Editing Waveguide (section)

=== Propagation modes and cutoff frequencies ===
A propagation [[Mode (electromagnetism)| mode]] in a waveguide is one solution of the wave equations, or, in other words, the form of the wave.{{sfn|Balanis|1989}}  Due to the constraints of the [[boundary conditions]], there are only limited frequencies and forms for the wave function which can propagate in the waveguide. The lowest frequency in which a certain mode can propagate is the [[cutoff frequency]] of that mode. The mode with the lowest cutoff frequency is the fundamental mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency.{{sfn|Cronin|1995}}{{rp|38}}

Propagation modes are computed by solving the [[Helmholtz equation]] alongside a set of boundary conditions depending on the geometrical shape and materials bounding the region. The usual assumption for infinitely long uniform waveguides allows us to assume a propagating form for the wave, i.e. stating that every field component has a known dependency on the propagation direction (i.e. <math>z</math>). More specifically, the common approach is to first replace all unknown time-varying fields <math>u(x,y,z,t)</math> (assuming for simplicity to describe the fields in [[Cartesian coordinate system|cartesian]] components) with their complex [[phasor]]s representation <math>U(x,y,z)</math>, sufficient to fully describe any infinitely long single-tone signal at frequency <math>f</math>, (angular frequency <math>\omega=2\pi f</math>), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like <math>U(x,y,z)=\hat{U}(x,y)e^{-\gamma z}</math>, where the <math>\gamma</math> term represents the propagation constant (still unknown) along the direction along which the waveguide extends to infinity. The Helmholtz equation can be rewritten to accommodate such form and the resulting equality needs to be solved for <math>\gamma</math> and <math>\hat{U}(x,y)</math>, yielding in the end an eigenvalue equation for <math>\gamma</math> and a corresponding eigenfunction <math>\hat{U}(x,y)_\gamma</math>for each solution of the former.{{sfn|Pozar|2012}}

The propagation constant <math>\gamma</math> of the guided wave is complex, in general. For a lossless case, the propagation constant might be found to take on either real or imaginary values, depending on the chosen solution of the eigenvalue equation and on the angular frequency <math>\omega</math>. When <math>\gamma</math> is purely real, the mode is said to be "below cutoff", since the amplitude of the field phasors tends to exponentially decrease with propagation; an imaginary <math>\gamma</math>, instead, represents modes said to be "in propagation" or "above cutoff", as the complex amplitude of the phasors does not change with <math>z</math>.{{sfn|Ramo|Whinnery|Van Duzer|1994}}