- 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

$v=\sqrt{\frac{T}{\mu}}$

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

$Z=\frac{T}{v}=\sqrt{T \mu}=\mu v$ - 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 $\xi (x,t) \to displacement$

Y is young’s modulus of the rod

$\rho $ is the density - Velocity of longitudinal wave in rod is

$v=\sqrt{\frac{Y}{\rho}$

**(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 $\mu$ of the medium.
- This property provides magnetic inertia of the medium.
- Elasticity of the medium is provided by the capacitive property called electrical permittivity $\epsilon $ of the medium.
- Permeability $\mu$ stores magnetic energy and the permittivity $\epsilon $ 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)

$ \bigtriangledown \times \textbf{H} =\epsilon \left ( \frac{\partial \textbf{E}}{\partial t} \right )$

$ \bigtriangledown \times \textbf{E} = – \mu \left ( \frac{\partial \textbf{H}}{\partial t} \right )$

$ \epsilon (\bigtriangledown \cdot \textbf{E})=\rho$

$ \bigtriangledown \cdot \textbf{H} =0$ - Here in above equations E is electric field , H is the magnetic field and $\rho$ is charge density