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# Sound

5.1. INTRODUCTION
You are on a holiday relaxing at the edge of a pond. And you observe the following: A whiff of air drops an ant on the water surface. While the ant struggles to keep abreast, a leaf falls close to it. The ant, wise enough, crawls on to the top of the leaf …. the only safe zone. You want to save the ant. You drop a pebble at the centre of the pond. Immediately, ripples form and reach the edge. The question here is … Will the ripples play the saviour’s role?
If you think that the ripples moving outward will carry the leaf to the edge, then you are mistaken. What actually happens is that the leaf simply bobs up and down at the very place.
In the above example there are some interesting observations.
a) The motion occurs in the form of waves or ripples.
A wave is produced by the vibrations of the particles of the medium through which it passes.
b) The particles of the medium (the upper layer of water in the pond), however do not move along with the waves. Instead they oscillate up and down. Such a motion is called wave motion.
5.2. WAVE MOTION
It is a motion where the energy or disturbance is transferred from one point to another in a medium, without the transfer of particles of the medium.
Examples
1) Formation of ripples on the water surface.
2) Propagation of sound wave through air or any other material medium.
Explanation of wave motion
If we drop a stone into a pond, we see circular water waves (ripples) spreading out in all directions on the surface of the water.
If we place a small leaf on the water surface, the leaf moves up and down about its original position but does not move away from or towards the source of disturbance along with the waves. This shows that the disturbances moves from one place to another but the water is not carried with it.

The water particles simply move up and down about their mean positions. The formation of ripples on the surface of water is an example of wave motion. The material in which the wave motion is produced called a medium. Water surface is the medium of wave motion in the above example.

In the above example, when the stone is thrown into the pond, the energy carried by the stone disturbs the water molecules close to it. By gaining the energy from the stone, the water molecules near the stone start vibrating up and down. These vibrating water molecules transfer some of the energy to the next set of water molecules which also start vibrating, and so on. In this way, water wave is formed. We can now say that, wave motion is a vibratory disturbance produced in one part of the medium that travels to another part involving the transfer of energy but not the transfer of any matter with it. The disturbance itself is called a wave.
Characteristics of wave motion
1. In wave motion the particles of the medium vibrate about their mean positions. The particles of the medium don’t move from one place to another.
2. During a wave motion, energy is transferred from one point of the medium to another. There is no transfer of matter through the medium.
3. During wave motion, the medium does not move as a whole only the disturbance travels through the medium.
4. A wave motion travels at the same speed in all directions in any medium. The speed of a wave depends upon the nature of the medium through which it travel.
5.3. TERMS RELATED TO WAVE MOTION
Mean and extreme positions
It is the original position of rest about which the particle oscillates. In the figure, O is the mean position and A and B are extreme positions.

Oscillations
In the above diagram, if a particle is at A, at the start, then one complete oscillation is the motion of particle from A to B and back from B to A. So, the oscillation of a particle is a to and fro motion. If the particle is at O at the start, then one complete oscillation is the motion from O to B, B to O, O to A and finally from A to O.
Amplitude (A)
The maximum displacement of the particles of a medium from their mean positions during the propagation of a wave is called amplitude of the wave. It is represented by the letter ‘A’.

Here A is the amplitude of the wave.
Unit: The S.I unit of amplitude is metre (m).
Note: The amplitude of a wave is a measure of its energy. Thus the greater the amplitude of a wave, the greater the energy carried by the wave.
Time period (T)
The time in which a vibrating body completes one vibration is called time period or the time required to produce one complete wave (or cycle) is called time period of the wave.
Ex: Suppose two waves are produced in 1 second. Then the time required to produce one wave will be 1/2 second or 0.5 second. In other words, the time period of this wave will be 0.5 second.
Note: The S.I unit of time period is second (s).
Frequency ($\nu$ )
The number of complete waves (or cycles) produced in one second is called frequency of wave or The number of vibrations per second is called frequency.
Unit: The S.I unit of frequency is hertz (which is written as Hz).
Hertz: When a vibrating body produces one vibration in one second, then its frequency is said to be one hertz.
1 hertz is equal to 1 vibration per second.

Ex: If 10 complete waves (or vibrations) are produced in one second, then the frequency of the waves will be 10 hertz (or 10 cycles per second).
Note:
1. The frequency of a wave is fixed and does not change even when it passes through different substances.
2. It does not depend on the wave velocity, amplitude and nature of the medium.
Relation between frequency and time period
The frequency of a wave is the reciprocal of its time–period.

Where f = frequency of the wave and T = time period of the wave.
Wavelength ( $\lambda$ )
Wavelength of a wave is the length of one wave.
The distance travelled by the wave in one time period is called the wavelength or the distance between two nearest particles which are in the same phase is called wavelength. It is denoted by the Greek letter $\lambda$ (lambda).

Unit: The S.I unit of wavelength is meter (m).
Phase
The points on a wave which are in the same state of vibration are said to be in the same phase.

In the above figure A, B are in the same phase and C, D are in the same phase.
Ex: All points that lie on the crests are in the same phase similarly all points that lie on the trough are in the same phase.
5.4. WAVE VELOCITY
It is the distance travelled by a wave in one second.
Note:
The wave velocity is different from the velocity of the particle in the wave. The velocity of wave in a medium is constant, whereas the velocity of an oscillating particle changes continuously.
The velocity of oscillating particle is zero at extreme positions and maximum at mean positions.
Expression for wave velocity

Suppose a wave travels a distance $\lambda$ in time T, then $\mathrm{v}=\frac{\lambda }{\mathrm{T}}$

[i.e., the relation between wave velocity, wavelength and time period]

Hence T is the time taken by one wave. We know that $\frac{1}{T}$
becomes the number of waves per second and this is
known as frequency (f) of the wave.
So the above equation can also be written as, v = f × $\lambda$
Where, v = velocity of the wave ; f = frequency ; $\lambda$ = wavelength
In other words velocity of a wave = frequency × wavelength
Thus, the velocity (or speed) of a wave in a medium is equal to the product of its frequency and wavelength.
The formula v = f × $\lambda$ is called wave equation.

5.5. CLASSIFICATION OF WAVES
Classification-1
Depending upon the requirement of medium for the propagation of wave, the waves are classified in the following two groups.
(i) Mechanical waves (or elastic waves]
(ii) Electromagnetic waves.
Mechanical waves
The waves which need a material medium for their propagation are called mechanical waves. The medium may be a solid, a liquid or a gas. (OR)
The wave which required material media for their transmission is called mechanical wave.
Note: Mechanical waves cannot travel through vacuum.
Example: Vibrations in a rope, vibrations in a stretched string, vibrations on the surface of water, sound waves in air etc., are the mechanical waves.

Electromagnetic waves (non-mechanical waves):
The waves which don’t need a material medium for their propagation are called electromagnetic waves.
(OR)
The waves which do not required material media for their transmission is called electromagnetic waves.
Note:
(i) electromagnetic waves can travel through vacuum.
(ii) electromagnetic waves travel in vacuum with the speed of light i.e $3×{10}^{8}$m/s
Example
Light waves, X-rays, Gamma rays, Radio waves, Microwaves etc., are the examples of electromagnetic waves.
Classification -2
On the basis of the relative directions of the propagation of the wave with respect to direction of the periodic changes in the medium (such as displacement, pressure etc.) the waves are classified into the following two groups.
(i) Longitudinal waves (L.W)
(ii) Transverse waves (T.W)
Longitudinal wave
A wave in which the particles of the medium oscillate (vibrate) to-and-fro (back and forth) in the same direction in which the wave is moving, is called a longitudinal wave.
(OR)
A wave in which the particle of the medium vibrate up and parallel to the direction of wave is longitudinal wave.
Example:
1. Sound waves in air
2. Waves in spiral spring
3. The waves produced in air when a sitar wire is plucked.
Representation of a longitudinal wave
In this figure, the direction of wave has been shown from P to Q, in the horizontal plane. The direction of vibrations of the particles is also along PQ, parallel to the direction of wave. That is, the particle of the medium vibrates back and forth in the horizontal direction.

