.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/compressible/plot_01_laplace_speed_of_sound.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_01_laplace_speed_of_sound.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_01_laplace_speed_of_sound.py:


Newton, Laplace and the speed of sound
======================================

Newton's *Principia* (1687) predicted the speed of sound assuming the
compressions are isothermal, :math:`a = \sqrt{p/\rho} = \sqrt{RT}`, about 15%
below measurement. Pierre-Simon Laplace (1816) argued that the compressions
are too fast to exchange heat, so they are adiabatic and the stiffness rises
by :math:`\gamma`: :math:`a = \sqrt{\gamma R T}`
(:func:`~engineeringkit.aerodynamics.speed_of_sound`). The Mach number
(:func:`~engineeringkit.aerodynamics.mach_number`) measures speed in units of
this local sound speed.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import R_AIR, isa_atmosphere, mach_number, speed_of_sound








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Isothermal versus adiabatic
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-33

.. code-block:: Python


    T = np.linspace(200, 320, 121)
    newton = np.sqrt(R_AIR * T)
    laplace = speed_of_sound(1.4, R_AIR, T)
    a_measured_20C = 343.2
    a_laplace = float(speed_of_sound(1.4, R_AIR, 293.15))
    print(f"at 293.15 K: Newton {np.sqrt(R_AIR * 293.15):.1f} m/s, Laplace {a_laplace:.1f} m/s, measured {a_measured_20C} m/s")
    assert np.allclose(laplace / newton, np.sqrt(1.4))





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

 .. code-block:: none

    at 293.15 K: Newton 290.1 m/s, Laplace 343.2 m/s, measured 343.2 m/s




.. GENERATED FROM PYTHON SOURCE LINES 34-36

The same airspeed is a higher Mach number at altitude
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-50

.. code-block:: Python


    h = np.linspace(0, 20000, 81)
    state = isa_atmosphere(h)
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.6))
    axes[0].plot(T, newton, label="Newton (isothermal)")
    axes[0].plot(T, laplace, label="Laplace (adiabatic)")
    axes[0].plot(293.15, a_measured_20C, "ko", label="measured, 20 °C")
    axes[0].set(xlabel="temperature [K]", ylabel="speed of sound [m/s]")
    axes[0].legend()
    axes[1].plot(mach_number(250.0, state.speed_of_sound), h / 1e3)
    axes[1].set(xlabel="Mach number at 250 m/s", ylabel="altitude [km]", title="ISA")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_01_laplace_speed_of_sound_001.png
   :alt: ISA
   :srcset: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_01_laplace_speed_of_sound_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.201 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_compressible_plot_01_laplace_speed_of_sound.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_01_laplace_speed_of_sound.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_laplace_speed_of_sound.ipynb <plot_01_laplace_speed_of_sound.ipynb>`

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

      :download:`Download Python source code: plot_01_laplace_speed_of_sound.py <plot_01_laplace_speed_of_sound.py>`

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

      :download:`Download zipped: plot_01_laplace_speed_of_sound.zip <plot_01_laplace_speed_of_sound.zip>`


.. only:: html

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

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