Phase of traveling wave
=======================

The phase of a wave or other periodic function of some real variable :math:`t` is an angle-like
quantity representing the fraction of the cycle covered up to :math:`t`. As the variable :math:`t`
completes a full period, the phase increases by :math:`360^\circ` or :math:`2 \pi`.

If a function :math:`h(x, t)` describes a traveling wave, then position :math:`x` and time :math:`t`
can only appear in the form of the wave phase described below.

**Notes:**

#. :math:`\omega = (\vec \omega \cdot {\vec e}_x)`, i.e. the angular frequency is a positive quantity
   if the wave travels in the positive direction of the :math:`x`-axis. Here :math:`{\vec e}_x` is the
   unit vector pointing in the positive direction of the :math:`x`-axis.

**Conditions:**

#. This law applies to a 1-dimensional traveling wave.
#. The constant phase shift is not taken into account.

**Links:**

#. `Wikipedia, similar formula <https://en.wikipedia.org/wiki/Sinusoidal_plane_wave#Alternative_representations>`__.

.. py:currentmodule:: symplyphysics.laws.waves.phase_of_traveling_wave

.. py:data:: wave_phase

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

Symbol:
    :code:`phi`

Latex:
    :math:`\varphi`

Dimension:
    :code:`angle`


.. 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:: position

    :attr:`~symplyphysics.symbols.classical_mechanics.position`, or spatial coordinate.

Symbol:
    :code:`x`

Latex:
    :math:`x`

Dimension:
    :code:`length`


.. py:data:: angular_frequency

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

Symbol:
    :code:`w`

Latex:
    :math:`\omega`

Dimension:
    :code:`angle/time`


.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time`.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: law

    :code:`phi = k * x - w * t`


    Latex:
        .. math::
            \varphi = k x - \omega t