Compression: The part of a longitudinal wave in which the density of the particles of the medium is higher than the normal density is called a compression.

Rarefaction: The part of a longitudinal wave in which the density of the particles of the medium is lesser than the normal density is called a rarefaction.

Note:
1. Compression and rarefaction are formed alternately in longitudinal wave.
2. The distance between the two successive compressions or between the two successive rarefactions is equal to one wavelength ($\lambda$).
3. Longitudinal waves can travel through all media i.e. solids, liquids and gases.
4. The phase difference between two successive compressions or rarefactions is equal to $2\mathrm{\pi }$ radians.
Graphical representation of longitudinal waves
Longitudinal waves in air represented graphically by plotting the density of air against distance from the source.
In other words, a longitudinal wave is represented by a density – distance graph.
In figure the horizontal line OX represents the normal density of air. All the points above this line represent greater density. In a compression of a longitudinal wave, the density of the particles is high. Here, A and C represent compressions. All the points below the line OX represent less density (than normal). In a rarefaction, the density of the particles is less than that in the normal. So here B and D represent rarefactions.

Graphical representation of a longitudinal wave by density – distance graph

Formation of Longitudinal wave in a slinky
A long flexible spring which can be compressed or extended easily is called slinky. The waves which travel along a spring (or slinky) when it is pushed and pulled at one end, are longitudinal waves. We will now describe how longitudinal waves are formed on a spring. Figure: (a) shows the normal position of a spring whose one end is fixed.
Now, if the free end of the spring is moved to and fro continuously, then longitudinal waves consisting of alternate compressions and rarefactions travel along the spring.
When a wave travels along the spring, then each turn of the spring moves back and forth by only a small distance in the direction of the wave. Since the particles of the medium (turns of the spring) are moving back and forth in the direction of the wave, the waves which travel across the spring are longitudinal waves.

In a spring, a compression is that part in which the coil (or turns) is closer together than normal. In figure (b), the regions marked ‘C’ are compressions.
In a spring, a rarefaction is that part in which the coils (or turns) are farther apart than normal. In figure (b), the regions marked as ‘R’ are rarefactions.

The Sound waves in air are longitudinal waves
When a sound wave passes through air, the particles of air vibrate back and forth parallel to the direction of sound wave. Thus, when a sound wave travels in the horizontal direction, then the particles of the medium also vibrate back and forth in the horizontal direction.
It should be remembered that the waves produced in air when a sitar wire is plucked are longitudinal waves, because those are sound waves.
Transverse wave
A wave in which the particles of the medium oscillate (vibrate) up and down, i.e perpendicular to the direction in which the wave is moving is called a transverse wave.
(OR) If the particles of the medium vibrate perpendicular to the direction of the propagation of the wave then the wave is called transverse wave (or) when the particles of a medium oscillate at right angles to the direction of propagation of a wave, then the wave so produced is called a transverse wave.
Examples:
1. Waves on the surface of water
2. Waves on a long stretched rubber tube
3. Waves along a stretched string.
4. Even the light waves and radio waves are transverse waves.
Representation of a transverse wave
In this figure, the direction of wave is from A to B but the vibration of the particles is along CD which is at right angles to the direction of wave AB. So, this is a transverse wave.

Crest and trough of a transverse wave
We know that when a transverse wave travels horizontally in a medium, the particles of the medium vibrate up and down in the vertical direction. When the vibrating particles move upward or above the line of zero disturbance, they form an ‘elevation’ or ‘hump’ and when the vibrating particles move downward or below the line of zero disturbance, they form a ‘depression’ or ‘hollow’.
Crest: The ‘elevation’ or ‘hump’ in a transverse wave is called crest. In other words, a crest is that part of the transverse wave which is above the line of zero disturbance of the medium. In figure XY is the line is zero disturbance.
Trough: The ‘depression’ or ‘hollow’ in a transverse wave is called trough. In other words, a trough is that part of the transverse wave which is below the line of zero disturbance.

Note:
(1) The distance between the two consecutive crests or between the two consecutive troughs is equal to one wavelength ($\lambda$).
(2) The distance between a crest and the adjoining trough is $\frac{\lambda }{2}$
(3) A transverse wave cannot travel inside the liquid and in gases.
(4) Propagation of a transverse wave, through a medium, results in the formation of crests and troughs.
Graphical representation of transverse waves:
A transverse wave is represented graphically by plotting the displacement of different particles of the medium from the line of zero disturbance against distance from the source. In other words, a transverse wave is represented by a displacement – distance graph.
When a transverse wave passes through a medium, then some particles of the medium are displaced above the line of zero disturbance whereas others are displaced below the line of zero disturbance. So a transverse wave is represented graphically by plotting the displacement of different particles of the medium from the line of zero disturbance against the distance from the source.
In figure the horizontal line OX represents the line of zero disturbance of the particles of the medium. All the particles above the line of zero disturbance have positive displacements and those below it have negative displacements. In figure A and C represent two crests and B and D represent two troughs of the transverse wave.

Graphical representation of a transverse wave by displacement-distance graph

Formation of transverse wave in a slinky
The waves produced by moving one end of a long spring (or slinky) up and down rapidly, whose other end is fixed, are transverse waves. The transverse wave produced on a slinky is shown in figure. As the wave passes along the slinky in the horizontal direction, the particles of slinky vibrate up and down at right angles to the direction of wave.

Formation of transverse wave on the surface of water
The water waves (or ripples) formed on the surface of water in a pond are also transverse waves. This is because of the fact that in water wave, the molecules of water move up and down in the vertical direction when the wave travels in the horizontal direction along the water surface. Since the water molecules vibrate up and down at the same place therefore, a cork or leaf placed on the surface of water moves up and down at the same place as water wave moves across the surface of the pond.

Note: When a stone is dropped in a pond of water, transverse waves are produced on the surface of water.

Diagram to show the crests and troughs of a transverse wave. Here, A and C are crests; B and D are troughs

DIFFERENCE BETWEEN TRANSVERSE WAVES AND LONGITUDINAL WAVES:

 Transverse waves Longitudinal waves 1.       In transverse waves, the particles of the medium vibrate at right angles to the direction of wave. 2.       2. Transverse waves consist of crests and troughs 3.       Transverse waves can be propagated by only through a solid or over surface of liquid but not in a gas. 4.       In transverse waves, the distance between two consecutive crests or consecutive troughs is equal to one wavelength. 5.       It is represented by displacement distance graph. 1.       In longitudinal waves, the particles of the medium vibrate parallel to the direction of wave. 2.       Longitudinal waves consist of compression and rarefactions. 3.       Longitudinal waves can be propagated through solids, liquids, as well as gases. 4.       In longitudinal waves, the distance between the two successive compressions or rarefactions is equal to one wavelength. 5.       It is represented by density distance graph.

