270
this see remember this z 0, how how does the
hydrostatic pressure vary well it is minus
285
00:37:35,990 --> 00:37:41,940
rho d z z is negative down.
So, therefore, this becomes positive here
286
00:37:41,940 --> 00:37:48,940
this way, giving a pressure we can say is
positive pressure; obviously, pressure can
287
00:37:51,510 --> 00:37:55,270
be positive, but the interesting point is
that what is happening here, well you end
288
00:37:55,270 --> 00:37:59,740
up getting a negative pressure that does not
make any sense, but any how let us put this
289
00:37:59,740 --> 00:38:06,740
way you end up getting here, rho g z that
is static pressure to be a minus rho g into
290
00:38:07,870 --> 00:38:14,870
A at this point.
Now, what is my dynamic pressure, you will
291
00:38:17,570 --> 00:38:23,760
find out that if I were to work, this is also
well see here, actually there is rho here,
292
00:38:23,760 --> 00:38:30,730
that I this rho has to be there, this rho,
this rho here, so this will be also rho g
293
00:38:30,730 --> 00:38:35,210
A because z is 0.
So, if you work it out here it will turn out
294
00:38:35,210 --> 00:38:42,210
to be like this exponentially down and this
merge is rho g A, everything is fine remember
295
00:38:51,690 --> 00:38:55,860
that we have defined in our theory and that
is the most important point when you are doing
296
00:38:55,860 --> 00:39:02,860
this application, my z is 0 at this level
this is my z equal to 0 level and my water
297
00:39:07,070 --> 00:39:14,060
is only below z equal to 0, my water is assumed
in theory only z equal to less than 0, this
298
00:39:14,060 --> 00:39:21,060
is my fluid domain.
However please understand this that I have
299
00:39:23,610 --> 00:39:30,610
now a crest here means, this part is water,
but my theory I always have only in a valid
300
00:39:31,030 --> 00:39:36,740
for z equal to less than 0, what happen here
I know, I want to know physically what is
301
00:39:36,740 --> 00:39:39,940
the pressure of this point for example, how
do I find out.
302
00:39:39,940 --> 00:39:45,120
Let us say, this is 5 meter I want I want
to find out what is my pressure 1 meter below
303
00:39:45,120 --> 00:39:52,120
the crest, now if I were to apply only rho
g z by this formula what do I get a completely
304
00:39:52,690 --> 00:39:56,160
absurd result of a negative pressure.
305
00:39:56,160 --> 00:40:03,160
No, that is not, but it is that is true, but
then this wave is of course, existing here
306
00:40:03,890 --> 00:40:10,360
see the wave is existing because the particles
are moving liquid cannot sustain sheer, question
307
00:40:10,360 --> 00:40:15,080
that is ask if perfectly correct, so why that
there was no motion there see obviously, if
308
00:40:15,080 --> 00:40:19,810
you look at the wave it is sustaining this
ship it is of course, sustaining the ship,
309
00:40:19,810 --> 00:40:23,930
if it did not sustain the ship, I would not
have a wave, why it is sustaining the ship,
310
00:40:23,930 --> 00:40:26,560
because the particles are moving.
So, there are dynamic involve and this is
311
00:40:26,560 --> 00:40:33,560
a question you cannot presume this mass of
water to be hydrostatic and if I take an hydrostatic
312
00:40:34,650 --> 00:40:40,770
pressure, I end up getting minus rho g z,
that is that absurd result come because, hydrostatically
313
00:40:40,770 --> 00:40:45,740
it is it is absurd to consider that water
mass can stay like that, it can only stay
314
00:40:45,740 --> 00:40:51,140
along with the hydrodynamics.
So, now if I were to add the dynamic pressure
315
00:40:51,140 --> 00:40:55,490
there there is also a problem supposing I
add dynamic pressures, so I apply this formula,
316
00:40:55,490 --> 00:41:02,250
so what would happen it goes like that, since
z e power of k z z equal to plus 1 whatever,
317
00:41:02,250 --> 00:41:06,950
now if you look at this point what will happen
this; obviously, is more than rho g A, now
318
00:41:06,950 --> 00:41:11,800
this is rho g A, so this will end up getting
some pressure that is not correct, because
319
00:41:11,800 --> 00:41:16,250
my pressure of this point is suppose to be
0, where is anomaly?
