clj4 said:
This is what you got at post 361 for E_y, you applied the zero boundary condition to it and you got k. All the calculations seemed correct, we double checked them together and they reproduced Gagnon (7,8) perfectly.
I assume you mean E_x, but yes.
clj4 said:
It may, or may not be the correct procedure since the correct way of dealing with waveguides is to use E(x,y)= E_x(x)*E_y(y) and to separate the original equation into two equations, one in x and the other one in y.
Whoa... hold on. We appear to be using different notations. (maybe that has lead to some confusion) I have used E to refer to the electric field (a vector field). And E_x,E_y,E_z to refer to the components of that vector. However, you seem to take E to just be a component and then further, the subscript to just refer to the separable function of just that subscript.
I hope this hasn't been the cause of some of the confusion.
Different notation is fine as long as all parties are aware of it (but it is easier if we just use one).
clj4 said:
So where is this thing going? Looks like you are going in circles. You need to either prove wrong:
1. Eq(5) (the partial differential equation that is at the origin of it all
or
2. Eq(9) , i.e. the expression that is 0 for SR and non-zero for GGT
So why do you keep coming back to (7,8)?
I'm trying to get us to agree on Gagnon's solution to the wave equation because that is where his error is.
--------------------
Where is their error?
I'll let Griffith's
Introduction to electrodynamics (3rd ed.) do the talking:
\nabla^2\vec{E} = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \vec{E}, \ \ \nabla^2\vec{B} = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \vec{B} (eq 9.41)
(pg 377) "Now the wave equations for
E and
B (Eq. 9.41) were derived from Maxwell's equations. However, whereas every solution to Maxwell's equations (in empty space) must obey the wave equation, the converse is
not true; Maxwell's equations impose extra constrains on
E and
B."
This makes sense because Maxwell's equations are 4 equations (with E and B coupled), which we reduced to just 2
uncoupled equations. Here's an easy example:
\vec{B}=0, E_x=0,E_y=0, E_z = \cos(kx - kt/c) is a solution to the wave equations (eq 9.41), but obviously aren't solutions to Maxwell's equations (because \frac{1}{c^2}\frac{\partial}{\partial t} \vec{E} \ne \nabla \times \vec{B} = 0).
That was about Maxwell's equations in a "normal Lorentz frame" and not a GGT frame, but the reasons behind it remain the same.
This is where Gagnon makes their error. Let me demonstrate.
\nabla \times E = -\frac{\partial}{\partial t}B (still true in a GGT frame, according to Gagnon's choice ... see ref 9)
-\frac{\partial E_x}{\partial z} + \frac{\partial E_z}{\partial x} = -\frac{\partial B_y}{\partial t}
-ikE_x = -\frac{\partial B_y}{\partial t}
B_y = -\frac{k}{w} E_x + function(x,y,z)
So we are not free to just make B whatever we want and ignore other boundary conditions. Remember, from the boundary condition on a waveguide B_\perp=0 we know B_y(x=0)=B_y(x=b)=0, where b is the width of the waveguide in the x direction. Yet, B_y cannot satisfy this. So their solution to the wave equation is not valid.
No where in the paper do they mention this boundary condition. Since they were solving for the electric field, I believe they just felt it was not relevant (as you yourself did when boundary conditions were first brought up). This is their error.