5.6. SOUND
Sound is a form of energy which emitted by a vibrating body that travels in the form of waves and causes the sensation of hearing. Sound cannot travel through vacuum.
Production of sound
We hear many sounds every day such as the sound of our school bell, an alarm clock, a barking dog and so on.
We talk and communicate with others by producing sounds.
Sound is produced when a body vibrates (moves back and forth rapidly). In other words, sound is produced by vibrating bodies.
Note: Vibrations of the bodies produces sound so vibrations are the causes of sound.
Propagation of sound – Sound needs a material medium to travel
The substance through which sound travels is called a medium. The medium can be a solid substance, a liquid or a gas. Solids, liquids and gases are called material media. Sound needs a material medium like solid, liquid or gas to travel. In other words, sound can travel through solids, liquids and gases but it cannot travel through vacuum (or empty space). Sound waves are called mechanical waves because they need a material medium (like solid, liquid or gas) for their propagation.
Note: Sound travels about 15 times faster in steel than in air.
Perception of sound
Sound is perceived by the vibration of the ear drum (Tympanic membrane). Sound moves in air into the form of longitudinal waves. The vibrating particles of the longitudinal wave make the ear drum to vibrate at the same frequency as the source. This causes the effect of hearing sounds.

Note: We can perceive the direction of sound with certain accuracy. Left and right location of sound is determined by the perception of difference in the arrival time at each ear. A give sound reaches at one ear first and then at the second ear after a certain time interval. Based on this time interval, we can identify the direction of the sound.
Scuba divers ( divers under sea) find it difficult to ascertain the direction from which an underwater sound is coming, because the sound waves are moving so quickly in water that they reach both the ears at the same time.
Persistence of hearing
The impact of sound should remain on the eardrum for a minimum interval to perceive the sound clearly. This minimum time interval is known as persistence of hearing. For the human ear, the persistence of hearing is equal to $\frac{1}{10}th$ of a second. If two sound fall on the eardrum within $\frac{1}{10}th$ of a second, they cannot be differentiated.

5.7. AUDIBLE AND INAUDIBLE SOUNDS
We cannot sense the sounds produced by all vibrating bodies. For example, we cannot sense the sound produced by a vibrating pendulum. This is because the frequency of the pendulum is very less. We will be able to sense the sounds having frequencies from 20Hzt 20,000H
Based on frequency, waves can be classified as:
i)Audible range    (ii) Infrasonics      (iii) Ultrasonics
Audible range
The range of frequencies that people with normal hearing can detect is called the audible range.
The highest frequency that humans can hear is the upper limit of your hearing. For young people this is usually between 16000 and 20,000 Hertz. There is also a lower limit. This is about 20 Hertz.
Thus, “the hearing range or the range of human audibility is 20 Hz to 20,000 Hz”.
Infrasonic Sound
Some vibrations produce sounds below the normal hearing range. These are known as infrasonic sounds (below 20 Hz). Earthquakes, underground nuclear explosions, and tides, all produce infrasonic sounds. These sounds, although they cannot be heard by humans, can be detected by a device called a seismometer.
Ultrasonic Sound
Vibrations above the normal hearing range produce ultrasonic sounds (above 20,000 Hz).
Many animals like bats, dolphins, dogs, produce and hear such sounds. For example, humans can hear only part of the high-pitched squeak of the mouse. The mouse squeak is also made up to ultrasonic sounds which can be heard by a hunting cat.
Many forms of animal life depend on ultrasonic sound for their survival.

Application
1. Investigation of Structure of Matter
Sending ultrasound through bulk of matter and studying the variation in their velocity inside it, valuable information regarding the constitution of complex molecules can be obtained.
2. Cleaning
Dirty odd shaped parts, spiral tubes, electronic components are placed in a cleaning solution and ultrasonic waves are passed through it. Due to high frequency vibrations particles of dirt get detached and drop out. The objects are thoroughly cleaned.

3. Detection of flaws (Cracks) in Metals
Ultrasound is sent through the metallic structure. They pass through unobstructed structure if the structure is homogeneous. In case of a crack inside the structure the ultrasound will be reflected back towards observer and received. Existence of reflection, confirms existence of a crack. Due to high frequency and short wavelength, even fine cracks can be detected.
4. Depth of Seas and Oceans
Ultrasound oscillator producing waves of frequency 40 kHz is provided inside ships. An ultrasound receiver is also provided inside the ship.
The transmitter oscillator transmits the waves towards bottom and reflected waves are received by the receiver.
A recorder finds the time interval (t). Knowing velocity (v) of ultrasound through sea water, the depth (d) can be calculated as $d=\frac{vt}{2}$.

A similar mini arrangement can be used for finding the depth of liquid level in a tank without opening its cover.
5. Sound Ranging
A special equipment called SONAR (SOund Navigation And Ranging) is used for this purpose.
6. Emulsions of Immiscible Liquids
When a strong beam of ultrasound is passed through a liquid, it is heated to a very high temperature. This fact is utilized in preparing homogeneous stable emulsion of immiscible liquids. Ultrasound treated honey does not crystallize.
7. Medical and Biological Effects
a. When ultrasound is passed through a body part having muscular pain or rigid joints, their high frequency vibrations produce soothing effect and relieve the pain.
b. Ultrasound sent through brain cures a mental patient.
c. A newly developed technique of three dimensional photographs with the help of ultrasound (ultrasonography) is being used by the physicians to locate the exact position of an eye tumour and its removal giving normal vision to the patient.
d. Harmful insects are killed by exposing them to ultrasound.

5.8. VELOCITY OF SOUND
The speed of sound: Sound takes some time to travel from the sound producing body to our ears. The speed of sound tells us the rate at which sound travels from the sound travels from the sound producing body to our ears.
Speed of sound in different media: The speed of sound is different in different media. The speed of sound is more in solids, less in liquids and least in gases (since solids are much more elastic than liquids and gases). The speed of sound is nearly 5100 m/s in steel, 1450 m/s in water and 330 m/s in air at ${0}^{0}$C.
Mathematical formula for speed of sound

1. Speed of sound

2. Laplace’s formula for velocity of sound, $v=\sqrt{\frac{\gamma P}{\rho }}$

Where

P = Pressure of the medium.

$\rho$ = density of the medium.

Factors affecting the speed of sound in air or in a gas
i) Density    ii) Temperature    iii) Humidity    iv) Direction of wind
1. Effect of density: as $\mathrm{v}\propto \frac{1}{\sqrt{\rho }}\mathrm{i}.\mathrm{e}$ The speed of sound is inversely proportional to the square root of the density of the medium.
Ex: The density of oxygen is 16 times the density of hydrogen, therefore the speed of sound in hydrogen is four times the speed of sound in oxygen.
2. Effect of temperature: The speed of sound increases with the increase in temperature of a gas. It is found that the velocity of sound in a gas is directly proportional to the square root of its absolute temperature i.e $\mathrm{v}\propto \sqrt{\mathrm{T}}$
The reason is that with the increase in temperature there is a decrease in the density and consequently, the speed of sound increases.
The speed of sound in air increases by about 0.6 m/s (or 60 cm per second) for each degree Celsius rise in temperature i.e ${\mathrm{V}}_{\mathrm{t}}={\mathrm{V}}_{0}+0.6\mathrm{t}$

Ex: Speed of sound in dry still air at 0°C is 330 m/s. At 25°C, the speed of sound in dry still air will be ${\mathrm{V}}_{25}={\mathrm{V}}_{0}+0.6\mathrm{t}=330+0.6×25=345\mathrm{m}/\mathrm{s}$

3. Effect of humidity: The speed of sound increases with the increase in humidity.
The presence of water vapour in the air changes its density. The presence of water vapour reduces the density of air i.e density of moist air < Density of dry air therefore, velocity of sound in moist air > velocity of sound in dry air.

