Cylindrical waveguides and Bessel functions

------------(1)

--------(2)

Dividing both sides by “fgh” and replacing the partial

derivatives by ordinary derivatives

Substituting this in the previous equation and multiply by

-----------(3)

Since the third term on the left side is only function of φ

therefore it can be set to a constant equal to “ –m2 ”

Letting

Multiplying both sides of equation(3) by “ f ” .

The above equation is recognized as classical Bessel

Differential equation.

Now the summary

Where,

Solution to these differential equations are

OR

And

OR

And

OR

For standing wave and periodic solution is

The values of the constants are calculated by boundary

conditions.

For TE-mode in a circular waveguide the function “ψ” is

replaced by the z-component of the vector potential “F” i-e

“Fz”.

Where

So now

And

Consider waves proapagating only in +z direction then

Where

OR

Where,