Displacement in underdamping
============================

In the presence of a damping force in the oscillating system, the system's behavior
depends on the value of the damping ratio. When it is less than :math:`1`, the system oscillates
with a slightly different frequency than in the undamped case, and its amplitude decreasing
to zero. This behavior is also known as *underdamping*.

**Conditions:**

#. The system is underdamped, i.e. its damping ratio :math:`\zeta < 1`.

.. py:currentmodule:: symplyphysics.laws.kinematics.damped_oscillations.displacement_in_underdamping

.. py:data:: displacement

    Displacement from rest, usually a function of time. See :attr:`~symplyphysics.symbols.classical_mechanics.euclidean_distance`.

Symbol:
    :code:`d`

Latex:
    :math:`d`

Dimension:
    :code:`length`


.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time` at which :attr:`~displacement` is measured.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: scaling_coefficient

    Scaling coefficient to be found using initial conditions.

Symbol:
    :code:`a`

Latex:
    :math:`a`

Dimension:
    :code:`length`


.. py:data:: exponential_decay_constant

    :attr:`~symplyphysics.symbols.basic.exponential_decay_constant` of the oscillator.

Symbol:
    :code:`lambda`

Latex:
    :math:`\lambda`

Dimension:
    :code:`1/time`


.. py:data:: damped_angular_frequency

    :doc:`Damped angular frequency <laws.kinematics.damped_oscillations.damped_angular_frequency>`
    of the oscillator. See :attr:`~symplyphysics.symbols.classical_mechanics.angular_frequency`.

Symbol:
    :code:`w_d`

Latex:
    :math:`\omega_\text{d}`

Dimension:
    :code:`angle/time`


.. py:data:: phase_shift

    :attr:`~symplyphysics.symbols.classical_mechanics.phase_shift` of the oscillations, i.e. the phase at :math:`t = 0`.

Symbol:
    :code:`phi`

Latex:
    :math:`\varphi`

Dimension:
    :code:`angle`


.. py:data:: law

    :code:`d = a * exp(-lambda * t) * cos(w_d * t + phi)`


    Latex:
        .. math::
            d = a \exp{\left(- \lambda t \right)} \cos{\left(\omega_\text{d} t + \varphi \right)}