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In a hollow rectangular wave guide of dimensions $a \times b$ for example, I know how to apply the boundary conditions to find the solutions. In particular, for TE (or TM) modes we have the expression $$k=\sqrt{\left(\frac{\omega}{c}\right)^2-\pi^2\left[\left(\frac{m}{a}\right)^2+\left(\frac{n}{b}\right)^2\right]} $$ as our dispersion relation. I understand that in order to excite a certain TE$_{mn}$ mode the driving frequency must exceed the cutoff frequency for such mode. My question is what does it mean to excite a mode? Is it that we may only find waves propagating with distinct frequencies that correspond to the excited modes?

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There are many ways to excite modes in a waveguide. The term is often used in a theoretical sense, like `let there be a wave...'

In a more practical sense, one can think of the one end of the waveguide being connected to a horn antenna, which receives some radiation from free-space and as a result excites modes in the waveguide. Does this answer your question?

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For some given $\omega$, one can only excite specific traveling modes corresponding to different values of $k$ as shown by the dispersion relation. This means that a traveling mode can be excited at any frequency as long as it exceeds the cutoff frequency $\omega_{mn}\equiv c\pi \sqrt{\big(\frac{m}{a}\big)^2+\big(\frac{n}{b}\big)^2}$ for that mode.

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I will give you a mostly non-mathenatical way to understand waveguide modes.

First, note that waves propagating in the space between two surfaces must bounce off the surfaces.

Next, note that when a wave reflects off a surface it forms a standing wave pattern with stationary (Bragg) surfaces parallel to the reflective surface.

A wave propagating between two parallel surfaces, then, forms standing waves with Bragg surfaces associated with both surfaces.

Those Bragg surfaces, on both sides of the structure, are stacked with a separation that depends on the wavelength of the propagating wavefront and the angle of incidence on the reflective surfaces. The math would show that in order for the system of Bragg surfaces to be stable, allowing a steady-state wave pattern in the space between the reflective surfaces, th he Bragg surfaces inside the space must coincide. That means, for light of any given wavelength, there is only a finite number of propagation angles that support a steady-state wave pattern. Those propagation angles correspond to the available propagation modes in the waveguide.

All the equations describing waveguide modes derive from these considerations.

Note also that a 3D waveguide can support a lot more modes than a 2D waveguide, such as "corkscrew" modes that reflect cyclically off of all of the faces of the waveguide.

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