# What is the general boundary condition of wave-guides?

By wave-guides I refer to the device with (perfectly) conducting walls that enclose EM wave inside. I'm reading this tutorial here http://farside.ph.utexas.edu/teaching/em/lectures/node105.html and found this interesting boundary condition for wave-guides:

##E_{\parallel} = 0## -- (1)

##B_{\perp} = 0## -- (2)

however what anchor ##\parallel## and ##\perp## are respecting is not mentioned in the tutorial but I suppose that it's the wall of wave-guide (please correct me if I get it wrong). By the way the tutorial assumes that the wave-guide is guiding the EM wave to propagate along the z-axis but the geometry of the cross-section is NOT specified.

My question is, are (1) and (2) the general boundary conditions for wave-guides? If so why isn't ##E_{\perp}## or ##B_{\parallel}## taken into consideration? I drew some pill-boxes but unfortunately I didn't come up with a convincing answer for myself.

In an engineering textbook I only learned to solve equations for rectangular wave-guide (with cross-section ##x \in [0, a], y \in [0, b]##) like assuming that the wave is propagating along the z-axis and ##E_z = 0## for TE wave and ##B_z=0## for TM wave then apply "separate variable" assumption to solve the rest of the unknowns.

Any help would be appreciated :)

HallsofIvy
Homework Helper
By wave-guides I refer to the device with (perfectly) conducting walls that enclose EM wave inside. I'm reading this tutorial here http://farside.ph.utexas.edu/teaching/em/lectures/node105.html and found this interesting boundary condition for wave-guides:

##E_{\parallel} = 0## -- (1)

##B_{\perp} = 0## -- (2)

however what anchor ##\parallel## and ##\perp## are respecting is not mentioned in the tutorial but I suppose that it's the wall of wave-guide (please correct me if I get it wrong).
Yes, those refer to the components of the E wave parallel and the B wave perpendicular to the wave guide. At the walls of the guide, those components must be 0.

By the way the tutorial assumes that the wave-guide is guiding the EM wave to propagate along the z-axis but the geometry of the cross-section is NOT specified.

My question is, are (1) and (2) the general boundary conditions for wave-guides? If so why isn't ##E_{\perp}## or ##B_{\parallel}## taken into consideration? I drew some pill-boxes but unfortunately I didn't come up with a convincing answer for myself. The component of B parallel and of E perpendicular to the walls of the guide are not affected by the guide.

In an engineering textbook I only learned to solve equations for rectangular wave-guide (with cross-section ##x \in [0, a], y \in [0, b]##) like assuming that the wave is propagating along the z-axis and ##E_z = 0## for TE wave and ##B_z=0## for TM wave then apply "separate variable" assumption to solve the rest of the unknowns.
And you got "sine" and "cosine" functions, right? The form of the equations and the
type of solutions are heavily affected by the geometry. For example, with a cylindrical wave guide, you could expect to get "Bessel's equation" for r and Bessel functions as solutions.

Any help would be appreciated :)

@HallsofIvy , thanks for the reply. So do conditions (1) & (2) hold for all shapes of wave-guides when the wall is (nearly) perfect conductor (i.e. the skin-depth of the propagating wave is much less than the thickness of wall) ?