wave equation, and general wave in physics

What are waves?

Mechanical and Electromagnetic waves

Waves motion

Transverse and longitudinal waves

Propagation of waves

General representation of waves

Terms use in describing waves and relationship between them

Equations of traveling of waves

Wavefronts

Progressive and stationary waves

Stroboscope

Properties of waves

Polarization  and  Polarization by reflection

What is wave?

A wave is a disturbance which travels through a medium, transferring energy from one point to another without making any permanent displacement of the medium itself.

For instance, when a stone is thrown or dropped into a pond or swimming pool, ripple or wave is seen spreading on the surface of the water from the point where the stone is dropped, but the water itself which act as medium here does not move in the direction of ripple, though the wave transfers energy from one point to the other

Wave also can be generated by plucking string fixed at both end. Light and sound are can also be shown as wave

Mechanical and Electromagnetic waves

What is mechanical wave?  It is type of wave that need material medium for its propagation. Such waves are, water wave, wave generated in string or rope and sound wave.

Define electromagnetic wave: Electromagnetic waves are type of waves that do not require material medium for its propagation. The examples are light rays, radio waves, X-ray and gamma-ray

Wave motion

The motion of the waves are by the particles of the medium, which move to and fro, or vibrate about a mean position when waves pass. The vibration is passed from one particle of medium to the next. The direction of in which this vibration take places is very important as it will bring about the classification of wave into transverse and longitudinal wave

Transverse and longitudinal wave

Transverse wave: It is a wave, that its direction of travel is perpendicular to the direction of vibration of the medium. Examples are the wave generated by plucking a string, water wave and all the electromagnetic waves.

Longitudinal wave: A wave is said to be longitudinal, if the direction of travel of the wave is the same with the direction of vibration of the medium. Example is sound wave.

Propagation of wave

The propagation of waves,  simple mean, how the waves travel, what pattern, does it take to travel, does it moves in a straight line or in up and down motion.

Propagation of transverse wave

From the definition of transverse waves, let take water wave as a case study for propagation of transverse waves. Suppose a wave is generated by dropping a stone in water, and a small piece of wood or cock floating in the path of the wave. The cock will be seen moving up and down in on spot as the wave travels pass. This up and down vibration of water particles is perpendicular to the direction of the wave along the surface.

Similarly particles of a plucked string vibrate up and down of from side to side, perpendicular to the direction of travel of the wave

Electromagnetic waves are also transverse wave but since no medium is required for their propagation , vibration cannot be particle vibration, because the wave is made of electric and magnetic vibration of very high frequency

Transverse waves can be represented by a series of up and down movements, that is some portion of the waves are displaced upward and while adjacent side is displaced downward The region of maximum upward displacement is called a CREST, while the maximum downward displacement is called TROUGH.

Propagation of Longitudinal wave

In longitudinal wave, the vibrating particles behave like a spiral spring with series of compressed region called Compression and spaced out region called rarefaction. Therefore particle of the medium can be represented by spiral spring in longitudinal wave motion.

Consider a sound wave (longitudinal wave) be propagated in air as show below, C=compression  and, R= rarefaction

• (a) Shows undisturbed air when no sound wave is being propagated through it , here the pressure of air is normal throughout.
• (b) It show a propagation of sound wave through air medium, at where is there is compression, the air pressure there is little above normal, and where there is rarefaction ,that is where the layers are further apart , the air pressure there is little below normal

General representation of waves

A transverse wave which is represented as with a series of crest and trough, when a graph of the wave is plotted, where y represents the direction of vibration of particles and x-represent the direction of propagation of the wave, similar picture of crest and trough is obtain, likewise when a graph of longitudinal wave is plotted, where y-represents the compression and x-represents the rarefaction.

Therefore both transverse and longitudinal waves can be represented by the diagram below. Terms use in describing waves and relationship between them

Amplitude (a): This can be defined as the maximum displacement of vibrating particles from their mean position (or rest) is called the amplitude a, of the wave, and is measured in meter

Period (T): It is time required for vibrating particles to make one complete cycle or oscillation, and is measured in seconds

Frequency (f):  It is the number of complete cycle or oscillation made by wave in one second

Wavelength (𝛌): The distance between the successive crest or successive trough is called wavelength 𝛌.

Wave speed (v): It is a distance wave travels per a second. It is measured in meter per second (m/s). For example if a wave travels a distance of x meter in t seconds, the speed is given by: Relationship between, T,f, 𝛌, and v

From definition of frequency and period, they are related by; The speed v, the wavelength 𝛌, and frequency f, are related by this equation:

V=f 𝛌

Equations of traveling of waves

The equation of traveling waves can be written as a sin or cosine function: Both sine  and cosine  are periodical function with amplitude =1 and period=2πrad . It means that it repeat after 2πrad. But angular velocity is defined as radians per second or rad, it similar to circular motion =3600 (one complete cycle)

So the sin and cos equation can be written as: In general , a traveling wave whose amplitude is A, and constant angular velocity is ,  can be written as: Where = a constant for wave that did not start from origin.

