
English: 
Hello welcome to my talk, All about Fluids. This talk is on the wave
characteristics. In this talk we will see how the water particles move in waves and
how we can calculate the wavelength, the wave speed and the group speed and
the comparison of the linear and nonlinear dynamic pressures, to the
static pressure in wave.
based on the linear wave theory in deep water, we can have the following results.
we can see how these are obtained in my talk on linear wave theory.

English: 
Hello welcome to my talk, All about Fluids, this talk is on the wave
characteristics. In this talk we will see how the water particles move in waves and
how we can calculate the wavelength, the wave wave speed and the group speed and
the comparison of the linear and nonlinear dynamic pressures, to the
static pressure in wave.
based on the linear wave theory in deep water, we can have the following results.
we can see how these are obtained in my talk on linear wave theory.

English: 
the wave elevation of the free surface can be expressed as ETA_0 for a wave with an
amplitude A and wave frequency Omega. here k is the wave number, which can be
decided with the wave frequency. the wave velocity potential function is
given by this, or in a complex form as this,
and an important relation called dispersion relation given by this
this dispersion relation links the wave frequency and the wave number
which indicates the waves of different frequencies would travel at different speeds

English: 
the wave elevation of the free surface can be expressed as ETA_0 for a wave with an
amplitude A and wave frequency Omega. here k is the wave number, which can be
decided with the wave frequency. the wave velocity potential function is
given by this, or in a complex form as this
and an important relation called dispersion relation given by this
this dispersion relation links the wave frequency and the wave number
which indicates the waves of different frequencies would travel at different
speeds

English: 
One important missing mentioned here is that the water wave does not only exist on
the water surface, but also in water. based on the velocity potential function
we can calculate the full wave elevation as this, ETA. So we can see in
different water depths, the amplitude of the wave elevation would be different.
more specifically the wave amplitude in water would decrease with the increase
of the water depth. Conventionally, the wave elevation is given
by ETA_0, which is the wave elevation at the free surface

English: 
One important missing mentioned here is that the water wave does not only exist on
the water surface, but also in water. based on the velocity potential function
we can calculate the full wave elevation as this, ETA. So we can see in
different water depths, the amplitude of the wave elevation would be different.
more specifically the wave amplitude in water would decrease with the increase
of the water depth. Conventionally, the wave elevation is given
by ETA_0, which is the wave elevation at the free surface

English: 
at z=0. this plot is an example of a regular
wave of an amplitude of 4m, and a period of 8s, you can see the
wave amplitude in water decreases with the increase of the water depth, so we
can see in different depths in water, the wave amplitudes are different.  at the
water surface, z=0, the amplitude is 4 meters and in 5m
water depth, the wave amplitude is 2.92 ms. in 10m, the wave
amplitude is 2.13 meters, and to 25 meters of the water depth,
the amplitude is less than 1 meter.

English: 
at z=0. this plot is an example of a regular
wave of an amplitude of 4m, and a period of 8s, you can see the
wave amplitude in water decreases with the increase of the water depth, so we
can see in different depths in water, the wave amplitudes are different.  at the
water surface, z=0, the amplitude is 4 meters and in 5m
water depth, the wave amplitude is 2.92 ms. in 10m, the wave
amplitude is 2.13 meters, and to 25  meters of the water depth,
the amplitude is less than 1 meter.

English: 
from the wave velocity potential function, the water particle velocity
components are calculated as this, and based on this velocity component, we can
calculate the water particle motion, for instance, consider the water particle on
the free surface, which corresponds to the reference position x and z are both zero.
and the inertia condition x = 0 and z = A,
so we can have that water particle position x and z given by these.

English: 
from the wave velocity potential function, the water particle velocity
components are calculated as this, and based on this velocity component, we can
calculate the water particle motion, for instance, consider the water particle on
the free surface, which corresponds to the reference position x and z are both zero.
and the inertia condition x=0 and z=A,
so we can have that water particle position x and z given by these.