320
00:41:16,250 --> 00:41:21,540
The anomaly is because that in developing
linear water wave theory, what we have said
321
00:41:21,540 --> 00:41:28,540
is that this height with respect to the length
is very small a consequence of for that was
322
00:41:28,990 --> 00:41:35,990
that, every quantity that is on the surface
is assumed to be same as it is on the z equal
323
00:41:37,800 --> 00:41:44,470
to 0 line; that means, if I were to take a
point here and if I were to take phi at z
324
00:41:44,470 --> 00:41:51,300
equal to eta this is taken same as phi z equal
to 0.
325
00:41:51,300 --> 00:41:57,530
Now; obviously, you look at a pressure, what
is pressure here, d phi by d T at z equal
326
00:41:57,530 --> 00:42:03,420
to eta, but that I have said is same as d
phi by d T at z equal to 0; that means, I
327
00:42:03,420 --> 00:42:09,500
have said the pressure in this region dynamic,
all the dynamic quantities are same as which
328
00:42:09,500 --> 00:42:15,150
means I made a presumption, that this actually
pressure is like this, this pressure is constant
329
00:42:15,150 --> 00:42:19,300
in this region.
And if I were to do this then, there is no
330
00:42:19,300 --> 00:42:24,010
problem because now if I add this line, with
this line what do I get I get a pressure here,
331
00:42:24,010 --> 00:42:29,830
this way because you see here up to this much
if I add this black line with this line, I
332
00:42:29,830 --> 00:42:35,620
get the pressure A and here I get the pressure
like this this is how much this is nothing
333
00:42:35,620 --> 00:42:42,620
but this plus this, that is this much plus
this much, gives you this much.
334
00:42:43,820 --> 00:42:49,750
So, this is exactly how the pressure varies
under linear water wave that this is extremely
335
00:42:49,750 --> 00:42:56,750
important for us to understand, because you
see you must have a physical pressure here
336
00:42:58,110 --> 00:43:05,110
and the pressure dynamic pressure is having
like this, once again I repeat this whereas,
337
00:43:05,220 --> 00:43:08,860
hydrostatic pressure is having like that,
now if I were to evaluate a pressure in this
338
00:43:08,860 --> 00:43:15,860
region or anywhere under wave you cannot use
hydrostatic pressure by itself, because if
339
00:43:17,550 --> 00:43:23,840
you did that you end up getting a wrong result,
but if you were to find pressure, you have
340
00:43:23,840 --> 00:43:29,470
to use hydrostatic and hydrodynamic pressure
simultaneously.
341
00:43:29,470 --> 00:43:33,940
Supposing I want to use hydrostatic pressure
as you have done sometime in ship strength,
342
00:43:33,940 --> 00:43:39,980
in ship strength what we do if you recall
you simply take a ship, you have this profile
343
00:43:39,980 --> 00:43:45,720
and you find this buoyancy curve based on
buoyancy that means, you are using only hydrostatic
344
00:43:45,720 --> 00:43:49,250
pressure.
What here you can see is that, supposing I
345
00:43:49,250 --> 00:43:54,640
use this my datum line this was my z equal
to 0, what would my hydrostatic pressure it
346
00:43:54,640 --> 00:44:01,640
would have been like that, well this is closer
to see now, if I were to see this color, this
347
00:44:02,760 --> 00:44:09,760
line is one and reality is this and this,
so at least this is not that bad in a sense
348
00:44:12,190 --> 00:44:16,180
that you will not go completely off, your
going to make this much of under over prediction
349
00:44:16,180 --> 00:44:23,180
fine, but you do not go completely off.
So, if you use only hydrostatic pressure,
350
00:44:23,460 --> 00:44:28,770
so you know that if you have to use only hydrostatic
pressure, then you must use you cannot use
351
00:44:28,770 --> 00:44:33,710
rho g z with z equal to 0 was the mean line,
then you must use a instantaneous water surface.
