The characteristics or properties of transverse electric (TE) and transverse magnetic (TM) in parallel planes or plates waves can be studied with the help of propagation constant g_{g} for these waves.

**(a) ****Propagation Constant in parallel planes
**

γ_{g } = √** K**^{2}_{g}** – **w^{2}με

γ_{g } = √(mπ /α)^{2} – w^{2}με

Where **K**_{g} = mπ /α

At very high frequency, so that

w^{2}με >> (mπ /α)^{2}

Thus γ_{g } = √-[-( mπ /α)^{2} + w^{2}με ]

γ_{g } = √-[ w^{2}με – ( mπ /α)^{2} ]

This shows that quantity under the radical will be negative and then γ_{g } will be pure imaginary that is

γ_{g } = j √ w^{2}με – (mπ /α)^{2}

Also γ_{g } = α_{g} + jb_{g}

Where α_{g} is **attenuation constant** and b_{g} is **phase constant.**

**Definition of ****attenuation constant α _{g} in parallel planes**: α

_{g}is defined as a constant which indicates the rate at which the wave amplitude reduces as it propagates from one point to another.

It is real part of propagation constant. It has units of dB/m or Neper/m.

**Definition of ****phase shift constant in parallel planes **b_{g} . b_{g} is defined as a measure of the phase shift in radians per unit length.

It is imaginary part of propagation constant, g_{g } with units radians/m.

Comparing the imaginary parts of above two equations, we get

**Phase Constant**

b_{g }= √w^{2}με – ( mπ /α)^{2} ]

Under these conditions, the fields will progress in the +z direction as waves and the attenuation of such waves will be zero for perfectly conducting planes

that is **Attenuation Constant **α_{g} = 0

**(b) ****Cut-Off Frequency**

As the frequency is decreased, there will be a stage at critical frequency,

f_{c} = w_{c}/2π, at which

w^{2}με = ( mπ /α)^{2}

or w_{c} = ( mπ /α) 1/√ με

or 2πf_{c }= ( mπ /α) 1/√με

or f_{c} = (m/2 α) (1/√ με) = (m/2 α) ( 1/√ μ_{0}ε_{0}) (if μ = μ_{0}, ε = ε_{0})

or f_{c} = m/2 α υ_{0}

Here f_{c} is the **cut – off frequency**.

For all frequencies less than f_{c} , the quantity under the radical of equation will be positive and γ_{g } will be a real number, that is γ_{g } = α_{g} + j 0 = α_{g}, as b_{g }= 0. This implies that fields will be attenuated exponentially in the +z direction and there will be no wave motions as b_{g }= 0.

**Definition of Cut-off Frequency** (f_{c} ). The frequency at which wave motion cases is known as the cut-off frequency of the guide.

**Another Definition.** It is defined as a frequency below which there exists only attenuation constant, α_{g} and phase shift constant, b_{g} = 0 and above which α_{g} = 0 and γ_{g} exists.

As f_{c} = m/2 α υ_{0}

Thus for each value of m, there is a corresponding cut-off frequency below which wave propagation cannot occur. Above the f_{c} , the wave propagation does occur and there will be no attenuation (α_{g} = 0) of the wave for perfectly conducting planes.

I will discuss more characteristics or properties of TE and TM waves in parallel planes in next article.