English: 
this corresponding to a circular motion for the water particle as this.
if we plot out this for the wave we can see the motion for the water particle at
the free surface is circular, and in different water depth, the water
particle motion is still circular, but the size of the circle decreases with the
water depth, that is the radius of the circle is proportional to the exponent of kz.
in this slide, we consider the wave of an amplitude 1 meter and a period of
10 seconds in deep water, we can show the fluid patterns and the maximum

English: 
this corresponding to a circular motion for the water particle as this.
if we plot out this for the wave, we can see the motion for the water particle at
the free surface is circular, and in different water depth, the water
particle motion is still circular, but the size of the circle decreases with the
water depth, that is the radius of the circle is proportional to the exponent of kz.
in this slide, we consider the wave of an amplitude 1 meter and a period of
10 seconds in deep water, we can show the fluid patterns and the maximum

English: 
horizontal and vertical velocities, based on the wave velocity potential
function: the horizontal and vertical velocity components are calculated as this
and the drawing given for this. so from the drawing we can see the water
particle velocities change according to
the wave phase, see here
for different phase, the velocity directions are different
and the water particle velocities reduce with the increase of the water depths
given by this. Thus the maximum velocity components
happen at the free surface. in this case the maximum horizontal velocity

English: 
horizontal and vertical velocities, based on the wave velocity potential
function: the horizontal and vertical velocity components are calculated as this
and the drawing given for this. so from the drawing we can see the water
particle velocities change according to the wave phase, see here
for different phase, the velocity directions are different
and the water particle velocities reduce with the increase of the water depths
given by this. Thus the maximum velocity components
happen at the free surface. in this case the maximum horizontal velocity

English: 
components correspond to the wave peak or trough, and the maximum vertical
velocity corresponds to the wave  phase  
at zero crossing.
the oceans have certain depths, especially the limited water depths in the
coastal areas, but what is the minimum water depth which can be taken
at the deep water for a given wave. based on the water particle velocities
in waves, we can see that the water particle velocity amplitudes are
proportional to the exponent of kz, meaning that the velocity component
amplitudes decrease with the increase of the water depth. at the water depths of

English: 
components correspond to the wave peak or trough, and the maximum vertical
velocity corresponds to the wave phase at zero crossing.
the oceans have certain depths, especially the limited water depths in the
coastal areas, but what is the minimum water depth which can be taken
at the deep water for a given wave. based on the water particle velocities
in waves, we can see that the water particle velocity amplitudes are
proportional to the exponent of kz, meaning that the velocity component
amplitudes decrease with the increase of the water depth. at the water depths of

English: 
half wave length, the exponent of kz is 0.0432,
equalling to 4.32%. this means at the water depth equalling to the
half of the wavelength, the water particle motion is very small, compared
to those at the free surface. this small motion would have a very small
effect on the overall wave motion, this is why the water depth equal or
larger than half of the wavelength can be regarded as the deep water condition.
for instance, for a wave of a period of 10 seconds, its wave length is 156 meters,

English: 
half wave length, the exponent of kz is 0.0432,
equalling to 4.32%. this means at the water depth equalling to the
half of the wavelength, the water particle motion is very small, compared
to those at the free surface. this small motion would have a very small
effect on the overall wave motion, this is why the water depth equal or
larger there half of the wavelength can be regarded as the deep water condition.
for instance for a wave of a period of 10 seconds, its wave length is 156 meter,

English: 
the water depth h=78 m or larger can be regarded as a deep water.
However, for a tsunami of a period of 20 minutes, its wave length supposed in the
deep water would be more than 2,000km, hence only the water depth
more than 1000 kilometers can be regarded as a deep water. in this regard for such a
tsunami, all oceans are shallow waters.
use the dispersion relation, we build the relation between the wave length and
the wave period: replacing the wave frequency, and the wave number with the

English: 
the the water depth h=78 m or larger can be regarded as a deep water.
however for a tsunami of a period of 20 minutes, its wave length supposed in the
deep water would be more than 2,000km, hence only the water depth
more than 1000 kilometers can be regarded as a deep water. in this regard for such a
tsunami, all oceans are shallow waters.
use the dispersion relation,  we build the relation between the wave length and
the wave period: replacing the wave frequency, and the wave number with the