352
00:44:33,710 --> 00:44:39,500
I gave an example of this with respect to
a both now, in a minute see we, so we end
353
00:44:39,500 --> 00:44:46,500
up finding this, that if I was to have a this
thing, the pressure depth is basically coming
354
00:44:48,520 --> 00:44:55,520
like this this my pressure, actual pressure
and if I were to ignore the dynamics and I
355
00:45:02,070 --> 00:45:06,810
use this as z 0 when my pressure would have
been this, now you think of this a boat what
356
00:45:06,810 --> 00:45:13,810
happens. Suppose there is a boat here, now
what happens, but let us consider in a calm
357
00:45:16,010 --> 00:45:23,010
water first the boat, it has a trough of t,
which means that below T I have this pressure
358
00:45:25,600 --> 00:45:30,820
rho g rho g T is supporting it, now what is
happening here you see or rather I should
359
00:45:30,820 --> 00:45:36,950
make it bigger see trough like this much.
Now, thing is that in this case what happens
360
00:45:36,950 --> 00:45:43,280
tell me, if the trough go down or go up under
crest, what would be the the you know the
361
00:45:43,280 --> 00:45:48,830
situation remember that you see that the trough
would have been same if the pressure was this
362
00:45:48,830 --> 00:45:54,740
much, this much here, but actually my pressure
is this much. So, what would happen the body
363
00:45:54,740 --> 00:46:01,040
will actually sink down in crest, because
I must have the pressure equal to this much
364
00:46:01,040 --> 00:46:08,000
to support that, I will explain this this
picture again once more.
365
00:46:08,000 --> 00:46:15,000
Now, perhaps I will take a better diagram
once again, because this diagram, I will take
366
00:46:18,470 --> 00:46:25,470
this case here, see the pressure actual pressure
on the wave under a crest looks like this
367
00:46:36,350 --> 00:46:43,350
this is my straight line and actual pressure
looks like that, that my actual pressure.
368
00:46:44,770 --> 00:46:51,770
Now I have a boat which has a trough of some
value, what does it mean; it means that in
369
00:46:52,990 --> 00:46:59,990
this here I had a hydrostatic pressure of
rho g into T which supported my mass.
370
00:47:00,940 --> 00:47:07,880
So, I must go down that much where my pressure
is rho g T, it is a flat bottom plane because;
371
00:47:07,880 --> 00:47:11,780
obviously, I must have as much trough were
here, because only this phase is contributing
372
00:47:11,780 --> 00:47:18,780
to my buoyancy in this case.
Now, here what has happened I have this body,
373
00:47:21,200 --> 00:47:28,200
now you see here, in this case my trough here
is this much, but my pressure this is my T
374
00:47:31,940 --> 00:47:38,940
and this is my rho g T this much, so if the
pressure was this much my trough will be in
375
00:47:40,150 --> 00:47:42,230
T, but actually my pressure is only the small
line.
376
00:47:42,230 --> 00:47:49,230
So, my rho g T pressure would come when this
bottom surface is here, which means the body
377
00:47:50,170 --> 00:47:57,170
must actually come down, because the pressure
below the crest have reduced rather than increased,
378
00:48:00,070 --> 00:48:07,070
so therefore, the ship must come down further
in order to support itself and the opposite
379
00:48:08,280 --> 00:48:12,240
would actually happen in trough.
So, therefore, what would happen of course,
380
00:48:12,240 --> 00:48:18,480
this actually happen small you know numbers
are not, so large, but there is a tendency
381
00:48:18,480 --> 00:48:24,250
for the trough to come down and go up more,
because of the hydrodynamic contribution dynamic
382
00:48:24,250 --> 00:48:30,570
pressure contribution, so the dynamic pressure
has therefore, a extremely important role
383
00:48:30,570 --> 00:48:34,080
to play.
Now we are interested in dynamic pressure,
384
00:48:34,080 --> 00:48:41,080
so much more, now let us look at this in the
same case also hydrostatic balance case, ultimately
385
00:48:41,270 --> 00:48:48,270
I want to find force for any dynamic system
I would like to find out the force. The net
386
00:48:48,790 --> 00:48:55,790
force acting on the body, because the module
seakeeping is because to study, what the waves
387
00:48:58,869 --> 00:49:03,750
does to the structure, what does it to.
