Group velocity from dispersion relation
=======================================

Waves can form a group, called wave packets. The speed with which a wave packet travels
is called group velocity. In other words, it is the speed with which the overall envelope
shape of the wave's amplitudes — called envelope of modulation of the wave — propagates
through space.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Group_velocity#Definition_and_interpretation>`__.

.. py:currentmodule:: symplyphysics.laws.waves.group_velocity_from_dispersion_relation

.. py:data:: angular_wavenumber

    :attr:`~symplyphysics.symbols.classical_mechanics.angular_wavenumber` of the wave.

Symbol:
    :code:`k`

Latex:
    :math:`k`

Dimension:
    :code:`angle/length`


.. py:data:: group_velocity

    :attr:`~symplyphysics.symbols.classical_mechanics.group_speed` of the wave packet.

Symbol:
    :code:`v_g`

Latex:
    :math:`v_\text{g}`

Dimension:
    :code:`velocity`


.. py:data:: angular_frequency

    :attr:`~symplyphysics.symbols.classical_mechanics.angular_frequency` of the wave as a function of :attr:`~symplyphysics.symbols.classical_mechanics.angular_wavenumber`, also called the
    *dispersion relation* of the wave.

Symbol:
    :code:`w(k)`

Latex:
    :math:`\omega{\left(k \right)}`

Dimension:
    :code:`angle/time`


.. py:data:: law

    :code:`v_g = Derivative(w(k), k)`


    Latex:
        .. math::
            v_\text{g} = \frac{d}{d k} \omega{\left(k \right)}