Waveguides and Cavity resonator Equation confusion

[Dhruv]

Homework Statement

This is not a homework question and I am asking this question out of curiosity. We have been taught rectangular wave guides and cavity resonator recently and I don't want to gulp the wave equations as it is, so the problem that I am facing ( I have written it on paper and uploaded the photo) . Don't know if my question is stupid but it really confuses me .

Homework Equations

The Attempt at a Solution

 

[baby_1]

Hello
I don't know exactly what is your problem but I assume you confuse why in rectangular wave guide we set d^2/dz^2=Gama^2 but in cavity d^2/dz^2=-Kz^2?
As you know in the wave guide we transfer a wave in Z direction and it goes through wave guide with it's wave coefficient (Beta) but in cavity a wave repeat simultaneously in the cavity in +Z and -Z direction to create a fix resonator so it should be in sinusoidal form

 

[Dhruv]

Hello
I don't know exactly what is your problem but I assume you confuse why in rectangular wave guide we set d^2/dz^2=Gama^2 but in cavity d^2/dz^2=-Kz^2?

Yes this is what I am confused about. Thank you for reply :) , it helped a lot . So this means I am making changes in equation because I know that in resonator my wave has to be a standing wave rather being a propagating one in +z or -z direction.

 

[baby_1]

Hello
Yes of course ,However you can try to solve wave equation in cavity and use separation variables to obtain it's answers after that you should add two new boundary condition ( E tangential in EZ(0)=EZ(d)=0 )that only a sinusoidal form can prove that.

 

[Dhruv]

Thank you :)