Wave Equation

From Maxwell’s second equation

Take curl on both side

From Maxwell’s first equation

……….1

……….2

Substitute 2 in 1

The vector identity

From Maxwell's third equation

For the charge free region

……….3

……….6

Substitute 5 in 4

……….5

Substitute 6 in 3

……….4

This is the wave equation for Electric field

From Maxwell's first Equation

Take curl on both side

From Maxwell’s second equation

….. 7

…… 8

Substitute 8 in 7

The vector identity

From Maxwell’s fourth equation

………. 8

………. 9

Substitute 10 in 9

Substitute 11 in 8

……. 10

……. 11

This is the wave equation for Magnetic field

The above two equation are combinedly called as wave equations

Wave Equation in free space

The wave equations are

In free space conductivity , ,

Substitute these values in the wave equations 1 and 2

……..1

……..2

Wave Equation in Phasor form

Consider a linear, isotropic charge free medium i.e. =0The Maxwell equation in Phasor form are

Take curl on both side of equation 2

The vector identity is

Equating 4 and 5

Substitute 3 in 6

……1

……3

……2

Substitute 1 in 4

……4

……5

……6

……7

……6

The equations 6 and 7 are known as Helmholtz’s equation or Vector Wave equation

If the wave propagating along direction and it has x component only

This is scalar wave equation, a linear differential equation with solution

It has component only and component is ‘0’

To Include the time factor, multiply by

Similarly for ‘H’

……1

……2

By solving the equation 1 and 2 we get the value of and

It’s unit is nepers per meter

It’s unit is radian per meter

Plane wave in Free space

Condition

Values

Attenuation constant

Phase constant or wave number

Velocity of propagation

Wave length

Phase difference between “E” and “H”

Phase difference between “E” and “H”

Attenuation constant

Phase constant or wave number

Velocity of propagation

Wave length

Plane waves in Good conductor

Condition

Phase difference between “E” and “H”

As E or H wave travels in a medium .It’s amplitude is attenuated by a factor

The distance at which the wave amplitude is decreased to a factor is called skin depth

The skin depth is a measure of the depth to which an EM wave can penetrate the medium

Skin Depth