Well in a hydrostatic case you know if there
388
00:49:03,750 --> 00:49:08,790
is a body there it gives you hydrostatic pressure,
that gives you buoyancy and that makes gives
389
00:49:08,790 --> 00:49:14,010
you a float, everything is based on that,
now in a dynamic cases same thing, I actually
390
00:49:14,010 --> 00:49:19,110
have to find out what is my pressure well,
what is the force and force is nothing but
391
00:49:19,110 --> 00:49:26,110
integrational pressure over the surface.
So, I need this pressure very well and therefore,
392
00:49:29,630 --> 00:49:34,860
I need to find the pressure very well of course,
the fact that, if there is a body there the
393
00:49:34,860 --> 00:49:39,400
wave itself would changes the different issue
altogether will come to that afterwards, but
394
00:49:39,400 --> 00:49:45,140
the fact that even if I did not assume that,
we still have to figure out the pressure under
395
00:49:45,140 --> 00:49:50,380
wave. And therefore, I need to have a very
good understanding about pressure and if I
396
00:49:50,380 --> 00:49:57,380
do not take the z etcetera properly I end
up getting pressure wrong, so I were to only
397
00:49:58,300 --> 00:50:05,300
use hydrostatic pressure I will repeat again,
you must use z 0 to be the variable axis from
398
00:50:10,190 --> 00:50:14,910
measured from the instantaneous water line.
If I was using a theoretically consistent
399
00:50:14,910 --> 00:50:20,840
value of z 0 to be the mean water line, then
I have to use hydrostatic and hydrodynamic
400
00:50:20,840 --> 00:50:27,840
pressure together not in isolation, because
in isolation any of them cannot exist, this
401
00:50:28,280 --> 00:50:33,450
is one of the most important part that you
know we we end up finding here.
402
00:50:33,450 --> 00:50:40,450
Now, let us see now, the same similar thing
will actually happen in a in a trough case
403
00:50:45,030 --> 00:50:52,030
I were to look at the trough case you know
like, let me see this trough case here here
404
00:50:58,080 --> 00:51:05,080
this pressure is opposite see this was like
that coming whereas, other pressure well this
405
00:51:07,520 --> 00:51:14,520
a part a part here it is this side. So, we
have to actually now add this two, so therefore,
406
00:51:20,950 --> 00:51:26,630
the situation become different and there what
would happen is that, see here again if I
407
00:51:26,630 --> 00:51:29,590
were to see remember there is no water here
this part.
408
00:51:29,590 --> 00:51:36,590
let me just see this, see here, essentially
what would happen therefore, this this this
409
00:51:38,710 --> 00:51:45,710
and this will make it zero here, so I will
end up getting if I were to have this this
410
00:51:48,750 --> 00:51:55,750
part, but with this respect I have this added.
So, I end up getting this minus this because,
411
00:51:58,369 --> 00:52:02,950
this on the other side will end up getting
some value like that.
412
00:52:02,950 --> 00:52:08,400
So, what I mean is that again the similar
kind of thing will occur you know like in
413
00:52:08,400 --> 00:52:12,360
trough like in crest, the only problem that
happens here is that, because there is no
414
00:52:12,360 --> 00:52:17,280
water here you do not have to really consider,
because this region there is no water.
415
00:52:17,280 --> 00:52:23,200
Now, I tell you why these things are important,
because we will find out that in many case
416
00:52:23,200 --> 00:52:29,240
of floating body structure ultimately, you
will have to find the pressure over the mean
417
00:52:29,240 --> 00:52:35,800
water surface even though the water is like
that, theory will tell us that we have to
418
00:52:35,800 --> 00:52:42,800
determine on the mean water surface. And that
is why it is always good to know the pressure
419
00:52:43,640 --> 00:52:49,619
and use it consistent theory of z equal to
0 on the mean free surface, if z equal to
420
00:52:49,619 --> 00:52:54,830
0 on the mean free surface, then you will
have no problem.
