- Any system whose particle motion are governed by classical wave equation is a system in which harmonic waves of any wavelength can travel with the speed v

- The value of v depends on the elastic and inertial properties of the system under consideration.

**(1) Transverse wave on a stretched string**

- Displacement of the string is governed by the equation

Where T is the tension and µ is the linear density (mass per unit length of the string)

- Velocity of wave on the string is

v=√(T/μ)

v is the velocity of the wave.

- Medium through which waves travel will offer impedance to these waves.

- If the medium is loss less i.e., it does not have any resistive or dissipative components, the impedance is solely determined by its inertia and elasticity.

- Characteristic impedance of string is determined by

Z=T/v=√(μT)=μv

- Since v is determined by the inertia and elasticity this shows that impedance is also governed by these two properties of the medium.

- For loss-less medium impedance is real quantity and it is complex if the medium is dissipative.

**(2) Longitudinal waves in uniform rod**

- Equation for longitudinal vibrations of a uniform rod is

- where ξ (x,t )→displacement

Y is young’s modulus of the rod

ρ is the density

- Velocity of longitudinal wave in rod is

v=√(Y/ρ)

**(3) Electromagnetic waves in space**

- When electric and magnetic field vary in time they produce EM waves.

- An oscillating charge has an oscillating electric and magnetic fields around it and hence produces EM waves.

- Example: - (1) Electrons falling from higher to lower energy orbit radiates EM waves of particular wavelength and frequency. (2) The motion of electrons in an antenna radiates EM waves by a process called
**Bramstrhlung**.

- Propagation of EM waves in a medium is also due to inertial and elastic properties of the medium.

- Every medium (including vacuum) has inductive properties described by magnetic permeability µ of the medium.

- This property provides magnetic inertia of the medium.

- Elasticity of the medium is provided by the capacitive property called electrical permittivity ε of the medium.

- Permeability µ stores magnetic energy and the permittivity ε stores the electric field energy.

- This EM energy propagates in the medium in the form of EM waves.

- Electric and magnetic fields are connected by Maxwell’s Equations (dielectric medium)

∇×**H**=ε (∂**E**)/∂t

∇×**E**= - μ (∂**H**⃗)/∂t

ε(∇∙**E**⃗)=ρ

∇∙**H**=0

- Here in above equations E ⃗ is electric field , H ⃗ is the magnetic field and ρ is charge density

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