.. _secpdewaveeqoned:

Simple wave equation in one dimension
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let us start with a simple wave equation

.. math:: :label: eqpdewaveeq

   \partial_t^2 u-c^2\nabla^2 u=0\,.

Upon multiplication with the test function :math:`w` and partial integration, one obtains the weak form

.. math:: :label: eqpdewaveeqweak

   \left(\partial_t^2 u,w\right)+\left(c^2\nabla u,\nabla w\right)-\left\langle c^2\nabla u\cdot\vec{n},w \right\rangle =0\,.

In pyoomph, we just can use :py:func:`~pyoomph.expressions.generic.partial_t` for the time derivative and do the spatial part as before in :numref:`secspatial`. For the bulk contributions, i.e. the :math:`(\cdot,\cdot)` terms, we write again a :py:class:`~pyoomph.generic.codegen.Equations` class:

.. literalinclude:: wave_eq.py
   :language: python
   :start-at: from pyoomph import *
   :end-at: self.add_residual(weak(partial_t(u,2),w)+weak(self.c**2*grad(u),grad(w)))

It is essentially the same approach as in :numref:`secode`. The problem class could read as follows

.. literalinclude:: wave_eq.py
   :language: python
   :start-at: class WaveProblem(Problem):
   :end-at: problem.run(20,outstep=True,startstep=0.1)

Note how the initial condition ``u_init`` depends on time ``t``, which is bound by ``var("time")`` again. Thereby, we ensure that we have a traveling wave solution. Besides the initial condition :math:`u(x,t{=}0)`, the additionally required first derivative :math:`\partial_t u(x,t{=}0)` is automatically evaluated. Indeed the result is, as expected, a traveling wave which is reflected at the boundaries, cf. :numref:`figpdewaveeq`. Without specifying the time dependency of the initial condition, :math:`\partial_t u(x,t{=}0)=0` would hold, yielding a different solution.

..  figure:: waveeq.*
	:name: figpdewaveeq
	:align: center
	:alt: Traveling wave solution
	:class: with-shadow
	:width: 60%

	Traveling wave solution, which is reflected at the boundaries.


Without the ``DirichletBC(u=0)`` terms, the :math:`\langle \cdot, \cdot \rangle` terms in :math:numref:`eqpdewaveeqweak` would become relevant. Since we do not add any contributions at the boundaries by some :py:class:`~pyoomph.generic.codegen.InterfaceEquations` or :py:class:`~pyoomph.meshes.bcs.NeumannBC`, the term :math:`\langle c^2\nabla u\cdot\vec{n},w \rangle` is zero. This can only hold for arbitrary :math:`w`, if :math:`\partial_x u=0`. Thereby, the wave will have free ends on both sides. The wave gets reflected, but without changing sign.


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	.. container:: downloadbutton

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