Last time I have discussed the few characteristics or properties of transverse electric and magnetic waves in parallel planes. Let us discuss more properties of TE and TM waves like cut-off wavelength, guide wavelength, phase velocity, group velocity:

**(a) ****Cut – off Wavelength**

ג_{c} = υ_{0}/f_{c}

Put the value of υ_{0} and f_{c} in above equation,

Or ג_{c} = 1/√ μ_{0}ε_{0} 2 α√ μ_{0}ε_{0}/m

** **ג_{c} = 2 α/m

The above is the expression for the **cut-off wavelength.**

**(b) ****Guide Wavelength**

**Definition.** It is defined as the distance required for the phase to shift through 2π radians.

That is** **ג_{g} = 2 π/b_{g }

Or ג_{g} = 2 π/ √ w^{2}με – ( mπ /α)^{2}

(because b_{g }= √ w^{2}με – ( mπ /α)^{2})

This is the expression for ** guide wavelength**

Now at ¦_{c} , w^{2}με = ( mπ /α)^{2}

By substituting value of ( mπ /α)^{2} in equation of ג_{g} , we get

ג_{g} = 2 π/ √ w^{2}με – w^{2}με

= 2 π / w√ με√(1 – w^{2}_{c} / w^{2} ) = 2 π / w√ μ^{0}ε^{0 }√(1 – w^{2}_{c} / w^{2} )

(if μ = μ_{0} , ε=ε_{0})

= 2 π υ_{0}/2 πf √(1 – w^{2}_{c} / w^{2} )

Or ג_{g} = ג/ √(1 – w^{2}_{c} / w^{2} )

Where ג = υ_{0f} is the free space wavelength.

As w_{c} = 2πf_{c} and w = 2πf

ג_{g} = ג/ √(1 – ¦_{c}^{2} / f )

The above equation also represents **guide wavelength** in terms of free space and cut-off frequencies.

**Derivation of the relation between **ג** ,**ג** _{g} and **ג

**ג**

_{c}(or**ג**

_{g }in terms of**,**ג

_{c}) :As ג = ג/Ö(1 – ג/(גc)^{2}

Therefore the expression of guide wavelength can also be written as

ג_{g} = ג/ Ö(1 – ג/(גc)^{2}

This is the relation between ג** _{g}, **ג

**and ג**

_{c}.

**Another Form :**

By squaring both sides of above equation, we get

ג^{2}_{g} = ג^{2}/1- ג/(גc)^{2}

ג/(ג_{g})^{2} = 1- ג/(ג_{c})^{2}

Dividing both sides by ג^{2}, we get

1/ ג^{2} = 1/ ג^{2}_{g} + ג^{2}_{c}

This expression also represents the relation between ג ,ג_{g} and ג_{c}

**(c) ****Phase (Wave) Velocity in parallel planes
**

**Definition . **It is the velocity of propagation of equiphase surface along the guide.

υ_{p} = w/b_{g}

Put the value of b_{g} in above expression

υ_{p} = w/√ w^{2}με – ( mπ /α)^{2}

when frequency becomes high enough so that w^{2}με >> ( mπ /α)^{2}

then υ_{p} => υ_{0 }= w/ √ w_{2}μ_{0}ε_{0}

or υ_{0 }=1/ √μ_{0}ε_{0} = 3 * 10^{8} m/s

Now at ¦_{c}, w^{2}_{c}μ_{0}ε_{0} = ( mπ /α)^{2} (if μ= μ_{0 }and ε = ε_{0})

Thus equation of υ_{p} will become

υ_{p} = w/ √ w_{2}μ_{0}ε_{0} – w_{2}μ_{0}ε_{0} = w/ √ w√μ_{0}ε_{0} √1-w^{2}_{c}/w^{2}

υ_{p} = υ_{0 }/√1-(f_{c}/f)^{2}

(υ_{0 }=1/ √μ_{0}ε_{0} , w= 2πf and w_{c} = 2πf_{c})

Or υ_{p} = υ_{0 }/ √1- ג^{2}/ ג^{2}_{c } (ג µ 1/f)

These are the expression for the **phase velocity.**

Phase velocity is also known as **Guide Velocity.**

These are the characteristics or properties of transverse electric and magnetic waves in parallel planes

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