The document discusses the wave equation for electromagnetic waves. It begins by deriving the wave equations for the electric and magnetic fields from Maxwell's equations. These are vector wave equations that describe wave propagation in free space. The equations are then specialized to the scalar wave equation for waves propagating in a single direction. Solutions to these wave equations describe plane electromagnetic waves in free space and good conductors. Characteristics of plane waves like attenuation constant, phase constant, velocity, wavelength, and phase difference between electric and magnetic fields are also defined.

2. 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

3. 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

4. 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

5. Wave Equation in Phasor form
Consider a linear, isotropic charge free medium
i.e. =0
The 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

6. 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’

7. ……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

8. 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”

9. Attenuation constant
Phase constant or wave
number
Velocity of propagation
Wave length
Plane waves in Good conductor
Condition
Phase difference
between “E” and “H”

10. 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