.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/wave/plot_01_dalembert_wave.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_pde_wave_plot_01_dalembert_wave.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_pde_wave_plot_01_dalembert_wave.py:


D'Alembert's traveling waves on a plucked string
================================================

Jean le Rond d'Alembert (1747) showed that every solution of
:math:`u_{tt} = c^2 u_{xx}` is a sum of two waves moving in opposite
directions, :math:`u = F(x - ct) + G(x + ct)`. A plucked string'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'Alembert's formula with a leapfrog finite-difference string
(:func:`mathematicskit.integrators.leapfrog_integrate`) run at Courant
number 1, where the scheme reproduces d'Alembert's solution exactly at
the grid points.

.. GENERATED FROM PYTHON SOURCE LINES 17-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import WaveEquation1D, dalembert_solution
    from mathematicskit.pde.visualizers import plot_spacetime


    def pluck(x):
        return np.maximum(0.0, 1.0 - np.abs(x - 0.3) / 0.1)


    string = WaveEquation1D(pluck, n=201, c=1.0)
    sol = string.solve(2.0, dt=string.dx, method="leapfrog", save_every=4)








.. GENERATED FROM PYTHON SOURCE LINES 32-34

The bump splits, reflects, and recombines
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-50

.. code-block:: Python


    fig1, axes = plt.subplots(4, 1, figsize=(7, 7), sharex=True)
    for ax, t in zip(axes, (0.0, 0.15, 0.5, 1.0), strict=False):
        k = int(np.argmin(np.abs(sol.t - t)))
        ax.plot(sol.x, sol.u[k], color="tab:blue", lw=2, label="leapfrog")
        ax.plot(sol.x, dalembert_solution(pluck, sol.x, sol.t[k], length=1.0), "k--", label="d'Alembert")
        ax.set_ylim(-1.1, 1.1)
        ax.set_ylabel(f"t = {sol.t[k]:.2f}")
    axes[0].legend(fontsize=8, loc="upper right")
    axes[-1].set_xlabel("$x$")
    fig1.suptitle("Two half-height waves: d'Alembert's solution")
    fig1.tight_layout()

    errors = [np.max(np.abs(u - dalembert_solution(pluck, sol.x, t, length=1.0))) for t, u in zip(sol.t, sol.u, strict=False)]
    print(f"max |leapfrog - d'Alembert| over the whole run: {max(errors):.2e}")




.. image-sg:: /api/gallery/pde/wave/images/sphx_glr_plot_01_dalembert_wave_001.png
   :alt: Two half-height waves: d'Alembert's solution
   :srcset: /api/gallery/pde/wave/images/sphx_glr_plot_01_dalembert_wave_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    max |leapfrog - d'Alembert| over the whole run: 7.44e-15




.. GENERATED FROM PYTHON SOURCE LINES 51-54

The characteristics in space-time
---------------------------------
The waves travel along the lines :math:`x \pm ct = \text{const}`.

.. GENERATED FROM PYTHON SOURCE LINES 54-61

.. code-block:: Python


    fig2, ax2 = plt.subplots(figsize=(6, 5))
    plot_spacetime(sol, ax=ax2, cmap="RdBu_r")
    ax2.set_title("u(x, t): straight characteristics at speed c")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/pde/wave/images/sphx_glr_plot_01_dalembert_wave_002.png
   :alt: u(x, t): straight characteristics at speed c
   :srcset: /api/gallery/pde/wave/images/sphx_glr_plot_01_dalembert_wave_002.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.415 seconds)


.. _sphx_glr_download_api_gallery_pde_wave_plot_01_dalembert_wave.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/pde/wave/plot_01_dalembert_wave.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_dalembert_wave.ipynb <plot_01_dalembert_wave.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_dalembert_wave.py <plot_01_dalembert_wave.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_dalembert_wave.zip <plot_01_dalembert_wave.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_