
English: 
Hello welcome to my talk All about Fluids, 
this talk is on the linear wave theory
linear wave is also called as Airy wave, and the linear wave theory is a
well-established theory, firstly published by George Airy in 1841.
the mostly important aspect in the linear wave theory is the establishment
of the wave velocity potential function, since it is the velocity potential
function which is widely applied in the wave-structure interaction, as well as to
provide the basis for the higher-order waves, such as the Stokes wave.

English: 
Hello welcome to my talk, All about Fluids. This talk is on the linear wave theory
linear wave is also called as Airy wave, and the linear wave theory is a
well-established theory, firstly published by George Airy in 1841.
the mostly important aspect in the linear wave theory is the establishment
of the wave velocity potential function, since it is the velocity potential
function which is widely applied in the wave-structure interaction, as well as to
provide the basis for the higher-order waves, such as the Stokes wave.

English: 
in the widely-used the boundary element method (BEM) for wave-structure
interaction, the wave velocity potential function is a must for describing the
fluid motion, a foundation for wave-structure interaction problem, thus it is
important to introduce how the wave of velocity potential function is obtained in this talk.
the ocean waves cover a large energy of different waves, with the period from
less than 0.1 second to more than 24 hours. the shortest waves are those
capillary and ripple waves, for which we can see in the small ponds when the

English: 
in the widely-used the boundary element 
method (BEM) for wave-structure
interaction, the wave velocity potential function is a must for describing the
fluid motion, a foundation for wave-structure interaction problem, thus it is
important to introduce how the wave of velocity potential function is obtained
in this talk.
the ocean waves cover a large energy of different waves, with the period from
less than 0.1 second to more than 24 hours. the shortest waves are those
capillary and ripple waves, for which we can see in the small ponds when the

English: 
wind blows over it; the second wave are those period of a second to tens of second and
in many cases these are the most interesting waves. This type of the wave
are generated by winds and sometimes generated by the ship's. Longer wave with
the period of minutes to tens of minute, as those generated from the
typhoons and earthquakes. Longer waves with the period of about 12 or 24 hours,
these are also called as tides, are caused by the relative motion between
the Earth and the moon and the Earth and  the Sun, for instance, the gravitation between
the Earth and the moon causes the tides on earth surface at the period about the

English: 
wind blows over it; the second wave are those periods of a second to tens of second and
in many cases these are the most interesting waves. This type of the wave
are generated by winds and sometimes generated by the ships. Longer wave with
the period of minutes to tens of minute, as those generated from the
typhoons and earthquakes. Longer waves with the period of about 12 or 24 hours,
these are also called as tides, are caused by the relative motion between
the Earth and the moon and the Earth and the Sun, for instance, the gravitation between
the Earth and the moon causes the tides on earth surface at the period about the

English: 
12 hours, as the tides twice a day are called the semi-diurnal tides
or 24 hours, the diurnal tides
it is well known that the winds are the main sources for the ocean waves.
the simple mechanism can be understood as when the winds blow over the ocean
surface, they generated the waves, however, the detailed mechanism is very
complicated and not fully understood yet. Generally, it is believed there are
three stages for establishment of the ocean waves
the first stage is the turbulent wind produces the random pressure
fluctuation at the sea surface, thus, the ripples, or more technological term, the

English: 
12 hours, as the tides twice a day are called the semi-diurnal tides
or 24 hours, the diurnal tides
it is well known that the winds are the main sources for the ocean waves.
the simple mechanism can be understood as when the winds blow over the ocean
surface, they generated the waves, however, the detailed mechanism is very
complicated and not fully understood
yet. generally it is believed there are
three stages for establishment of the ocean waves
the first stage is the turbulent wind produces the random pressure
fluctuation at the sea surface, thus, the ripples, or more technological term, the

English: 
capillary waves are generated by the pressure fluctuation.
here as the wind and the wave interaction,   the energy is transferred from wind to wave.
the winds keep blowing on the sea surface to cause the wave to become larger as the waves
growth, the pressure differences get larger, which could cause the wave to increase.
so this is the wind and the wave interaction, but the man energy flux is
still from wind to wave, and the dynamic interaction among
the waves could generate longer waves and transfer the energy from the short
wave to the longer wave, with the quadruplet wave and wave interaction

