Velocity of a wave

Velocity of a wave

Introduction : The progressive wave such as mechanical and electromagnetic wave has the parameter such as wavelength ,frequency, amplitude, etc. In this topic we are going to learn about velocity of  mechanical wave. The velocity of the mechanical wave plays a vital role in understanding the nature and mathematical concept. the velocity of  a mechanical wave depends upon elastic property and density of the medium. Let us study this in details.

Velocity of a wave

Velocity of transverse wave (on string) : The velocity of a transverse wave depends on the elastic property that is the restoring force in the medium. It also depends upon the linear mass density of the medium. In case of a string , the restoring force is the tension in the string.

Velocity of a transverse wave in string is directly proportional to square root of tension.

V ∝ √T ----(1)

Velocity of a wave in string is inversely proportional to square root of linear mass density.

From (1) and (2)

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Velocity of longitudinal wave : Velocity of a longitudinal wave depends on the nature of compression and rarefaction. The compression and rarefaction is due to elastic property of the medium. Thus at compression the density of the medium is greater and at rarefaction it is quite lower. The conclusion gives the result that velocity of a longitudinal wave depends on the elasticity and  linear mass density.

Mathematically, 

where E  the modulus of elasticity of medium and ρ is the density of medium.

The sound waves travel in the medium. Therefore, according to Newton the heat is  developed during compression and cools during rarefaction. Thus the process is isothermal. Hence, isothermal elasticity to be considered .If the volume or bulk elasticity of air is  determined under isothermal change then it is called isothermal bulk modulus which is equal to the atmospheric pressure P. hence the above equation changes to ,

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Laplace's correction : According to laplace ,newton's equation was not sufficient to explain the concept of velocity of a longitudinal wave. He was of the saying that the formation of  compression and rarefaction is not a slow process but it is very fast process and heat dissipation is not done completely. Thus the formation of rarefaction and compression was a adiabatic process and hence adiabatic elasticity should be considered. 

Adiabatic modulus of elasticity is given by E = үP.

where,

P = pressure of the air medium

γ = ratio of specific heat of air at constant pressure (Cp) to specific heat of air at constant volume (Cv) 

γ  = Cp / Cv

Thus the newtons equation changes to,

_{│<<<Superposition of wave│   │Factors affecting velocity>>>│} 

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