.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/seismology/plot_01_poisson_two_elastic_waves.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_seismology_plot_01_poisson_two_elastic_waves.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_seismology_plot_01_poisson_two_elastic_waves.py:


Poisson's two elastic waves (1829)
==================================

Siméon Denis Poisson showed that an elastic solid carries two kinds of wave:
a faster compressional wave and a slower shear wave. When the two Lamé
constants are equal (a Poisson solid) the speed ratio is exactly sqrt(3) and
Poisson's ratio is 1/4. With no rigidity the shear wave vanishes, as in a fluid.

.. GENERATED FROM PYTHON SOURCE LINES 10-36



.. image-sg:: /api/gallery/seismology/images/sphx_glr_plot_01_poisson_two_elastic_waves_001.png
   :alt: Bulk modulus 130 GPa: no rigidity, no S wave
   :srcset: /api/gallery/seismology/images/sphx_glr_plot_01_poisson_two_elastic_waves_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.seismology import p_wave_speed, poisson_ratio, s_wave_speed

    density, bulk = 3300.0, 1.3e11
    shear = np.linspace(0, 1.0e11, 201)
    vp = p_wave_speed(bulk, shear, density)
    vs = s_wave_speed(shear, density)

    # Poisson solid: lambda = mu, so the bulk modulus is 5 mu / 3.
    mu = 6e10
    vp_poisson, vs_poisson = p_wave_speed(5 * mu / 3, mu, density), s_wave_speed(mu, density)
    assert np.isclose(vp_poisson / vs_poisson, np.sqrt(3))
    assert np.isclose(poisson_ratio(vp_poisson, vs_poisson), 0.25)
    assert vs[0] == 0 and vp[0] > 0

    fig, ax = plt.subplots()
    ax.plot(shear / 1e9, vp / 1e3, label="P (compressional)")
    ax.plot(shear / 1e9, vs / 1e3, label="S (shear)")
    ax.axvline(mu / 1e9, color="k", ls=":", lw=1)
    ax.set(xlabel="shear modulus [GPa]", ylabel="wave speed [km/s]", title=f"Bulk modulus {bulk / 1e9:.0f} GPa: no rigidity, no S wave")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.075 seconds)


.. _sphx_glr_download_api_gallery_seismology_plot_01_poisson_two_elastic_waves.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/seismology/plot_01_poisson_two_elastic_waves.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_poisson_two_elastic_waves.ipynb <plot_01_poisson_two_elastic_waves.ipynb>`

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

      :download:`Download Python source code: plot_01_poisson_two_elastic_waves.py <plot_01_poisson_two_elastic_waves.py>`

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

      :download:`Download zipped: plot_01_poisson_two_elastic_waves.zip <plot_01_poisson_two_elastic_waves.zip>`


.. only:: html

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

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