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 = υ₀/f_c

Put the value of υ₀ and f_c in above equation,

Or                        ג_c = 1/√ μ₀ε₀   2 α√ μ₀ε₀/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/ √ w²με – ( mπ /α)²

(because b_g = √ w²με – ( mπ /α)²)

This is the expression for  guide wavelength

Now at ¦_c ,                                 w²με = ( mπ /α)²

By substituting value of ( mπ /α)² in equation of ג_g , we get

ג_g = 2 π/ √ w²με – w²με

= 2 π / w√ με√(1 – w²_c / w² ) = 2 π / w√ μ⁰ε⁰ √(1 – w²_c / w² )

(if μ = μ₀ , ε=ε₀)

= 2 π υ₀/2 πf √(1 – w²_c / w² )

Or                        ג_g = ג/ √(1 – w²_c / w² )

Where ג = υ_0f is the free space wavelength.

As                        w_c = 2πf_c and w = 2πf

ג_g = ג/ √(1 – ¦_c² / 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_c (or ג_g  in terms of ג ,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

ג²_g = ג²/1- ג/(גc)²

ג/(ג_g)² = 1- ג/(ג_c)²

Dividing both sides by ג², we get

1/ ג² = 1/ ג²_g +  ג²_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/√ w²με – ( mπ /α)²

when frequency becomes high enough so that w²με >> ( mπ /α)²

then                    υ_p => υ₀ = w/ √ w₂μ₀ε₀

or                         υ₀ =1/ √μ₀ε₀ = 3 * 10⁸ m/s

Now at ¦_c,                      w²_cμ₀ε₀ = ( mπ /α)²                           (if μ= μ₀ and ε = ε₀)

 

Thus equation of υ_p will become

υ_p = w/ √ w₂μ₀ε₀ – w₂μ₀ε₀ = w/ √ w√μ₀ε₀ √1-w²_c/w²

υ_p = υ₀ /√1-(f_c/f)²

(υ₀ =1/ √μ₀ε₀ , w= 2πf and w_c = 2πf_c)

Or                                    υ_p = υ₀ / √1- ג²/ ג²_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