.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/oceanography/plot_03_gerstner_trochoidal_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_oceanography_plot_03_gerstner_trochoidal_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_oceanography_plot_03_gerstner_trochoidal_wave.py:


Gerstner's trochoidal wave (1802)
=================================

Franz Josef von Gerstner found the first exact nonlinear water wave. Each
particle moves on a circle whose radius shrinks as ``exp(-k b)`` with its
depth ``b``, and the free surface is a trochoid with sharp crests and flat
troughs. Because the crests are narrow, the mean surface lies ``k a**2 / 2``
below the level of the orbit centres. The wave travels at the deep-water speed
``sqrt(g / k)``.

.. GENERATED FROM PYTHON SOURCE LINES 12-45



.. image-sg:: /api/gallery/oceanography/images/sphx_glr_plot_03_gerstner_trochoidal_wave_001.png
   :alt: Trochoidal surface and particle orbits
   :srcset: /api/gallery/oceanography/images/sphx_glr_plot_03_gerstner_trochoidal_wave_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.constants import GRAVITY
    from earthsciencekit.oceanography import deep_wave_speed, gerstner_orbit_radius

    wavelength, amplitude = 100.0, 8.0
    k = 2 * np.pi / wavelength
    assert np.isclose(np.sqrt(GRAVITY / k), deep_wave_speed(wavelength, GRAVITY))


    def surface(rest_x, depth, phase=0.0):
        radius = gerstner_orbit_radius(depth, amplitude, k)
        return rest_x - radius * np.sin(k * rest_x - phase), -depth + radius * np.cos(k * rest_x - phase)


    rest = np.linspace(0, wavelength, 4001)
    x, z = surface(rest, 0.0)
    assert np.isclose(np.trapezoid(z, x) / wavelength, -k * amplitude**2 / 2, rtol=1e-6)

    fig, ax = plt.subplots(figsize=(8, 4.5))
    rest = np.linspace(-wavelength, 2 * wavelength, 6001)
    for depth in [0, 10, 20, 35]:
        ax.plot(*surface(rest, depth), color="tab:blue", lw=1.5 if depth == 0 else 0.6)
    theta = np.linspace(0, 2 * np.pi, 100)
    for centre in np.arange(0, 2 * wavelength, wavelength / 4):
        for depth in [0, 10, 20, 35]:
            radius = gerstner_orbit_radius(depth, amplitude, k)
            ax.plot(centre + radius * np.cos(theta), -depth + radius * np.sin(theta), color="0.6", lw=0.5)
    ax.set(xlim=(0, 2 * wavelength), xlabel="distance [m]", ylabel="height [m]", aspect="equal", title="Trochoidal surface and particle orbits")

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.086 seconds)


.. _sphx_glr_download_api_gallery_oceanography_plot_03_gerstner_trochoidal_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/oceanography/plot_03_gerstner_trochoidal_wave.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_gerstner_trochoidal_wave.ipynb <plot_03_gerstner_trochoidal_wave.ipynb>`

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

      :download:`Download Python source code: plot_03_gerstner_trochoidal_wave.py <plot_03_gerstner_trochoidal_wave.py>`

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

      :download:`Download zipped: plot_03_gerstner_trochoidal_wave.zip <plot_03_gerstner_trochoidal_wave.zip>`


.. only:: html

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

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