Attenuation coefficient in metal in rectangular waveguide for transverse electric waves
=======================================================================================

A coaxial waveguide is an electrical cable consisting of a central conductor and a
shield arranged coaxially and separated by an insulating material or an air gap. It is
used to transmit radio frequency electrical signals. The specific resistance of a
coaxial waveguide depends on the radius of the outer conductor and the radius of the
inner conductor, as well as on the relative permeability of the insulator material,
frequency of signal and specific conductivity of conductor.

**Conditions:**

#. Waves propagating in the waveguide must be transverse electric waves.
#. Second index :math:`n \ge 1`.

..
    TODO: find link

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

.. py:data:: attenuation_coefficient

    :attr:`~symplyphysics.symbols.electrodynamics.attenuation_coefficient` in metal.

Symbol:
    :code:`alpha`

Latex:
    :math:`\alpha`

Dimension:
    :code:`1/length`


.. py:data:: surface_resistance

    :attr:`~symplyphysics.symbols.electrodynamics.electrical_resistance` of the surface.

Symbol:
    :code:`R_s`

Latex:
    :math:`R_\text{s}`

Dimension:
    :code:`impedance`


.. py:data:: first_index

    The first index shows how many half-wavelengths fit across the width of the cross
    section.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`dimensionless`


.. py:data:: second_index

    The second index shows how many half-wavelengths fit across the height of the cross
    section.

Symbol:
    :code:`n`

Latex:
    :math:`n`

Dimension:
    :code:`dimensionless`


.. py:data:: width

    Width, or first dimension of the cross section. See :attr:`~symplyphysics.symbols.classical_mechanics.length`.

Symbol:
    :code:`a`

Latex:
    :math:`a`

Dimension:
    :code:`length`


.. py:data:: height

    Height, or second dimension of the cross section. See :attr:`~symplyphysics.symbols.classical_mechanics.length`.

Symbol:
    :code:`b`

Latex:
    :math:`b`

Dimension:
    :code:`length`


.. py:data:: medium_resistance

    :attr:`~symplyphysics.symbols.electrodynamics.electrical_resistance` of the medium filling the waveguide.

Symbol:
    :code:`R`

Latex:
    :math:`R`

Dimension:
    :code:`impedance`


.. 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:`alpha = 2 * R_s / R / (a * sqrt(1 - (lambda / (2 * lambda_c))^2)) * ((1 + a / b) * (lambda / (2 * lambda_c))^2 + (1 - (lambda / (2 * lambda_c))^2) * a / b * (a / b * n^2 + m^2) / ((a / b * n)^2 + m^2))`


    Latex:
        .. math::
            \alpha = \frac{2 \frac{R_\text{s}}{R}}{a \sqrt{1 - \left(\frac{\lambda}{2 \lambda_\text{c}}\right)^{2}}} \left(\left(1 + \frac{a}{b}\right) \left(\frac{\lambda}{2 \lambda_\text{c}}\right)^{2} + \frac{\left(1 - \left(\frac{\lambda}{2 \lambda_\text{c}}\right)^{2}\right) \frac{a}{b} \left(\frac{a}{b} n^{2} + m^{2}\right)}{\left(\frac{a}{b} n\right)^{2} + m^{2}}\right)