English: 
capillary waves are generated by the pressure fluctuation.
here as the wind and the wave interaction, the energy is transferred from wind to wave.
the winds keep blowing on the sea surface to cause the wave to become larger as the waves
growth, the pressure differences get larger, which could cause the wave to increase.
so this is the wind and the wave interaction, but the man energy flux is
still from wind to wave, and the dynamic interaction among
the waves could generate longer waves and transfer the energy from the short
wave to the longer wave, with the quadruplet wave and wave interaction

English: 
or the triad wave and wave interaction, and finally the wave will (could)
be faster than the wind speed.
To study the linear wave, a Cartesian coordinate is used, with the origin of
the coordinate located on the surface of the calm water and z axis pointing
up vertically, thus the wave elevation can be described as z equals to ETA_0,
and x axis points to the wave propagation 
direction. For a long crested wave,
it can be regarded as a 2D wave, which is independent of y.

English: 
or the triad wave and wave interaction, and finally the wave will (could)
be faster than the wind speed.
To study the linear wave, a Cartesian coordinate is used, with the origin of
the coordinate located on the surface of the calm water and z axis pointing
up vertically, thus the wave elevation can be described as z equals to ETA_0,
and x axis points to the wave propagation direction. For a long-crested wave,
it can be regarded as a 2D wave, which is independent of y.

English: 
for the linear wave, the flow can be regarded as the irrotational and the
incompressible flow, thus its velocity potential function satisfies the Laplace
equation, and in this talk we will see how the potential function is solved
from the Laplace equation and the corresponding boundary conditions.
Here the seabed condition is given as this, this is essentially non-penetration
condition, that means the fluid cannot penetrate into the seabed
under the assumptions of irrotational and
incompressible flow,  the conventional

English: 
for the linear wave, the flow can be regarded as the irrotational and the
incompressible flow, thus its velocity potential function satisfies the Laplace
equation, and in this talk we will see how the potential function is solved
from the Laplace equation and the corresponding boundary conditions.
Here the seabed condition is given as this, this is essentially non-penetration
condition, that means the fluid cannot penetrate into the seabed
under the assumptions of irrotational and incompressible flow, the conventional

English: 
Bernoulli's equation must be satisfied, with the time-dependent
constant C0, this constant must be chosen carefully for different
applications. In this application C0 must be chosen as the atmospheric
pressure p0, so the Bernoulli equation can be written as this.
The uniqueness and correctness for
C0
equalling to p0 can be justified when we reduce the wave amplitude to 0, to a calm
water, in that case, the corresponding velocity potential function Phi would be

English: 
Bernoulli's equation must be satisfied, with the time-dependent
constant C0, this constant must be chosen carefully for different
applications. In this application C0 must be chosen as the atmospheric
pressure p0, so the Bernoulli equation can be written as this.
The uniqueness and correctness for C0
equalling to p0 can be justified when we reduce the wave amplitude to 0, to a calm
water, in that case, the corresponding velocity potential function Phi would be

English: 
equal to zero or constant, thus p=p0 at z=0 at the surface in
calm water. if we choose any other constant for C0 in this case, it would
be not correct in such an application.
the more complicated boundary conditions for wave theory is the free
surface boundary conditions. In this slide the kinematic boundary condition
is discussed. the kinematic boundary condition for free surface can be
derived from the substantial derivative of the parameter, z-ETA_0
which must vanish on the free surface as this, it is because on the free surface,
z-ETA_0 is zero.

English: 
equal to zero or constant, thus p=p0 at z=0 at the surface in
calm water. if we choose any other constant for C0 in this case, it would
be not correct in such an application.
the more complicated boundary conditions for wave theory is the free
surface boundary conditions. In this slide the kinematic boundary condition
is discussed. the kinematic boundary condition for free surface can be
derived from the substantial derivative of the parameter, z-ETA_0
which must vanish on the free surface as this, it is because on the free surface,
z-ETA_0 is zero.

