More characteristics or properties of transverse electric and magnetic waves in parallel planes

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

גc = υ0/fc

Put the value of υ0 and fc in above equation,

Or                        גc = 1/√ μ0ε0   2 α√ μ0ε0/m

                 גc = 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                  גg = 2 π/bg

Or                                    גg = 2 π/ √ w2με – ( mπ /α)2

(because bg = √ 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 גg , we get

גg = 2 π/ √ w2με – w2με

= 2 π / w√ με√(1 – w2c / w2 ) = 2 π / w√ μ0ε0 √(1 – w2c / w2 )

(if μ = μ0 , ε=ε0)

= 2 π υ0/2 πf √(1 – w2c / w2 )

Or                        גg = ג/ √(1 – w2c / w2 )

Where ג = υ0f is the free space wavelength.

As                        wc = 2πfc and w = 2πf

גg = 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 ,גg and גc (or גin terms of ג ,גc) :

As                                    ג = ג/Ö(1 – ג/(גc)2

 

Therefore the expression of guide wavelength can also be written as

גg = ג/ Ö(1 – ג/(גc)2

This is the relation between גg, ג and גc.

Another  Form :

By squaring both sides of above equation, we get

ג2g = ג2/1- ג/(גc)2

ג/(גg)2 = 1- ג/(גc)2

Dividing both sides by ג2, we get

1/ ג2 = 1/ ג2g +  ג2c

This expression also represents the relation between ג ,גg and גc

(c) Phase (Wave) Velocity in parallel planes

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

υp = w/bg

Put the value of bg in above expression

υp = w/√ w2με – ( mπ /α)2

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

then                    υp => υ0 = w/ √ w2μ0ε0

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

Now at ¦c,                      w2cμ0ε0 = ( mπ /α)2                           (if μ= μ0 and ε = ε0)

 

Thus equation of υp will become

υp = w/ √ w2μ0ε0 – w2μ0ε0 = w/ √ w√μ0ε0 √1-w2c/w2

υp = υ0 /√1-(fc/f)2

0 =1/ √μ0ε0 , w= 2πf and wc = 2πfc)

Or                                    υp = υ0 / √1- ג2/ ג2c                      (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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