.. include:: /project-links.txt
.. include:: /abbreviation.txt

.. getthecode:: beat.py
    :language: python

===============
 Acoustic Beat
===============
In acoustics, a beat is an interference pattern between two sounds of slightly different
frequencies, perceived as a periodic variation in volume whose rate is the difference of the two
frequencies.

Theory
------

The sum of two sinusoidal signals is:

.. math::

    \cos(2\pi f_1 t) + \cos(2\pi f_2 t) = 2 \cos\left(2\pi \frac{f_1 + f_2}{2} t\right) \cos\left(2\pi \frac{f_1 - f_2}{2} t\right)

If the frequencies are close enough, we can rewrite this expression using :math:`f2 = f1 + f_\delta`

.. math::

    \cos(2\pi f_1 t) + \cos(2\pi f_2 t) = 2 \cos\left(2\pi \frac{2 f_1 + f_\delta}{2} t\right) \cos\left(2\pi \frac{f_\delta}{2} t\right)

where :math:`f_\text{beat} = f2 - f1 = f_\delta`

Simulation
----------

.. code-block:: python

    import numpy as np
    import matplotlib.pyplot as plt
    from matplotlib.ticker import MultipleLocator
    
    from Musica.Math.MusicTheory import *
    
    delta_frequency = Frequency(5)
    frequency1 = Frequency(100)
    frequency2 = Frequency(float(frequency1) + float(delta_frequency))
    beat_frequency = Frequency(float(delta_frequency) / 2)
    
    t = np.arange(0, beat_frequency.period, frequency1.period / 100)
    
    y1 = np.cos(t * frequency1.pulsation)
    y2 = np.cos(t * frequency2.pulsation)
    yb = 2 * np.cos(t * beat_frequency.pulsation)
    
    figure = plt.figure(1, (20, 10))
    
    axe = plt.subplot(111)
    axe.set_title('Beat')
    axe.set_xlabel('Time')
    axe.set_ylabel('Amplitude')
    axe.grid()
    axe.plot(t, y1)
    axe.plot(t, y2)
    axe.plot(t, y1 + y2)
    axe.plot(t, yb)
    
    plt.show()
    


.. image:: beat.png
  :align: center