English: 
wave period and wave names as this, so we can have the equation for the wave
length LAMBDA equals 1.56*T^2: the wave length is proportional to the wave
period squared. When the wave length and the wave period are known, and the way you
phase speed Vp can be easily calculated as this, so in deep water the wave phase
speed Vp is the proportional to wave period.
this equation means longer waves travel faster. If water waves have colors for
different wave frequencies as those in light, then the colored waves could

English: 
wave period and wave names as this, so we can have the equation for the wave
length LAMBDA equals 1.56*T^2: the wave length is proportional to the wave
period squared. When the wave length and the wave period are known, and the way you
phase speed Vp can be easily calculated as this, so in deep water the wave phase
speed Vp is the proportional to wave period.
this equation means longer waves travel faster. If water waves have colors for
different wave frequencies as those in light, then the colored waves could

English: 
travel in different speeds. Thus the colors would be separated during the wave
propagation, which is a called dispersion.
if we have a narrow band of the component waves, traveling in similar
directions, the resulting waves would travel in groups. Wave grouping is an
important concept since it is related to the wave energy transport. the wave
group velocity is the wave energy transport velocity.
consider two waves trains of different
wave amplitudes A1, and A2,
and frequencies: Omega 1 and Omega 2, traveling in the same direction,

English: 
travel in different speeds. Thus the colors would be separated during the wave
propagation, which is a called dispersion.
if we have a narrow band of the component waves, traveling in similar
directions, the resulting waves would travel in groups. Wave grouping is an
important concept since it is related to the wave energy transport. the wave
group velocity is the wave energy transport velocity.
consider two waves trains of different wave amplitudes A1, and A2,
and frequencies: Omega 1 and Omega 2, traveling in the same direction,

English: 
the total wave can be expressed as this, we can see the envelope of these two waves.
and from this expression, this part can be 
regarded as the carrier wave, while this
part in the square brackets is the envelope.
So the envelope is dependent on the
differences of the wave numbers DELTA k and DELTA Omega, so based on this, we can
define the group velocity, Vg, as DELTA Omega divided by DELTA k,  and its

English: 
the total wave can be expressed as this, we can see the envelope of these two waves.
and from this expression, this part can be regarded as the carrier wave, while this
part in the square brackets is the envelope.
So the envelope is dependent on the
differences of the wave numbers DELTA k and DELTA Omega, so based on this, we can
define the group velocity, Vg, as DELTA Omega divided by DELTA k, and its

English: 
limiting form is given by the DELTA  Omega approaches to 0
now we examine the envelope of the two wave trains, when we reduce the difference of the
frequencies, you can see the changes of
the envelope,
when the frequency difference is very small, and the resulting wave is
actually a regular wave, thus in the extreme case, DELTA Omega approaches to 0,
these two wave trains are of the same frequencies, so the total wave is
actually a sinusoidal wave of a single frequency.
hence the group velocity can be calculated for a single sinusoidal wave,

English: 
limiting form is given by the DELTA_Omega approaches to 0
now we examine the envelope of the two wave trains, when we reduce the difference of the
frequencies, you can see the changes of the envelope,
when the frequency difference is very small, and the resulting wave is
actually a regular wave, thus in the extreme case, DELTA Omega approaches to 0,
these two wave trains are of the same frequencies, so the total wave is
actually a sinusoidal wave of a single frequency.
hence the group velocity can be calculated for a single sinusoidal wave,

English: 
not necessary for two or more wave trains. and in the deep water the
group velocity can be obtained as this, so the group velocity is half of the
wave phase velocity.
consider a wave of amplitude of 1m and the period 10 seconds in deeper
water, so the plot of the water particle motion, we have shown before as this, and
the wave length is calculated by this formula in deep water, the wave length is
156 meters and the phase velocity is calculated as this, group of velocity as this.
and for a comparison, the water particle velocity components are also

English: 
not necessary for two or more wave trains. and in the deep water the
group velocity can be obtained as this, so the group velocity is half of the
wave phase velocity.
consider a wave of amplitude of 1m and the period 10 seconds in deeper
wate,r so the plot of the water particle motion, we have shown before as this, and
the wave length is calculated by this formula in deep water, the wave length is
156 meters and the phase velocity is calculated as this, group of velocity as this.
and for a comparison, the water particle velocity components are also

