Jabarkan Mathematical analysis dari Cutoff frequency dalam bahasa sunda

The starting point is the wave equation (which is derived from the Maxwell equations),

which becomes a Helmholtz equation by considering only functions of the form

Substituting and evaluating the time derivative gives

The function  here refers to whichever field (the electric field or the magnetic field) has no vector component in the longitudinal direction - the "transverse" field.  It is a property of all the eigenmodes of the electromagnetic waveguide that at léast one of the two fields is transverse. The z axis is defined to be along the axis of the waveguide.

The "longitudinal" derivative in the Laplacian can further be reduced by considering only functions of the form

where  is the longitudinal wavenumber, resulting in

where subscript T indicates a 2-dimensional transverse Laplacian.  The final step depends on the géometry of the waveguide.  The éasiest géometry to solve is the rectangular waveguide.  In that case the remainder of the Laplacian can be evaluated to its characteristic equation by considering solutions of the form

Thus for the rectangular guide the Laplacian is evaluated, and we arrive at

The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular géometry crossection with dimensions a and b:

where n and m are the two integers representing a specific eigenmode.  Performing the final substitution, we obtain

which is the dispersion relation in the rectangular waveguide. The cutoff frequency  is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber  is zero. It is given by

The wave equations are also valid below the cutoff frequency, where the longitudinal wave number is imaginary. In this case, the field decays exponentially along the waveguide axis.