Basics of Wave Propagation
6th June 2023
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Outline
 Time Harmonic Fields
 Helmholtz Wave Equation
 Loosy Materials
 Poynting Vector
 Wave Transmission and Reflection
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Time Harmonic Fields
 Any time varying Field
can be written in form of
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Maxwell’s Equation
For a sinusoidal time variation of
replacing
with
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Wave Equation
Homogeneous medium:
ε, µ and σ are constant throughout the medium
Isotropic medium:
If ε is scalar constant so that D and E will have same
direction everywhere.
Homogeneous wave equation: source free
Non Homogeneous wave equation: include all sources
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Maxwell’s Equation
For a sinusoidal time variation of
replacing
with
Taking curl of first equation
and using the identity
We obtain the wave equation
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Solution for Free space condition
Consider a free space medium
or
Perfect dielectric (no conduction current)
ρv=0 & J =0
Maxwell’s Equation:
Wave equation:
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Time-Dependent Wave Equation
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Time-Dependent Wave Equation
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Time-Harmonic Wave Equation
The Time-Harmonic Wave Equation can be obtained by
starting with the time-harmonic Maxwell’s equations and
following steps similar to those in last section
or
with the time dependent equation and then transforming
the resulting time dependent wave equations to time –harmonic
wave equations.
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Time-Harmonic Wave Equation
Maxwell’s Equation:
Time-Harmonic Wave Equation in terms of electric field E:
Time-Harmonic Wave Equation in terms of electric field E for LHS:
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Time-Harmonic Wave Equation
Three sources:
-Charge distribution in free space (gradient of charge density
-Applied current density
-Induced current density +displacement current densities
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Time-Harmonic Wave Equation
Source free:
-
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Uniform Plane Wave Equation
A uniform plane wave:
-is a wave (i.e. a solution to the wave equation) in which
the electrical and magnetic field intensities are directed in fixed
directions in space and
are constant in magnitude and phase on planes
perpendicular to the direction of propagation.
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Uniform Plane Wave Equation
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Uniform Plane Wave Equation
Complete equation in time domain form
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Uniform Plane Wave Equation
Complete equation in time domain form
In boundless space
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Uniform Plane Wave Equation
Phase velocity of wave:
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Uniform Plane Wave Equation
Phase velocity of wave:
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Uniform Plane Wave Equation
Phase velocity of wave in free space:
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Wave Equation
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Wave Equation
1-dimensional time dependent wave equation:
1-dimensional time harmonic wave equation:
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Helmholtz Wave Equation
Compare wave equation:
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Helmholtz Wave Equation
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Helmholtz Wave Equation
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Helmholtz Wave Equation
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Wave Equation
Similarly, magnetic field satisfies the equation
Where
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Is the Laplacian operator
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Wave Equation
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Wave Equation
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Relation between E and H
Maxwell’s equation to obtain the magnetic field intensity (Faraday’s Law)
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Relation between E and H
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Intrinsic Impedance of material
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Poynting Vector
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Poynting Vector
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Poynting Vector
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Poynting Vector
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Poynting Vector: Physical Interpretation
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Lossy material
For lossy dielectric
For lossless dielectric
Complex permittivity
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Lossy material
Loss Tangent =
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Lossy material
Lossy dielectrics
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Propagation constant
For lossy dielectric
So the Propagation constant
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Propagation of Plane waves in low loss
dielectrics
For loss tangent is small
Real part>> Imaginary part
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Propagation of Plane waves in conductors
For losses high
Real part << Imaginary part
Attenuation and phase constants are equal and very high, wave is attenuating rapidly
Propagating wave can be:
Propagation in conducting media can exist within short distance
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Skin Depth
The distance δ, through which the wave amplitude attenuated by a
factor 1/e of its original amplitude is called skin depth or depth of
penetration of the medium that is
Phase velocity in good conductor
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Skin depth
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Group Velocity
Group velocity is the velocity of a wave packet consisting of
narrow range or band of frequencies.
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Group Velocity
Phase velocity
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Group Velocity
Group velocity
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Group Velocity
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Group Velocity
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Polarization of Plane wave
The figure traced by the tip of the electric field vector as a function
of time at a fixed point in space.
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Polarization of Plane wave
The figure traced by the tip of the electric field vector as a function
of time at a fixed point in space.
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Plane waves at interfaces
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Incident and Reflected Field
Incident field (Ei,Hi) is in medium 1 in direction az
Electric field of incident EM wave
Magnetic field of incident EM wave
Reflected field (Er,Hr) is in medium 1 in direction -az
Electric field of reflected EM wave
Magnetic field of incident EM wave
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Transmitted field
Transmitted field (Et,Ht) is in medium 2 in direction az
Electric field of transmitted EM wave
Magnetic field of transmitted EM wave
The resultant field in medium 1 and medium 2 becomes
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Plane wave at Normal incidence
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Plane wave at Normal incidence
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Plane wave at Normal incidence
Reflection coefficient
Zero (no reflection) to 1 (total reflection)
Transmission coefficient
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Apply BC
Boundary condition
Tangential components of E and H are continuous across boundary i.e
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Reflection & Transmission Coefficient
and
Reflection coefficient
Transmission coefficient
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Normal Incidence on a Conductor
Case 1
When medium 1 is a perfect dielectric (lossless, σ1=0)
and medium 2 is perfect conductor (σ2=∞)
η2=0 and hence Γ=-1 and τ=0; E2=0
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Standing wave
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Standing wave
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Standing wave
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Normal Incidence on a Conductor
Case 1
But
Hence
Or
Since
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Normal Incidence on a Conductor
Case 1
Similarly
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Standing wave pattern
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Medium 1 and 2 both lossless
Case 2
When meduim 1 and medium 2 are lossless
Sub Case-i
• Both reflected and transmitted wave exists
• incident and transmitted wave amplitudes are unequal
Maxima occur at
Minima occur at
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Standing Wave
Case-II
Maxima and minima in medium-1 roles are reversed compared to case-I
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Standing Wave Ratio
• H1 minimum occurs when E1 is maximum and vice-versa
• Transmitted wave in medium 2 is purely travelling wave and
there no maxima or minima
• Standing wave ratio (SWR) is
Alternately,
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Summary

v
  f .
 
1
Velocity,
Propagation Constant
    i  j (  j )
Attenuation constant   Re{ }    [ 1  (  ) 2  1] Np/m
2

Phase constant
  Im{ }  


[ 1  ( )  1 rad/m
2

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Summary
Intrinsic Impedance
j

  j
Wavelength
2
v



f

m
1Np= 8.686 dB
1dB= 0.115 dB
Average Poynting Vector P=Re{ExH*} W/m2
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