English: 
calculated: the maximum water particle velocities are 0.628 m/s,
It can be seen the water particle velocity is much smaller than
the wave phase velocity and the group velocity.
and it should be pointed out the wave phase and group velocity are
only dependent on the wave period and independent of the wave amplitude, while
water particle velocity components are dependent on both wave period and wave amplitude
based on the Bernoulli's equation, the gauge pressure can be calculated as this

English: 
calculated: the maximum water particle velocities are 0.628 m/s,
It can be seen the water particle velocity is much smaller than
the wave phase velocity and the group velocity.
and it should be pointed out the wave phase and group velocity are
only dependent on the wave period and independent of the wave amplitude, while
water particle velocity components are dependent on both wave period and wave amplitude
based on the Bernoulli's equation, the gauge pressure can be calculated as this

English: 
so the gauge pressure consists of three parts: the static pleasure, which is
linearly proportional to the water depth; the linear dynamical pressure, is given
with the differentiation of the wave velocity potential function with regard
to time, so in this plot we can see the static pressure and the total linear
pressure corresponding to the wave crest and wave trough, the static pressure is
independent of the wave, but the linear dynamic pressure is very relative to the
wave, and it is positive at the wave crest and the negative at the wave trough

English: 
so the gauge pressure consists of three
parts: the static pleasure, which is
linearly proportional to the water depth; the linear dynamical pressure, is given
with the differentiation of the wave velocity potential function with regard
to time, so in this plot we can see the static pressure and the total linear
pressure corresponding to the wave crest and wave trough, the static pressure is
independent of the wave, but the linear dynamic pressure is very relative to the
wave, and it is positive at the wave crest and the negative at the wave trough

English: 
in this slide the linear and non-linear  dynamical pressures are compared
for a comparison we assume the wave has an amplitude of 5m and a period
8 seconds. This is a very large wave and the slope of the wave is about 1/10,
choosing such a large wave is for getting a large nonlinear dynamic
pressure. The linear dynamic pressure is calculated for corresponding to the
wave crest. thus the linear dynamic pressure is

English: 
in this slide the linear and non-linear dynamical pressures are compared
for a comparison we assume the wave has an amplitude of 5m and a period
8 seconds. This is a very large wave and the slope of the wave is about 1/10,
choosing such a large wave is for getting a large nonlinear dynamic
pressure. The linear dynamic pressure is calculated for corresponding to the
wave crest. thus the linear dynamic pressure is

English: 
positive and decreases with the exponent of kz.
the non-linear dynamic pressure is calculated as this. By considering the
water particle velocity component, nonlinear dynamic pressure is independent of
the wave phase, which is negative and decreases with exponent of 2*kz, thus
nonlinear dynamic pressure decreases quicker than the linear dynamic pressure,
with the increase of the water depth. The largest dynamic pressures
happen at the free surface: linear dynamic pressure at 49,000 Pascals and
the nonlinear dynamic pressure at -7700 Pascal's

English: 
positive and decreases with the exponent of kz.
the non-linear dynamic pressure is calculated as this. By considering the
water particle velocity component, nonlinear dynamic pressure is independent of
the wave phase, which is negative and decreases with exponent of 2*kz, thus
nonlinear dynamic pressure decreases quicker than the linear dynamic pressure,
with the increase of the water depth. The largest dynamic pressures
happen at the free surface: linear dynamic pressure at 49,000 Pascals and
the nonlinear dynamic pressure at - 7700 Pascal's

English: 
for a comparison the static pressure is also plotted in the figure. it can be
seen the static pressure is linearly increased very quick with the water
depth, it reaches 60,000 Pa at the water depths of
slightly larger than 6 meters.

English: 
for a comparison the static pressure is also plotted in the figure. it can be
seen the static pressure is linearly increased very quick with the water
depth, it reaches 60,000 Pa at the water depths of
slightly larger than 6 meter.