English: 
Consider z is an independent variable,
so it is substantial derivative with regard to time
is same as the conventional derivative, thus differentiation with time t is
actually the velocity component on z-direction, w. however ETA_0 is a
parameter or a function depending on other variables, such as x, z and time t
thus its substantial derivative is  given by this
so substitute the velocity component based on the velocity potential function
we have equation as this. this is the exact kinematic boundary condition for

English: 
Consider z is an independent variable,
so it is substantial derivative with regard to time
is same as the conventional derivative, thus differentiation with time t is
actually the velocity component on z-direction, w. however ETA_0 is a
parameter or a function depending on other variables, such as x, z and time t
thus its substantial derivative is given by this,
so substitute the velocity component based on the velocity potential function
we have equation as this. this is the exact kinematic boundary condition for

English: 
the free surface. by dropping the higher-order terms, we can have the
linearized kinematic boundary conditions for the free surface
it should be noted that this linearized boundary condition is approximated
at z=0.
the dynamic boundary condition for the free service can be derived from the
Bernoulli's equation. Bernoulli's equation can be written as in this form.
on the free surface where z equals to ETA_0, so the pressure p equals to
the atmospheric pressure p0

English: 
the free surface. by dropping the higher-order terms, we can have the
linearized kinematic boundary conditions for the free surface,
it should be noted that this linearized boundary condition is approximated at z=0.
the dynamic boundary condition for the free service can be derived from the
Bernoulli's equation. Bernoulli's equation can be written as in this form.
on the free surface where z equals to ETA_0, so the pressure p equals to
the atmospheric pressure p0,

English: 
so we can have the expression for the dynamical boundary condition on the
free surface as this. Again where we drop the high-order terms,
we can get the linearised dynamic boundary condition for the free surface
this is a linearized dynamic boundary condition for free surface.
It is approximated, again, at z=0.
put the kniematic condition and the dynamic boundary condition
together, we can have an equation for the free
surface as this.  from the appearance of the equation, it can be seen
it is a typical wave equation

English: 
so we can have the expression for the dynamical boundary condition on the
free surface as this. Again where we drop the high-order terms,
we can get the linearised dynamic boundary condition for the free surface
this is a linearized dynamic boundary condition for free surface.
It is approximated, again, at z=0.
put the kinematic condition and the dynamic boundary condition
together, we can have an equation for the free
surface as this. from the appearance of the equation, it can be seen
it is a typical wave equation.

English: 
because the equation for the free surface is a wave equation, thus we
can assume the 2D wave elevation has a form as this, where A is the wave
amplitude, k is the wave number, defined as 2*Pi divided by the wavelength.
and Omega the wave circular frequency. we can see the wave form for a fixed t which
is the wave in space, thus we can see the wave shape, as if we take a snapshot
of the wave and the distance between the peaks here is the wave length. we can

English: 
because the equation for the free surface is a wave equation, thus we
can assume the 2D wave elevation has a form as this, where A is the wave
amplitude, k is the wave number, defined as 2*Pi divided by the wavelength.
and Omega the wave circular frequency.  we can see the wave form for a fixed t which
is the wave in space, thus we can see the wave shape, as if we take a snapshot
of the wave and the distance between the peaks here is the wave length. we can

English: 
also fix the position and observe the wave elevation at the fix point, this is the
wave we use a wave prop to measure the wave at a fixed point and the time
between two peaks is the wave period T. In mathematics, this wave elevation can be
expressed as this, by using the Euler formula as this. here Re means the real part of the complex.
to simplify the mathematical manipulation, the wave elevation can be
further expressed as this, it should be remembered the actual value of the wave
elevation is the real part of the expression

English: 
also fix the position and observe the wave elevation at the fix point, this is the
wave we use a wave prop to measure the wave at a fixed point and the time
between two peaks is the wave period T. In mathematics, this wave elevation can be
expressed as this, by using the Euler formula as this. here Re means the real part of the complex.
to simplify the mathematical manipulation, the wave elevation can be
further expressed as this,  it should be remembered the actual value of the wave
elevation is the real part of the expression

