.. _sphx_glr_api_gallery_pde_wave:

The wave equation
-----------------

A plucked string: d'Alembert's traveling-wave solution against a
symplectic leapfrog finite-difference solve.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Jean le Rond d&#x27;Alembert (1747) showed that every solution of u_{tt} = c^2 u_{xx} is a sum of two waves moving in opposite directions, u = F(x - ct) + G(x + ct). A plucked string&#x27;s triangular bump therefore splits into two half-height copies that run apart, reflect upside down at the fixed ends, and recombine. This script compares d&#x27;Alembert&#x27;s formula with a leapfrog finite-difference string (mathematicskit.integrators.leapfrog_integrate) run at Courant number 1, where the scheme reproduces d&#x27;Alembert&#x27;s solution exactly at the grid points.">

.. only:: html

  .. image:: /api/gallery/pde/wave/images/thumb/sphx_glr_plot_01_dalembert_wave_thumb.png
    :alt:

  :doc:`/api/gallery/pde/wave/plot_01_dalembert_wave`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">D'Alembert's traveling waves on a plucked string</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/pde/wave/plot_01_dalembert_wave