Hence, the velocity of sound in moist air is greater than the velocity of sound in dry air. That is why, sound travels faster on a rainy day than on a dry day.
4. Effect of wind: The speed of sound increases or decreases according to the direction of wind.
(i) If the wind blows in the same direction in which the sound travels, the velocity of sound increases i.e velocity of sound = velocity of sound in still air + velocity of wind.

(ii) If the wind blows in the opposite direction in which the sound travels, the velocity of sound decreases i.e velocity of sound = velocity of sound in still air – velocity of wind

Factors which do not affect the speed of sound in air
There is no effect on the speed of sound in air due to the following factors.
i) Change in frequency
ii) Change in amplitude
iii) Change in pressure

iv) Change in factors like phase, loudness, pitch, quality of sound etc.
Effect of pressure
We know, $\mathrm{v}=\sqrt{\frac{\gamma \mathrm{P}}{\rho }}$

Thus, if the temperature of a gas remains constant, a change in pressure of the gas remains constant, a change in pressure of the gas changes its density in the same ratio i.e if pressure P of the gas is doubled, the volume becomes half, so density (m/v) gets doubled. So $P/\rho$ remains unchanged.
Consequently, the velocity of sound is independent of the pressure of the gas provided the temperature remains constant.
Comparison of speed of sound with speed of light: The speed of light in air is $3×{10}^{–8}m{s}^{–1}$which is about a million times larger as compared to the speed of sound in air i.e
Apart from this, the speed of light decreases in a denser medium (speed of light in water is $2.25×{10}^{8}{\mathrm{ms}}^{-1}$, in glass is $2×108{\mathrm{ms}}^{-1}$), while the speed of sound is more in solids, less in liquids and still less in gases (speed of sound in steel is nearly $5100m{s}^{–1}$, in water is nearly $1450m{s}^{–1}$and in air is nearly 3$330m{s}^{–1}$).
Lightning is seen much earlier than the thunder is heard: In thundering, the light is seen much earlier than the sound of thunder is heard although they are produced simultaneously, as light takes almost negligible time in comparison to sound in reaching us from the thunder.

5.9. REFLECTION SOUND
When a wave going from one medium comes back to the same medium after striking the second medium, reflection of the wave is said to take place. Sound wave also suffer reflection like light waves. The reflection of sound can be demonstration as follows:
Take a drawing board and fix it on the floor. Put two metallic or cardboard tubes. These tubes are making some angle with each other. Put a clock near the end of one tube and a screen between the two tubes so that sound of clock may not be heard directly by the ear placed in front of the second tube. The second (like tick-tick) waves pass through the tube are reflected by the drawing board. The reflected sound waves enter the second tube and
are heard by the ear placed in front of the second tube.
Go on adjusting the position of the second tube. A stage will come when the loudness of the sound heard through the second tube is maximum. At this stage, the incident angle $\angle i$ =  reflected angle $\angle r$ , which is the law of reflection.
Applications of Reflection of Sound
1. Megaphone
Megaphone is a device used to address public meetings. It is horn-shaped. When we speak through megaphone, sound waves are reflected by the megaphone. These reflected sound waves are directed towards the people without much spreading.
2. Hearing Aid
Hearing aid is used by a person who is hard of hearing. The sound waves falling on hearing aid are concentrated into a narrow beam by reflection. This narrow beam of sound waves is made to fall on the diaphragm of the ear. Thus, diaphragm of the ear vibrates with large amplitude. Hence, the hearing power of the person is improved.
3. Sound Boards
Sound boards are curved surfaces (concave) which are used in big hall to direct the sound waves towards the people sitting in a hall. The speaker is placed at the focus of the sound board. Sound waves from the speaker are reflected by the sound board and these reflected waves are directed towards the people.

5.10. ECHO
It is a common experience that when we shout in front of a high mountain or a deep valley or a deep wall or in a big empty hall, we hear our sound back after a few seconds. This is known as echo. An echo is simply a reflected sound.
Thus, the sound heard after reflection from a rigid obstacle is called an echo. (or)
The repetition of sound caused by the reflection of sound waves is called an echo.

Minimum distance of hear an echo
The human ear can hear two sounds separately only if they reach the ear after an interval of 1/10th of a second. This is a natural feature of the human ear. If we take the speed of sound roughly as 340 m/s, the distance travelled by sound in 1/10th of a second would be 34 m. This means that we are able to hear the original and the reflected sound if we are at a distance of 17 m or more from the reflecting surface. This results in hearing the reflected wave known as echo.
Calculation of minimum distance to hear an echo:

We know that:

Speed of sound = 340 m/s (in air)

Time taken = $\frac{1}{10}$ s (persistence of hearing)
(time gap between the original sound and its echo) Distance travelled = ?
Now, putting these values in the above formula, we get:

Distance travelled

Thus, the distance travelled by sound in going from us (the source of sound) to the sound reflecting surface (wall), and then coming back to us should be 34 metres. So our distance from the sound reflecting surface (like a wall, etc,) to hear an echo should be half of 34 metres which is 34/2 = 17 metre. From this, we conclude that the minimum distance from a sound reflecting surface (like wall, etc.) to hear an echo is 17 metre.
Relation between speed of sound, time of hearing echo and distance from reflecting body.
If t is the time in which an echo is heard, d is the distance between the source of sound and the reflecting body, and v is the speed of sound, then the total distance travelled by the sound is 2d.
S = Source of sound

d = Distance between source of sound and reflecting body

O = Observer

$\therefore$In the time t seconds. distance travelled by sound = 2d

Hence, in time 1 seconds. distance travelled by sound $=\frac{2d}{t}$

But, distance travelled by sound in 1 second. = Speed of sound

Condition for the formation of an echo:
The minimum distance between the source of sound and the reflecting body should be 17 meters.
Reverberation
If a sound is made in a big hall, the sound waves are reflected from the walls, ceiling and floor of the hall, and produce many echoes. The echo time however, so short that the many echoes overlap with the original sound. Due to this original sound seems to be prolonged and lasts for a longer time. In other words, a sound made in a big hall persists (on lasts) for a longer time.
The repeated multiple reflection of sound in any big enclosed space is called reverberation.
(or)
The persistence of sound in a big hall due to repeated reflection from the walls ceiling and floor of the hall is called reverberation. Thunder that follows lightning flash during a storm is an excellent example of reverberatory or multiple echoes. Reverberations are also produced in a closed room.

Note: To reduce the reverberation to almost zero level, the walls and the ceiling of the hall are covered with sound absorbing materials, such as rough plaster, fibre board or loose woolen or cotton cloth.
SONAR
Another application of reflection of sound in Sound Navigation and Ranging (SONAR). The process of locating the position of objects submerged in water e.g., submarines and icebergs, by the reflection of sound from them is called echo depth sounding. It is also used for finding the depth of the sea. The instrument used for this purpose is called SONAR.

