Phase speed of wave in rectangular waveguide
============================================

The phase speed of a wave in a rectangular waveguide depends on the ratio of the
wavelength of the signal to the critical wavelength of the waveguide and the
electromagnetic properties of the insulator within the waveguide.

**Notation:**

#. :math:`c` (:code:`c`) is :attr:`~symplyphysics.quantities.speed_of_light`.

..
    TODO: fix file name
    TODO: find link

.. py:currentmodule:: symplyphysics.laws.electricity.circuits.waveguides.phase_velocity_of_wave_in_rectangular_waveguide

.. py:data:: phase_speed

    :attr:`~symplyphysics.symbols.classical_mechanics.phase_speed` of the wave in the waveguide.

Symbol:
    :code:`v`

Latex:
    :math:`v`

Dimension:
    :code:`velocity`


.. py:data:: relative_permittivity

    :attr:`~symplyphysics.symbols.electrodynamics.relative_permittivity` of the insulator.

Symbol:
    :code:`epsilon_r`

Latex:
    :math:`\varepsilon_\text{r}`

Dimension:
    :code:`dimensionless`


.. py:data:: relative_permeability

    :attr:`~symplyphysics.symbols.electrodynamics.relative_permeability` of the insulator.

Symbol:
    :code:`mu_r`

Latex:
    :math:`\mu_\text{r}`

Dimension:
    :code:`dimensionless`


.. py:data:: wavelength

    :attr:`~symplyphysics.symbols.classical_mechanics.wavelength` of the signal.

Symbol:
    :code:`lambda`

Latex:
    :math:`\lambda`

Dimension:
    :code:`length`


.. py:data:: critical_wavelength

    Critical :attr:`~symplyphysics.symbols.classical_mechanics.wavelength` of the system. See :ref:`Critical wavelength of waveguide <critical_wavelength_waveguide_def>`.

Symbol:
    :code:`lambda_c`

Latex:
    :math:`\lambda_\text{c}`

Dimension:
    :code:`length`


.. py:data:: law

    :code:`v = c / sqrt(epsilon_r * mu_r * (1 - (lambda / lambda_c)^2))`


    Latex:
        .. math::
            v = \frac{c}{\sqrt{\varepsilon_\text{r} \mu_\text{r} \left(1 - \left(\frac{\lambda}{\lambda_\text{c}}\right)^{2}\right)}}