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%matplotlib inline
from numpy import exp, cos, linspace 
import pylab
from ipywidgets import interact

Title of investigation

Want to understand $f(t) = \exp(-\alpha t) \cos(\omega t)$



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def f(t, alpha, omega):
    """Computes and returns exp(-alpha*t) * cos(omega*t)"""
    return exp(-alpha * t) * cos(omega * t)

We can execute the function for valuel of $\alpha$ and $\omega$:



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f(t=0.1, alpha=1, omega=10)

Although sometimes a plot is more instructive:



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def plot_f(alpha, omega):
    ts = linspace(0, 5, 400)    # many points in the interval [0, 5]
    ys = f(ts, alpha, omega)
    pylab.plot(ts, ys, '-')



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plot_f(alpha=0.1, omega=12)

A wide range of convenience tools, for example graphical interaction elements called Widgets:



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interact(plot_f, alpha=(0, 2, 0.1), omega=(0, 20, 0.5))

Conclusion

So we observe: Parameter $\alpha$ is responsible for damping, and $\omega$ for the frequency.