A sonar consists of a transmitter and a receiver. The transmitter is the source of ultrasonics. With the help of crystals that can be made to vibrate electrically, ultrasonics of frequency upto 1000 MHz can be produced. The ultrasonic waves from the transmitter travel under water in all directions. When they strike an object they are reflected. The receiver receives the reflected ultrasonic waves from the object. It records not only the reflected waves but also the time ‘t’ taken by them to travel from the ship and back from the object.
Knowing the speed ‘v’ of sound waves in water (), the depth of the object is found using the relation, $\mathrm{h}=\frac{\mathrm{vt}}{2}$.
The sonar not only indicates the distance of the object, but also its approximate direction.

11. CHARACTERISTIC OF SOUND
We here different types of sound around us. Some sounds are pleasant to our ears, whereas some unpleasant.
Musical Sound
Musical sound is produced by periodic vibrations. It has regular wave forms. Musical sound is produced by musical instruments like sitar, violin and drum. It produces a pleasant effect on the ear. Musical sound is one which produces pleasing sensation.
Thus, sounds which are pleasing to the ears are called music.
Noise:
Noise is produced by non-periodic vibrations. It has irregular wave forms. Noise is produced by machines in a factory, moving train by the traffic at a busy crossing, horns by trucks and buses are the example of noise. It produces an unpleasant effect on the ear.
DIFFERENCE BETWEEN MUSICAL SOUND AND NOISE

 Musical sound Noise 1. It has a pleasant effect on the ears 2. Made by regular and periodic vibrations 3. Has a definite frequency 4. It is produced by musical instruments 5. Has an regular wave form 6. Can be reproduced 1. It has un-pleasant effect on the ear 2. Made by irregular and non- periodic vibrations 3. Does not have a definite frequency 4. It is produced by machines in a factory moving buses, cars and trains etc., 5. Has an irregular wave form 6. Cannot be reproduced

Wave representation

Characteristics of musical sounds: The three characteristics of musical sounds by which they can be distinguished from one another one as follows.
1. Loudness or intensity
2. Pitch or shrillness and
3. Quality or timbre
1. Loudness or Intensity
Loudness: Loudness is the property of sound which distinguishes the degree of sensation produced in the ear. It is a characteristic of both musical sound and noise loudness depend on the intensity of sound.
The amount of sound energy passing each second through a unit area, is called intensity of sound.
It is our common experience that when we shout, we use more energy and hence produce a louder sound. Similarly, when we whisper, we use less energy and hence produce a lower sound.
Loudness is the characteristic of a musical sound by which a loud sound can be distinguished from a faint sound even though both have the same pitch.
Factors affecting the loudness of sound
1) Amplitude of vibration of the sources
It has been found that greater the amplitude of vibration of the source, the greater is the intensity (and hence loudness) of sound and vice versa. It we strike a drum softly its skin vibrates with a smaller amplitude and a soft sound is produced. If we strike the same drum harder, its skin vibrates with a greater amplitude and a louder sound is produced. This shows that greater the amplitude, the louder is the sound.
Mathematically and experimentally, loudness of sound is directly proportional to the square of the amplitude of the wave.

Surface area of the vibrating body
The greater the surface area of the vibrating body, the greater is the loudness of sound.
A large drum (having larger surface area) will produce a louder sound than a small drum (having smaller surface area)

Distance from the vibrating body
Loudness depends on the distance between the listener and the source. The lesser the distance between the listener and the source, the louder is the sound heard by the listener and vice versa. The loudness of sound varies inversely as the square of the distance,

i.e
Thus, nearer is the source of sound, louder is the sound.

Loudness decreases with an increase in the distance between the ear and the source
4) Density of the medium
The loudness of sound is directly proportional to the density of the medium through which it propagates, i.e greater the density of the medium, louder is the sound.
Note: Loudness is more in dry air than in moist air which is lighter.
5) Presence of resonant bodies
The loudness of sound is increased due to the presence of other resonant bodies near the source of sound. For example, sound appears to be much louder inside a hall than in open air. It is because the walls, roof, floor, etc, reflect the sound. Consequently, the loudness is increased.
6) Motion of the medium
If wind is blowing in the direction of propagation of sound, loudness is increased. On the other hand, if wind is blowing in a direction opposite to the propagation of sound, loudness is decreased.
Measurement of Loudness of Sound
The loudness of sound is measured in decibels (dB). The softest sound which human ears can hear is said to have a loudness of 0 dB (zero decibel). The loudness of sound of people talking quietly is about 65 dB, the loudness of sound in a very noisy factory is about 100 dB and the sound of a jet aircraft 50 metres away is said to have a loudness of about 130 dB.
Note:
1. Loudness increases with the amplitude of vibrating body.
2. Loudness increases with the increase in surface area of vibrating bodies.

3. Loudness decreases with the increase in distance from the source of sound.
2. Pitch or shrillness:
Pitch is the characteristic of sound which enables us to distinguish a shrill sound from a dull sound.
(1) A shrill sound has a high pitch and a dull sound has a low pitch. Women and children have shrill high pitched voices, whereas men have grave law pitched voices.
(2) All musical sound and notes have definite pitch. Pitch is the position of a note on a chosen musical scale.
(3) We have sa, re, ga, ma, pa, dha, nee, in Indian music. ‘nee’ is higher than all the other notes.
Factor on which pitch of the sound depend:
Pitch of a sound depends upon the frequency (or wavelength).
The sound produced by an object vibrating with a low frequency (or high wavelength, since frequency  its pitch is low and the sound is described as grave or flat sound.
The sound produced by an object vibrating with a high frequency (or low wavelength), its pitch is high and the sound is described as shrill sound. Thus, higher the frequency of a musical sound, higher is its pitch and vice versa.
Note:
1. Greater the frequency of a sound the higher will be its pitch. A sound having a frequency 512Hz is higher pitch than a sound of frequency 256 Hz
2. Faster the vibration of the sound producing source, the higher is the frequency and higher is the pitch.
3. The voice of children and ladies is shrill as compared to the voice of men. This is because children and ladies have short vocal chords which therefore vibrate with a high frequencies to produce high pitched voice. Whereas men have long vocal chords which vibrate with a low frequency to produce low–pitched voice.
4. The wings of the bee and mosquito vibrate with a high frequency. Hence, the sound produced by them is shrill and high–pitched.
3. Quality or timbre:
The quality of a sound is that property by virtue of which two sounds of the same pitch and loudness produced by two different musical instruments of people can be distinguished.
(OR)
Quality (or timbre) is that characteristic of musical sound which enables us to distinguish between the sounds of same pitch and loudness produced by different musical instruments (and different singers].
(1) Two notes of the same pitch and loudness can sound different. The sound produced on a veena is different from the same pitched sound produced by a flute.
(2) The difference in the quality of various tones is indicated by a difference in their wave forms.

(3) The sounds produced by different ‘singers’ such as S.P. Balasubrahmanyam, Udit Narayan, Lata Mangeshkar, Asha Bhonsle. Can be distinguished from one another on the basis of their quality or timbre.
(4) We can even recognize a person from his voice (even without seeing him) on the basis of the unique quality or timbre of his voice.

5.12. MUSICAL INSTRUMENTS
We are familiar with different musical instruments. The quality of sound of a musical instrument depends on the type of air vibrations produced by the instrument on the basis of this fact, the musical instruments are mainly divided into four groups.
(a) Wind instrument (or) reed instrument (sushir vadya)
In wind instruments, the sound is produced by vibrating column of air inside a tube (pipes)
Example
Flute, shahnai, bag pipes, bugles, mouth organ, harmonium and nadaswaram etc.


