.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/compressible/plot_02_mach_cone.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_aerodynamics_compressible_plot_02_mach_cone.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_aerodynamics_compressible_plot_02_mach_cone.py:


Ernst Mach's shock photographs and the Mach cone
================================================

In 1887 Ernst Mach and Peter Salcher photographed a supersonic bullet and saw
a conical wave trailing from its nose. A source moving faster than sound
outruns its own wavefronts; their envelope is a cone of half-angle
:math:`\mu = \arcsin(1/M)` (:func:`~engineeringkit.aerodynamics.mach_angle`).
This example draws the wavefronts emitted along the path and checks that the
envelope matches the Mach angle.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import mach_angle








.. GENERATED FROM PYTHON SOURCE LINES 20-25

Wavefronts from a moving source
-------------------------------

A pulse emitted at time :math:`t_i` is a circle of radius
:math:`a(t - t_i)` centred where the source was.

.. GENERATED FROM PYTHON SOURCE LINES 25-47

.. code-block:: Python


    a = 1.0
    fig, axes = plt.subplots(1, 3, figsize=(11, 3.4))
    for ax, M in zip(axes, [0.6, 1.0, 2.0], strict=True):
        t_now = 4.0
        for ti in np.arange(0, t_now, 0.4):
            circle = plt.Circle((M * a * ti, 0), a * (t_now - ti), fill=False, color="C0", lw=0.8)
            ax.add_patch(circle)
        x_src = M * a * t_now
        ax.plot(x_src, 0, "ko")
        if M >= 1:
            mu = mach_angle(M)
            length = 5
            ax.plot([x_src, x_src - length * np.cos(mu)], [0, length * np.sin(mu)], "C3")
            ax.plot([x_src, x_src - length * np.cos(mu)], [0, -length * np.sin(mu)], "C3")
            print(f"M = {M}: Mach angle {np.degrees(mu):.1f} deg")
        ax.set(xlim=(-4.5, 9), ylim=(-4.5, 4.5), title=f"M = {M}")
        ax.set_aspect("equal")
        ax.set_xticks([])
        ax.set_yticks([])
    fig.tight_layout()




.. image-sg:: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_02_mach_cone_001.png
   :alt: M = 0.6, M = 1.0, M = 2.0
   :srcset: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_02_mach_cone_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    M = 1.0: Mach angle 90.0 deg
    M = 2.0: Mach angle 30.0 deg




.. GENERATED FROM PYTHON SOURCE LINES 48-53

The envelope is the Mach cone
-----------------------------

The tangent from the source to the oldest wavefront makes angle
:math:`\arcsin(r/d) = \arcsin(a t/(M a t)) = \arcsin(1/M)`.

.. GENERATED FROM PYTHON SOURCE LINES 53-59

.. code-block:: Python


    for M in [1.2, 2.0, 5.0]:
        d, r = M * a * 4.0, a * 4.0
        assert np.isclose(np.arcsin(r / d), mach_angle(M))

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.103 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_compressible_plot_02_mach_cone.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/aerodynamics/compressible/plot_02_mach_cone.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_mach_cone.ipynb <plot_02_mach_cone.ipynb>`

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

      :download:`Download Python source code: plot_02_mach_cone.py <plot_02_mach_cone.py>`

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

      :download:`Download zipped: plot_02_mach_cone.zip <plot_02_mach_cone.zip>`


.. only:: html

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

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