Editing Cutoff frequency (section)

===Mathematical analysis===
The starting point is the wave equation (which is derived from the [[Maxwell equations]]),
<math display="block">
  \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial{t}^2}\right)\psi(\mathbf{r},t)=0,
</math>
which becomes a [[Helmholtz equation]] by considering only functions of the form 
<math display="block">
  \psi(x,y,z,t) = \psi(x,y,z)e^{i \omega t}.
</math>
Substituting and evaluating the time derivative gives
<math display="block">
  \left(\nabla^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0.
</math>
The function <math> \psi </math> 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 
<math display="block">
  \psi(x,y,z,t) = \psi(x,y)e^{i \left(\omega t - k_{z} z \right)},
</math>
where <math>k_z</math> is the longitudinal [[wavenumber]], resulting in
<math display="block">
  \left(\nabla_{T}^2 - k_{z}^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0,
</math>
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 
<math display="block">
  \psi(x,y,z,t) = \psi_{0}e^{i \left(\omega t - k_{z} z - k_{x} x - k_{y} y\right)}.
</math>
Thus for the rectangular guide the Laplacian is evaluated, and we arrive at
<math display="block">
  \frac{\omega^2}{c^2} = k_x^2 + k_y^2 + k_z^2
</math>
The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry cross-section with dimensions {{mvar|a}} and {{mvar|b}}:
<math display="block"> k_{x} = \frac{n \pi}{a},</math>
<math display="block"> k_{y} = \frac{m \pi}{b},</math>
where {{mvar|n}} and {{mvar|m}} are the two integers representing a specific eigenmode. Performing the final substitution, we obtain
<math display="block"> \frac{\omega^2}{c^2} = \left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2 + k_{z}^2,</math>
which is the [[dispersion relation]] in the rectangular waveguide. The cutoff frequency <math>\omega_{c}</math> is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber <math>k_{z}</math> is zero. It is given by
<math display="block"> \omega_{c} = c \sqrt{\left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2}</math>
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 wave|evanescent]].