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# What type of visa do pilots who fly from one country to another need? [duplicate]

I am little curious about international travel. I would like to know what type of visa allows pilots and cabin crews to fly from one country to another? I mean suppose I am pilot of Japan airline and I have trip a for Netherlands. How could I enter to the Netherlands with Japanese citizenship?

• Why is triplane one of the tags? – ThatCoolCoder Feb 11 at 5:51
• Wild guess: typo. Edited it away. – Jpe61 Feb 11 at 6:25
• I suppose you are asking about a situation where the pilot leaves the aircraft and exits the airport into the foreign country? International airports are "neutral ground" so as long as you stay on the "international" side, no visa is required. – Jpe61 Feb 11 at 6:29
• @Jpe61: That is too simple. You might not need a visa, or you might need a simpler "transit" visa, or just an ordinary visa - countries are free to make their own rules, and those could depend on the nationality of the pilot. I'd imagine that quite a few countries would ban El Al pilots altogether. – MSalters Feb 11 at 15:15
• Note: it is not a duplicate. The "duplicate question" is about not leaving the plane (or the airport). From the comment of OP in the answer, it seems OP want to do touristy things, so a different case. @Encipher: maybe you should add that information in the question. In any case. I'm voting to reopen the question. – Giacomo Catenazzi Feb 12 at 10:32

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27

There are two reasons why. First, for any glue to stick to something, that something needs to be wettable by the glue. A cured silicone surface is bristling with uncrosslinked silicone units which have extremely low surface energy and hence are very difficult if not impossible to wet by glues like epoxy. This means the cured epoxy exhibits no adhesion to the ...

24

The photo you have is not for the standard jigsaw blade. Standard blades teeth point upward and cut upward for the obvious reasons: less wobble, more control, center of force near the handle. This is a revers cut blade designed to cut on downstroke to cut laminated work like countertops, so that the cut is not going to splinter the Formica or other laminate ...

16

If you think what happens as each tooth cuts material then the blade is put into tension as it cuts, this means that the blade is likely to stay straight, but any unevenness between the teeth side to side may cause drift. If the blade cuts on the downstroke it will buckle as it is relatively thin for its length. A blade designed to cut on the downstroke ...

11

To enlarge slightly upon Solar Mike's response, a jigsaw that cuts on the upstroke tends to yank the saw shoe down into firm contact with the workpiece. This guarantees that the cut angle will match the shoe angle; for a shoe angle of 90 degrees this means a nice square cut.

11

One of the mechanism that affect corrosion is known in the literature as Stress Corrosion Cracking (SCC). The idea is that tensile stressed regions are prone to crack development. Crack development essentially maximizes the area that corrosion can develop. Since corrosion, degrades the properties of the material, this accelerates further the crack ...

9

Silicone is not a single substance, it is a very very wide range of different types of materials with widely varying properties! All materials that have several siloxane monomers in them can reasonably be called silicones. This can take almost any shape; foams, glues, coatings, rubbers, oils, powders, gels. 3D printed solids, even. Some silicones will adhere ...

7

Let me see if I understood it correctly: You have a rubber block under a uniaxial load (compression or tension). That block may or may not be constrained on one pair of sides (the load is applied along the block's x-axis and the block can't deform in the y-axis but can in the z-axis, for example). If this is the case, then the 1.33 is easy enough to explain. ...

5

Solar Mike's answer is accurate. Carbon fiber has a resin to fiber ratio which provides the optimum strength. This is typically measured by weight. The amount of resin is applied to the fiber prior to enclosing it for vacuum application. Once the vacuum begins, all of the air is removed from the fiber, forcing the resin into the voids, ensuring the ...

5

To expand on Solar Mike's answer and to help explain part of niels nielsen's answer, the saw blade is normally designed to essentially squeeze the material between the teeth and the shoe. This generally makes for a more controlled cut. Circular saws, saws-alls, and most other mechanical saws work on this principle, too. If you were to use the blade you have ...

5

Material Properties For the linear-elastic analysis you describe, the key pieces of material information are the elastic modulus (E) to calculate deflection and the yield stress (Fy) to check if the material remains elastic under the given loading. For steel, E is commonly around 200 GPa. The Fy can vary significantly depending on the grade of steel, so you'...

4

Steels are usually considered "as cast" or wrought. The microphotograph is overetched and either high magnification or coarse grained , likely had a heat-treatment. Cast or wrought would be identified by inclusion morphology , best seen in unetched condition. Modern steels are very clean so minimum inclusions to look for. Modern steels are also ...

4

I don't think of a screen protector as a mono layer, but a lamination. Deformable (softer) materials (possibly above and also) below the rigid layer (this layer also generally provides adhesion). That way point loads become distributed (potentially to the protector and also) to the device. Example: a thin clear sticky low molecular weight polymer (1), an ...

4

In the real world ,I don't believe I have seen CORROSION at STRESSED areas. Corrosion may be accelerated at STRAINED areas; the strains introduce energy to the microstructure which may promote corrosion. Cracking is another story - which you did not ask about . Hydrogen cracking of high strength steels is likely the most common case . It can often occur with ...

4

You left off a "T" ; it is "time, temperature, transformation". It is a basis to evaluate hardenability of alloy steels. A sample is austenitized, then rapidly cooled to some temperature ,and held at that temperature for a specific time . Then rapidly cooled to room temperature and the microstructure examined . And after one finishes the ...

3

In this case, we say that $\epsilon_{axial}$ ($z$-direction in your diagram) is positive by convention. Then, for a normal material with a Poisson's ratio $\nu \ge 0$, $\epsilon_{transverse}$ ($x$- and $y$-directions in your diagram) will be negative. The fact that the material is isotropic means we have no way of distinctly labeling one direction $x$ and ...

3

cell phone housings and trim, Lego blocks and all other similar types of plastic toys, camera housings, plates, bowls, and dishes of plastic as well as plastic knives, forks and spoons, automobile dashboards, PC and tablet housings, refrigerator drawers, plumbing pipe, fittings, fixtures and sinks, electrical enclosures and conduit fitments, booster seats ...

3

Strain is determined as $\epsilon = (l - l_o)/l_o$. The 0.2% offset strain is the point where $\epsilon = 0.002$. The 0.2% offset stress is determined by using a line that has the same slope as the initial stress strain ... the initial Young's modulus. In materials that are fully elastic up to and beyond this point, the initial slope remains constant. In ...

3

CTE characterizes dimensional change, with no load. It corresponds to a relatively fundamental physical principle, making it easy to use in a design calculation Heat Deflection test characterizes the deformation under heat AND load, which includes a variety of phenomena happening at the same time. The result is dependent on the particular geometry and ...

3

Usually the method to measure the toughness is either the Charpy or Izod Test. They are very similar and the main difference are the boundary condition of each one (Charpy fixed both ends, Izod: vertical cantilever). The setup is the following (and the differences) are presented below. Figure : Charpy and Izod impact testing (source Green-mechanic) As you ...

2

There are two reasons: One is to remove trapped air bubbles and two to pack the fibres as densely as possible within the structure or weave of the fibre pattern wanted.

2

I confirm that in your case the yield stress is an adapted limit value. However, there is a unit error in your computation. You forgot to convert the area in m2 which means that the results is not 362 Pa but 362x106 Pa = 362 MPa. Assuming that your reasoning is correct to compute the effort, your are above the limit of 170 MPa.

2

A rupturing pressure vessel consists of an interplay between rupture of the walls, that causes a reduction in pressure via venting, while rupture of the walls also weakens the ability of the structure to sustain more pressure. Which one declines faster, and to what extent, determines what happens next. If the rupture on the wall is small and does not ...

