Wave Motion (Notes)

Wave Motion

Wave: 

Wave is the transfer of energy in the form of disturbance produced in a medium. In wave motion there is no net displacement of the particles of the medium. The particles of the medium vibrate simple harmonically about their mean position and only energy is hande over from particle to particle. The property like elasticity and inertia are responsible for the propagation of wave in a medium.

Types of Waves:

Depending upon the modes of vibration of the particles of the medium, waves are divided into two types.

1. Longitudinal Wave:

The wave in which the particles of the medium vibrate along the direction of propagation of wave is called longitudinal wave. It is propagated in the medium in the form of compression and rarefaction.
Compressionis the region of larger density whereas rarefactionis the region of smaller density in the medium due to propagation of wave. Ex: sound waves, compressional waves on spring etc.



Fig:- Longitudinal Wave




2. Transverse Wave:

   The wave in which particles of the medium vibrates along the direction perpendicular to the direction of propagation of wave is called transverse wave. It is propagated in medium in form of crest and trough.
   Crestis the region of maximum displacement of the particle above mean position during propagation of wave.
   Trough is the region of maximum displacement of the particle below mean position during propagation of wave. Ex: water ripples, wave on streched string, light wave (light wave is non mechanical transverse wave) etc.

   
   
   

Terms used in Wave Motion

Amplitude:

The maximum displacement of the particles of the medium from mean position during the propagation of the wave is called amplitude of the wave. It is denoted by 'a'. It's SI unit is meter 'm'.

Time Period:

The time taken by the oscillating particles of the medium to complete one oscillation during the propagation of wave is called time period. It is denoted by 'T'. It's SI unit is second 's'.

Frequency:

The number of complete oscillations made by the particles of the medium in one second during the propagation of wave is called frequency of the wave. It is denoted by 'f'.
i.e.

$f=\frac{1}{T}$

. Its SI unit is Hertz 'Hz'.

Wavelength:

The distance between any two consecutive particles of the medium that are in same phase during the propagation of wave is called wavelength.
OR
The distance travelled by the wave during the time at which any particle of the medium completes one oscillation is called wavelength. It is denoted by 'λ'. Its SI unit is meter 'm'.

Wave velocity:

The distance travelled by a wave in one second is called wave velocity. Since 'λ' distance is travelled by the wave in time 'T', then wave velocity is given by,
$V=\frac{\mathrm{\lambda \; <!--greek\; small\; letter\; lambda-->}}{T}$
or, $v=f\mathrm{\lambda \; <!--greek\; small\; letter\; lambda-->}$ $\because \; <!--latex:\; \backslash because-->f=\frac{1}{T}$

Phase:

Phase of an oscillating particle represents the state of motion (postion and direction of motion) of the particle with respect to mean position. It is measured in terms of fraction of angle 2π or fraction of time period 'T'.

Relation Between Path Difference and Phase difference:

When the distance between the two particles is λ, phase difference = 2π

When the distance between the two particles is 1, phase difference = $\frac{2\times \; <!--multiplication\; sign-->\mathrm{\pi \; <!--greek\; small\; letter\; pi-->}}{\mathrm{\lambda \; <!--greek\; small\; letter\; lambda-->}}$

When the distance between two particles is 'x', phase direction = $\frac{2\times \; <!--multiplication\; sign-->\mathrm{\pi \; <!--greek\; small\; letter\; pi-->}}{\mathrm{\lambda \; <!--greek\; small\; letter\; lambda-->}}\times \; <!--multiplication\; sign-->x$

Φ) = $\frac{2\times \; <!--multiplication\; sign-->\mathrm{\pi \; <!--greek\; small\; letter\; pi-->}}{\mathrm{\lambda <!--greek\; small\; letter\; lambda-->}}$× path difference.

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Progressive Wave:

The wave which travels from one region of space to another region of space is called progressive wave.

Equation of Progressive Wave or general wave equation:

Consider a progressive wave travelling from left to right. i.e. along positive x direction. During the propagation of wave, the particles of the medium vibrate simple harmoniclly about their mean position. The displacement of the particle at 'O' at any time 't' can be expressed as;
       $y=\mathrm{asin}\mathrm{\omega \; <!--greek\; small\; letter\; omega-->}\mathrm{t }$
where 'a' is amplitude and 'ω' is angular frequency of wave.

Consider a particle at point 'P' at a distaance x from 'O'. Since the wave is travelling from left to right, the particle at 'P' starts to vibrate later than the particle at 'O'. i.e. there is phase difference between particle at 'O' and 'P'.

Let ф be the phase difference between the particles at point 'O' and at 'p'. Then the displacement of the particle at 'P' at any time 't' can be expressed as;

$y=\mathrm{asin}(\mathrm{\omega \; <!--\; greek\; small\; letter\; omega\; -->}t-\mathrm{\varphi \; <!--\; greek\; phi\; symbol\; -->})$

But, phase difference$\mathrm{\varphi \; <!--\; greek\; phi\; symbol\; -->}=\frac{2\times \; <!--\; multiplication\; sign\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{\mathrm{\lambda \; <!--\; greek\; small\; letter\; lambda\; -->}}$ $\mathrm{\varphi \; <!--\; greek\; phi\; symbol\; -->}=\frac{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{\mathrm{\lambda \; <!--\; greek\; small\; letter\; lambda\; -->}}x$