Resonant frequencies of stretched string with fixed ends
========================================================

For a string with fixed ends there is only a limited set of frequencies at which standing waves will
occur on it. Each possible frequency is a resonant frequency, and the corresponding wave pattern is
an oscillation mode. The oscillation mode corresponding to :math:`N = 1` is called the fundamental mode or
the first harmonic, the mode corresponding to :math:`N = 2` is the second harmonic, and so on.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/String_vibration#Frequency_of_the_wave>`__.

.. py:currentmodule:: symplyphysics.laws.waves.resonant_frequencies_of_stretched_string_with_fixed_ends

.. py:data:: resonant_frequency

    Resonant frequency of the :math:`m`-th harmonic. See :attr:`~symplyphysics.symbols.classical_mechanics.temporal_frequency`.

Symbol:
    :code:`f`

Latex:
    :math:`f`

Dimension:
    :code:`frequency`


.. py:data:: harmonic_number

    An integer called harmonic number. See :attr:`~symplyphysics.symbols.basic.positive_number`.

Symbol:
    :code:`N`

Latex:
    :math:`N`

Dimension:
    :code:`dimensionless`


.. py:data:: phase_velocity

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

Symbol:
    :code:`v`

Latex:
    :math:`v`

Dimension:
    :code:`velocity`


.. py:data:: string_length

    :attr:`~symplyphysics.symbols.classical_mechanics.length` of the string.

Symbol:
    :code:`l`

Latex:
    :math:`l`

Dimension:
    :code:`length`


.. py:data:: law

    :code:`f = N * v / (2 * l)`


    Latex:
        .. math::
            f = \frac{N v}{2 l}