2

Although the deflections equation @kamran reported, are indeed what most textbooks give, the problem is that they only hold true for relatively small deflections. I also agreewith @alephzero that there are a few errors in the calculations. e.g. I calculate $Ix = 0.0491 cm^4$. By my calculations, you would need a steel rod with almost 11cm diameter to support ...

2

Machinability index is measure for cutting processes which shows how easy it is to remove material. It is used by comparing the cutting speed to a reference value for steel. I.e.: you try to find what is the speed that a cutting tool will last 20 minutes while cutting a given material. $V_m$ you compare $V_m$ with the reference speed $V_r$ (which uses steel ...

2

Here is a detailed list for metals both ferrous like steel and cast iron and, nonferrous like copper and their alloys Machinability Rating Chart. machinabilty index

2

The data sheet says it has a 71ksi yield, I see no reason to doubt it. Cold drawn is likely stronger near the surface which would be an advantage for bending resistance. SAE 1045 means it means it is carbon steel with 0.45 %carbon. The primary feature or the material you list is a precision dimension surface . It will have low toughness but I can't imagine ...

2

Geick Technical Formulae has tables of common values, but that info is spread across several tables. One you missed is electrical resistance. The book also contains tables for density etc

2

that is a scanner that converts the writing on the board into a digital file for remote sharing, and then erases the board. These devices use either a rack-and-pinion gear arrangement with the rack running along the top edge of the board, or a flexible cable-and-pulley system with an electric motor coupled to the pulley through a set of reduction gears. You ...

2

Well mass and high-temperature behavior are really the key benefits with carbon/ceramic brakes. High temperature: Their high temperature properties make them more dependable during heavy use as they barely suffer from brake fade, they are also not prone to warping due to carbon fibers extremely low coefficient of thermal expansion (CTE). The CTE is actually ...

2

I'm not an expert, but the disk in a disk brake system needs to absorb enourmous amounts of heat to dissipate the energy of motion. It then needs to transfer that heat to the air. Polymers generally are poor heat conductors and have relatively low heat capacity. Add to this lower strength and you have a poor choice for disk material. As Jim Clark mentions in ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
⚠ Draft Standards Track: ERC

# ERC-5851: On-Chain Verifiable Credentials

### Interface for contracts that manage verifiable claims and identifiers as Soulbound tokens.

Authors Yu Liu (@yuliu-debond), Junyi Zhong (@Jooeys) 2022-10-18 https://ethereum-magicians.org/t/eip-5815-kyc-certification-issuer-and-verifier-standard/11513 EIP-721, EIP-1155, EIP-1167, EIP-1967, EIP-3475

## Abstract

This proposal introduces a method of certifying that a particular address meets a claim, and a method of verifying those certifications using on-chain metadata. Claims are assertions or statements made about a subject having certain properties that may be met conditions (for example: age >= 18), and are certified by issuers using a Soundbound Token (SBT).

## Motivation

On-chain issuance of verifiable attestations are essential for use-case like:

• Avoiding Sybil attacks with one person one vote
• Participation in certain events with credentials
• Compliance to government financial regulations etc.

We are proposing a standard claims structure for Decentralized Identity (DID) issuers and verifier entities to create smart contracts in order to provide on-chain commitment of the off-chain verification process, and once the given address is associated with the given attestation of the identity verification off-chain, the issuers can then onboard other verifiers (i.e. governance, financial institution, non-profit organization, web3 related cooperation) to define the condition of the ownership of the user in order to reduce the technical barriers and overhead of current implementations.

The motivation behind this proposal is to create a standard for verifier and issuer smart contracts to communicate with each other in a more efficient way. This will reduce the cost of KYC processes, and provide the possibility for on-chain KYC checks. By creating a standard for communication between verifiers and issuers, it will create an ecosystem in which users can be sure their data is secure and private. This will ultimately lead to more efficient KYC processes and help create a more trustful environment for users. It will also help to ensure that all verifier and issuer smart contracts are up-to-date with the most recent KYC regulations.

## Specification

The keywords “MUST”, “MUST NOT”, “REQUIRED”, “SHALL”, “SHALL NOT”, “SHOULD”, “SHOULD NOT”, “RECOMMENDED”, “NOT RECOMMENDED”, “MAY”, and “OPTIONAL” in this document are to be interpreted as described in RFC 2119 and RFC 8174.

### Definitions

• Zero-Knowledge Proof (ZKP): a cryptographic device that can convince a verifier that an assertion is correct without revealing all of the inputs to the assertion.

• Soulbound Token (SBT): A non-fungible and non-transferrable token that is used for defining the identity of the users.

• SBT Certificate: An SBT that represents the ownership of ID signatures corresponding to the claims defined in function standardClaim().

• Verifiable Credential (VC): A collection of claims made by an issuer. These are temper evident credentials that allow the holders to prove that they posses certain characteristics (for example, passport verification, constraints like value of tokens in your wallet, etc) as demanded by the verifier entity.

• Claim: An assertion that the DID Holder must fulfill to be verified.

• Holder: The entity that stores the claim, such as a digital identity provider or a DID registry. The holder is responsible for validating the claim and providing verifiable evidence of the claim.

• Claimer: The party making a claim, such as in an identity verification process.

• Issuer: The entity that creates a verifiable credential from claims about one or more subjects to a holder. Example issuers include governments, corporations, non-profit organizations, trade associations, and individuals.

• Verifier: An entity that validates data provided by an issuer of verifiable credentials, determining its accuracy, origin, currency and trustworthiness.

Claims MUST be exposed in the following structures:

Each claim requirement MUST be exposed using the following structure:

    /** Metadata
*
* @param title defines the name of the claim field
* @param _type is the type of the data (bool,string,address,bytes,..)
*/
string title;
string _type;
string description;
}

#### 2. Values Information

This following structure will be used to define the actual claim information, based on the description of the Metadata structure, the structure is the same as Values structure of EIP-3475.

   struct Values{
string stringValue;
uint uintValue;
bool boolValue;
}

#### 3. Claim structure

Claims (eg. age >= 18, jurisdiction in allowlist, etc.) are represented by one or many instances of the Claim structure below:

    /** Claims
*
* Claims structure consist of the conditions and value that holder claims to associate and verifier has to validate them.
* @notice the below given parameters are for reference purposes only, developers can optimize the fields that are needed to be represented on-chain by using schemes like TLV, encoding into base64 etc.
* @dev structure that defines the parameters for specific claims of the SBT certificate
* @notice this structure is used for the verification process, it contains the metadata, logic and expectation
* @notice logic can represent either the enum format for defining the different operations, or they can be logic operators (stored in form of ASCII figure based on unicode standard). like  e.g:
("⊄" = U+2284, "⊂" = U+2282,  "<" = U+003C , "<=" = U + 2265,"==" = U + 003D, "!="U + 2260, ">=" = U + 2265,">" =  U + 2262).
*/
struct Claim {
string logic;
Values expectation;

}

description of some logic functions that can be used are as follows:

Symbol Description
does not belong to the set of values (or range) defined by the corresponding Values
condition that the parameter belongs to one of values defined by the Values
< condition that the parameter is greater than value defined by the Values
== condition that the parameter is strictly equal to the value defined by the Values structure

#### Claim Example

{
"title":"age",
"type":"unit",
"description":"age of the person based on the birth date on the legal document",
"logic":">=",
"value":"18"
}

Defines the condition encoded for the index 1 (i.e the holder must be equal or more than 18 years old).