English: 
since the fluid satisfied 2D Laplace equation at a form as this
a general solution has a form as this.
substitute the general solution into the Laplace equation, we have the
equation as this. thus the Laplace equation is transformed to an equation
for the function capital Z, which depends on z only.
the equation for the capital Z has a general form as this, and we can

English: 
since the fluid satisfied 2D Laplace equation at a form as this
a general solution has a form as this
substitute the general solution into the Laplace equation, we have the
equation as this. thus the Laplace equation is transformed to an equation
for the function capital Z, which depends on z only.
the equation for the capital Z has a general form as this, and we can

English: 
simply check this general solution satisfied the equation as this. However,
in this general solution, the capital C and D must be decided using the relevant
boundary condition.
now the wave velocity potential function can be expressed as this.
In deep water, we can take h as an infinite and the no-
penetration seabed condition would lead to this equation.
this equation will require D must be 0. Therefore, the velocity
potential function is now as this, with the constant capital C to be decided.

English: 
simply check this general solution satisfied the equation as this. however
in this general solution, the capital C and D must be decided using the relevant
boundary condition.
now the wave velocity potential function can be
expressed as this.
In deep water, we can take h as an infinite and the no-
penetration seabed condition would lead to this equation.
this equation will require D must be 0. therefore the velocity
potential function is now as this, with the constant capital C to be decided.

English: 
now the wave elevation at the reference point, z =0 is
calculated as this, and compared to the wave elevation expression as this one,
so we can obtain the constant C. As such in the deep water, the wave
velocity potential function is the finally given as this, so its real part
of the wave velocity potential function is this.  this is the velocity potential
function for wave of a wave amplitude A and wave frequency Omega
now we substitute the wave potential function into the free surface

English: 
now the wave elevation at the reference point, z =0 is
calculated as this, and compared to the wave elevation expression as this one,
so we can obtain the constant C. As such in the deep water, the wave
velocity potential function is the finally given as this, so its real part
of the wave velocity potential function is this.  this is the velocity potential
function for wave of a wave amplitude A and wave frequency Omega
now we substitute the wave potential function into the free surface

English: 
boundary condition, we can obtain the dispersion relation as this.
This dispersion relation links the wave frequency and the wave number together.
this is a very important relation for deep water
waves, which can be used for deciding the wavelength, wave propagation speed
etc
in this slide using the dispersion relation in deep water, we can obtain the
relation between the wave length and the wave period. Replacing Omega
with the wave period T and replacing the wave number with the wavelength, we have

English: 
boundary condition, we can obtain the dispersion relation as this.
This dispersion relation links the wave frequency and the wave number together.
this is a very important relation for deep water
waves, which can be used for deciding the wavelength, wave propagation speed, etc
in this slide using the dispersion relation in deep water, we can obtain the
relation between the wave length and the wave period. Replacing Omega
with the wave period T and replacing the wave number with the wavelength, we have

English: 
the equation as this, so this will lead to wave length calculation formula,
so it can be seen in deep water the wave length is proportional to the wave period
squared, so when we know the wave length, and the wave period, we can easily
calculate the wave speed Vp, this is a phase speed, calculated as this.
so it can be seen the wave speed is proportional to the wave period,
this means the longer waves travel faster. if the water waves have different
the colors for different wave frequencies as those in light, then the
colored wave could travel at different speeds, thus the colors would be

English: 
the equation as this , so this will lead to wave length calculation formula,
so it can be seen in deep water the wave length is proportional to the wave period
squared, so when we know the wave length, and the wave period, we can easily
calculate the wave speed Vp, this is a phase speed, calculated as this.
so it can be seen the wave speed is proportional to the wave period,
this means the longer waves travel faster. if the water waves have different
the colors for different wave frequencies as those in light, then the
colored wave could travel at different speeds, thus the colors would be

English: 
separated during the wave propagation

English: 
separated during the wave propagation.