This constant angular distance ,which is known as phase constant can be related to linear distance as: So wave equation can be written as: where K= wave number

or substituting the, then we have   Since Then we can further write the wave equation as: Wavefronts

What is wavefront?  It can be defined as real or imaginary surface of wave  representing corresponding points of the wave that are in the same phase.

Consider a wave generated by dropping a spherical object in water, a circular pattern is observed emerging from the center of disturbance.  All particle the circumference of a given circle are at the same disturbance, therefore there are said to be in the same phase. So the common circle of all particle which vibrate in phase constitute what is called wavefront.

Progressive and stationary waves

What is progressive wave? A wave, whether longitudinal or transverse wave which travels continuously in a medium in the same direction without any change amplitude is called progressive or traveling wave.

What is stationary wave? It can be defined as a wave obtained when two progressive waves of equal amplitude and frequency, traveling in opposite directions, combined together. The combination of these two waves moving in opposite direction but have the same frequency and amplitude result from the interference between the two waves.

Standing /stationary can also be created by reflection of a wave when it hit an obstacle. The reflected wave returns along the same path with the incident wave. Then the reflected and incident waves combine to give raise to standing or stationary waves

Node:  is point on a stationary wave where there is no movement of the medium

Antinode: It is a point on a stationary wave where there is a maximum movement of the medium

Note that the wavelength of the wave is twice the distance between two successive nodes.

Example of stationary waves

1.  waves obtained by plucking a string fixed at both ends.
2.  waves set up in open and closed pipes

Stroboscope

Stroboscope: It is an instrument used in making a waves to appear stationary

Properties of waves

The properties of waves are reflection, refraction, diffraction, interference and polarization

Reflection: It is change in the direction of the a wavefront at an interface between two difference media so that the wavefront returns into the medium from where it originated. It can also be said that Reflection of light is a situation whereby light bounce back when it hits polished, smooth or shiny objects such as mirror, glass, water and metals.
The angle at which reflection of light occurs is the same angle at which the light hit the object

The light rays that hit the object is called incident rays while the one that reflected is called the reflected rays the common examples, are reflection of light, sound and water waves.

Refraction: Refraction of wave is the changing of the direction of traveling of wave, when it moves from one medium to another. The refraction or bending of paths of waves are accompanied by change in wavelength and the speed because the speeds of waves depend on the medium through which it travels.

Diffraction: Diffraction occur when waves bend around small obstacle, or waves spread out after they pass small opening. Bending of waves depend on the relative size of the light wavelength to the size of the opening. If the opening is much larger than the wavelength of light, the bending will be almost unnoticeable. Diffraction occurs in all waves, whether electromagnetic waves or mechanical waves

Interference:  Interference is superimposing (0r placing one on top of the other) of two waves traveling in the same direction. There are two type of interference: Constructive or addictive and destructive interference.

What is constructive interference?  It is superimposing of the two waves that are in phase,( that is one of the waves must have traveled a full wavelength further than other), and traveling the same direction. The resulting wave has amplitude twice the amplitude of the others.

What is destructive interference? When two waves traveling in the same direction, which are not in phase, (that is one of the waves must have traveled half a wavelength further than other), are superimposed, the waves cancel each other, this type of interference is called destructive one

Polarization  and  Polarization by reflection

A light wave that is vibrating in more than one plane is unpolarized light. Polarized light waves are light waves in which the vibrations occur in a single plane. The process of transforming unpolarized light into polarized one is called polarization

To polarizes light waves, we pass an unpolarized light , through a light polarizing filter, such as tourmaline, calcite, and quartz.  Another polarizer , but manufactured one is Polaroid

(A) (B) From the diagram above a light emerging from the crystal Q is polarized by putting a second crystal R behind Q and rotating R, when Q and R have their axes parallel to each other the light passing through R appears almost as bright as that passing through Q. When R is further slowly rotated about the line of vision the light emerging from it appears darker and darker and this is completely extinguished, as shown by second diagram. When the axis of R is perpendicular to that of Q. On further rotation in the same direction the emergence light become brighter and brighter and again becomes brightest when the axes of Q and B again parallel

This simple experiment leads to conclusion that light consist of transverse waves. Thus we can establish whether light is polarized or not by simply rotating a polarizer through which the light is passing.

If the transmitted light
(i). Shows no change in a density the incident light was polarized
(ii). The light Varies slightly in intensity per the revolution, the incident light was partly plane polarized
(ii). The light cut off completely  twice per  revolution ,the incident light was plane polarized

Polarization by reflection
It was discovered in 1808 by the French  physicist Etienne Malus ,that when an ordinary or  un-polarized light is incident at an angle of about 570 on the polished  surface of a glass plate the reflected light is plane polarized

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