### Interface specification

#### Verifier

/// @notice getter function to validate if the address claimer is the holder of the claim defined by the tokenId SBTID
/// @dev it MUST be defining the conditional operator (logic explained below) to allow the application to convert it into code logic
/// @dev logic given here MUST be the conditiaonl operator, MUST be one of ("⊄", "⊂", "<", "<=", "==", "!=", ">=", ">")
/// @param claimer is the EOA address that wants to validate the SBT issued to it by the issuer.
/// @param SBTID is the Id of the SBT that user is the claimer.
/// @return true if the assertion is valid, else false
/**
example ifVerified(0xfoo, 1) => true will mean that 0xfoo is the holder of the SBT identity token defined by tokenId of the given collection.
*/
function ifVerified(address claimer, uint256 SBTID) external view returns (bool);

#### Issuer

/// @notice getter function to fetch the on-chain identification logic for the given identity holder.
/// @dev it MUST not be defined for address(0).
/// @param SBTID is the Id of the SBT that the user is the claimer.
/// @return the struct array of all the descriptions of condition metadata that is defined by the administrator for the given KYC provider.
/**
ex: standardClaim(1) --> {
{ "title":"age",
"type": "uint",
"description": "age of the person based on the birth date on the legal document",
},
"logic": ">=",
"value":"18"
}
Defines the condition encoded for the identity index 1, defining the identity condition that holder must be equal or more than 18 years old.
**/

function standardClaim(uint256 SBTID) external view returns (Claim[] memory);

/// @notice function for setting the claim requirement logic (defined by Claims metadata) details for the given identity token defined by SBTID.
/// @param SBTID is the Id of the SBT-based identity certificate for which the admin wants to define the Claims.
/// @param claims is the struct array of all the descriptions of condition metadata that is defined by the administrator. check metadata section for more information.
/**
example: changeStandardClaim(1, { "title":"age",
"type": "uint",
"description": "age of the person based on the birth date on the legal document",
},
"logic": ">=",
"value":"18"
});
will correspond to the functionality that admin needs to adjust the standard claim for the identification SBT with tokenId = 1, based on the conditions described in the Claims array struct details.
**/

function changeStandardClaim(uint256 SBTID, Claim[] memory _claims) external returns (bool);

/// @notice function which uses the ZKProof protocol to validate the identity based on the given
/// @param SBTID is the Id of the SBT-based identity certificate for which admin wants to define the Claims.
/// @param claimer is the address that needs to be proven as the owner of the SBT defined by the tokenID.
/**
example: certify(0xA....., 10) means that admin assigns the DID badge with id 10 to the address defined by the 0xA.... wallet.
*/
function certify(address claimer, uint256 SBTID) external returns (bool);

/// @notice function which uses the ZKProof protocol to validate the identity based on the given
/// @param SBTID is the Id of the SBT-based identity certificate for which the admin wants to define the Claims.
/// @param claimer is the address that needs to be proven as the owner of the SBT defined by the tokenID.
/* eg: revoke(0xfoo,1): means that KYC admin revokes the SBT certificate number 1 for the address '0xfoo'. */
function revoke(address certifying, uint256 SBTID) external returns (bool);

#### Events

    /**
* standardChanged
* @notice standardChanged MUST be triggered when claims are changed by the admin.
* @dev standardChanged MUST also be triggered for the creation of a new SBTID.
e.g : emit StandardChanged(1, Claims(Metadata('age', 'uint', 'age of the person based on the birth date on the legal document' ), ">=", "18");
is emitted when the Claim condition is changed which allows the certificate holder to call the functions with the modifier, claims that the holder must be equal or more than 18 years old.
*/
event StandardChanged(uint256 SBTID, Claim[] _claims);

/**
* certified
* @notice certified MUST be triggered when the SBT certificate is given to the certifying address.
* eg: Certified(0xfoo,2); means that wallet holder address 0xfoo is certified to hold a certificate issued with id 2, and thus can satisfy all the conditions defined by the required interface.
*/

/**
* revoked
* @notice revoked MUST be triggered when the SBT certificate is revoked.
* eg: Revoked( 0xfoo,1); means that entity user 0xfoo has been revoked to all the function access defined by the SBT ID 1.
*/
}

TBD

## Backwards Compatibility

• This EIP is backward compliant for the contracts that keep intact the metadata structure of previous issued SBT’s with their ID and claim requirement details.
• For e.g if the DeFI provider (using the modifiers to validate the ownership of required SBT by owner) wants the admin to change the logic of verification or remove certain claim structure, the previous holders of the certificates will be affected by these changes.

## Test Cases

Test cases for the minimal reference implementation can be found here for using transaction verification regarding whether the users hold the tokens or not. Use Remix IDE to compile and test the contracts.

## Reference Implementation

The interface is divided into two separate implementations:

• EIP-5851 Verifier is a simple modifier that needs to be imported by functions that are to be only called by holders of the SBT certificates. Then the modifier will call the issuer contract to verifiy if the claimer has the SBT certifcate in question.

• EIP-5851 Issuer is an example of an identity certificate that can be assigned by a KYC controller contract. This is a full implementation of the standard interface.

## Security Considerations

1. Implementation of functional interfaces for creating KYC on SBT (i.e changeStandardClaim(), certify() and revoke()) are dependent on the admin role. Thus the developer must insure security of admin role and rotation of this role to the entity entrusted by the KYC attestation service provider and DeFI protocols that are using this attestation service.

Copyright and related rights waived via CC0.


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In this talk, we explain the computation of the Bergman kernel for Hartogs triangle domains
given by $$extract_itex$$|z_1|^k<|z_2|<1$$/extract_itex$$ or $$extract_itex$$|z_1|<|z_2|^k<1$$/extract_itex$$ for any positive integer $$extract_itex$$k$$/extract_itex$$.
We also generalize this domain to higher dimensional cases and discuss the existence of the zeros of the Bergman kernel.


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# pygpcca.GPCCA.optimize

GPCCA.optimize(m)[source]

Full G-PCCA [Reuter18] spectral clustering method with optimized memberships.

It also has the option to optimize the number of clusters (macrostates) m as well.

If a single integer m is given, the method clusters the dominant m Schur vectors of the transition_matrix. The algorithm generates a fuzzy clustering such that the resulting membership functions chi are as crisp (characteristic) as possible, given m.

Instead of a single number of clusters m, a tuple or a dict {'m_min': int, 'm_max': int} containing a minimum and a maximum number of clusters can be given. This results in repeated execution of the G-PCCA core algorithm for $$m \in [m_{min},m_{max}]$$. Among the resulting clusterings, the sharpest/crispest one (with maximal crispness) will be selected.

Parameters

The number of clusters or a range where a search for potentially optimal cluster numbers is performed. Valid options are:

See minChi() for selection of good (potentially optimal) number of clusters.

Return type

GPCCA

Returns

Returns self and updates the following attributes:


text:
Zeguxizhai suanxue

work by Li Shanlan

discussed in biography

...mathematics was not held in high esteem, and he had to find various other employments, such as tutoring. Almost all the treatises that he wrote in this period were later collected into his Zeguxizhai suanxue (1867; “Mathematics from the Zeguxi Studio”). These treatises are characterized by extensive use of infinite series expansions for trigonometric and logarithmic...
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Zeguxizhai suanxue
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