421
00:52:54,830 --> 00:53:01,830
So, for our study purpose, we would always
take z this mean this is x and this is my
422
00:53:02,650 --> 00:53:09,650
z equal to 0, that is z equal to will be always
my mean surface, so essentially today I will,
423
00:53:18,500 --> 00:53:23,210
in fact, close on this pressure part and we
will next lecture we will have one more lecture
424
00:53:23,210 --> 00:53:30,170
on linear water wave theory, we will discuss
about the other aspects of energy, energy
425
00:53:30,170 --> 00:53:36,760
flux group speed, etcetera and perhaps we
will also work out some simple elementary
426
00:53:36,760 --> 00:53:40,690
calculations.
You know who these numbers are used to determine,
427
00:53:40,690 --> 00:53:47,690
you can actually see from here itself as an
example, this speed as I said c is equal to
428
00:53:50,340 --> 00:53:57,340
omega by k and the wave, we have done is omega
by 2 in terms of in terms of you know like
429
00:53:59,400 --> 00:54:06,400
let me shift time it was something like 2
pi by t and omega g, omega square is g k;
430
00:54:08,100 --> 00:54:14,820
so well well I was trying to write it in terms
of time let me just figure it out.
431
00:54:14,820 --> 00:54:17,700
No, no that is
432
00:54:17,700 --> 00:54:24,700
See, c is omega by k omega square equal to
g k, in any case you can relate that c to
433
00:54:25,420 --> 00:54:30,070
be in proportional to something like T that
we have done at some point of time. So, what
434
00:54:30,070 --> 00:54:37,070
I am saying is that now, if see it turns out
that if T was 8 second my lambda was 100 meter
435
00:54:40,050 --> 00:54:47,050
and c was about 12.5 meter per second.
But you will see that, if my T is equal to
436
00:54:47,190 --> 00:54:54,190
say a tsunami wave 20 minutes or 30 minutes
say 30 into 60 second, you work out lambda
437
00:54:54,790 --> 00:54:59,830
you will find out, it will turn out to be
order of several 1000 kilometer and you will
438
00:54:59,830 --> 00:55:04,830
find out c will turn out to be, in order of
something like 100 kilometer per hour, 500
439
00:55:04,830 --> 00:55:09,540
kilometer per hour, this is exactly why the
speed is high. But people do not realize and
440
00:55:09,540 --> 00:55:15,650
you must realize and I will tell you, I will
end this term with this, that how do you explain
441
00:55:15,650 --> 00:55:19,859
physically, so high c, what is the meaning
of this, so high c what you like you know
442
00:55:19,859 --> 00:55:20,619
physically see.
443
00:55:20,619 --> 00:55:25,140
The reason is very simple, if the wave is
very long like that, let us say this is actually
444
00:55:25,140 --> 00:55:31,600
you know like say 100 kilometer, what happen
after 20 second is water has come down here
445
00:55:31,600 --> 00:55:36,580
and this has moved up here.
So, water till move out from here to here
446
00:55:36,580 --> 00:55:42,140
in 20, the form appeared to or moved up from
here to here is 20 minutes, so after 20 minutes
447
00:55:42,140 --> 00:55:47,869
this place would appear to be having trough
water. So, this it is like an oscillation,
448
00:55:47,869 --> 00:55:52,520
so this oscillation was like this at one time
next instant it become like that, so you see
449
00:55:52,520 --> 00:55:57,280
around this water appear to have come move
to this place and; obviously, if there was
450
00:55:57,280 --> 00:56:01,710
a coastline here you suddenly find out 20
minutes as water moved up and rushed it.
451
00:56:01,710 --> 00:56:06,960
So, therefore, the concept is on the form
only which can travel very fast like a long
452
00:56:06,960 --> 00:56:13,960
line you just shift it this way, so this much
distance is moving in that time, this is how
453
00:56:14,500 --> 00:56:19,580
you explain never the particle moving, particle
cannot move 100 kilometer per hour.
454
00:56:19,580 --> 00:56:25,600
Anyhow, so this is about the speed we in our
course we will find out that our range of
455
00:56:25,600 --> 00:56:30,380
wave speed will lie between something like
5 second to may be 20 second.
456
00:56:30,380 --> 00:56:34,869
We are not interested in tsunami, we are interested
in capillary wave, none of them have any effect
457
00:56:34,869 --> 00:56:41,260
on the ship, we will close today’s talk
of case and begin tomorrow on the other aspect
458
00:56:41,260 --> 00:56:42,940
of water wave theory, thank you.