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 least 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 geometry of the waveguide. The easiest geometry 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 geometry cross-section with dimensions and : where and 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 and the wave is thus evanescent. See also Full width at half maximum High-pass filter Miller effect Spatial cutoff frequency (in optical systems) Time constant References External links Calculation of the center frequency with geometric mean and comparison to the arithmetic mean solution Conversion of cutoff frequency fc and time constant τ Mathematical definition of and information about the Bessel functions Filter theory