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<!DOCTYPE html>
<html>
<head>
	<title>Parameteric Amplification Applet</title>
	<meta charset="utf-8" />
	<meta name="viewport" content="width=device-width, initial-scale=1" />
	<link rel="stylesheet" href="assets/css/main.css" />
    <script type="text/javascript">
    window.languagePluginUrl = 'https://cdn.jsdelivr.net/pyodide/v0.15.0/full/';
    </script>
    <script src="https://cdn.jsdelivr.net/pyodide/v0.15.0/full/pyodide.js"></script>
</head>

<body class="subpage">

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				<a href="applets.html" class="logo">Back to Applets</a>
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				<div class="inner" id="container">
				<p></p>
				<header>
				<h2>Parametric Amplification</h2>
				</header>
				<p id='text'>
				Parametric amplification is a simple concept that appears both in intuitive, everyday situations,
				as well as advanced technologies like lasers and quantum computers. From what I can tell, the
				concept is rarely introduced to undergraduates who are nevertheless well-equipped to understand it.
				Mathematically, parametric amplification falls under the category of nonlinear dynamics.
				This field focuses on equations that are famously hard to solve, but parametric amplification
				can readily be understood in terms of a quite straightforward example.
				If you've ever swung on a playground swing before (and no one was pushing you!),
				you were using your legs to parametrically amplify your motion. Let's understand
				what's going on before we define what "parametric" means.

				First, you and the swing comprise a pendulum, which has equation of motion
				\[
				\ddot{\theta} = \omega_0 \sin \theta
				\]
				where \(\theta\) is the angle you are from hanging vertical and \(\omega_0\) is the typical \(\sqrt{g/l}\), where \(g\) is the gravitational constant
				and \(l\) is the length of the swing. Let's consider just small oscillations where \(\sin x \approx x\) and let's also add a damping term with damping constant
				\(\beta\).
				\[
				\ddot{\theta} + 2 \beta \dot{\theta} + \omega_0 \theta = 0
				\]

In classical mechanics courses, students typically study the this system with a driving term added to the right hand side, often a sinusoidal force.

\[
\ddot{\theta} + 2 \beta \dot{\theta} + \omega_0 \theta = f_0 \sin(\omega t)
\]

But there's no net force or torque on the pendulum - the pendulum can't push against itself! So we are actually not driving the syste

So what do you do on a swing when you want to start moving? You start swinging your legs back and forth, extending and tucking them in over and over.
				</p>

				<script> <!-- this JS script runs on startup -->
    const output = document.getElementById("output");
    const text = document.getElementById("text");


	//the follwing is python code that contains almost all the python we want to do
	code = `
from js import document # for interacting with JS
import io, base64 # for saving the figure

from numpy import zeros, complex128, pi, array, sin, cos, linspace, exp, real
import matplotlib.pyplot as plt

def paramp(isOn, omega_s, omega_p, omega_0):

    N = 2

    t = linspace(0,100,1001)

    x = zeros([len(t),N],dtype=complex128)

    dt = t[1] - t[0]

    x0 = array([1,1])*0.0

    x[0,:] = x0

    def f(x,t, isOn, omega_s, omega_p, omega_0):

        x1 = x[0]
        x2 = x[1]

        beta = 1/2

        RHS = 1*exp(1j*omega_s*t)*beta**2

        dx1dt = x2
        dx2dt = -2*beta*x2 - ((omega_0)*(1 - 0.1*isOn*sin(omega_p*t + 0)))**2 * sin(x1) + RHS

        return array([dx1dt,dx2dt])

    for i in range(0, len(t)-1):#

        k1 = f(x[i,:], t[i], isOn, omega_s, omega_p, omega_0)
        k2 = f(x[i,:] + k1*dt/2, t[i] + dt/2, isOn, omega_s, omega_p, omega_0)
        k3 = f(x[i,:] + k2*dt/2, t[i] + dt/2, isOn, omega_s, omega_p, omega_0)
        k4 = f(x[i,:] + k3*dt, t[i] + dt, isOn, omega_s, omega_p, omega_0)

        x[i+1,:] = x[i,:] + (dt/6)*(k1 + 2*k2 + 2*k3 + k4)

    return t, x[:,0]


def plot_paramp(omega_s, omega_p, omega_0):

    t, x1 = paramp(True, omega_s, omega_p, omega_0)

    fig = plt.figure(1)

    ax1 = fig.add_subplot(1, 1, 1)
    ax1.grid()
    ax1.plot(t,real(x1),label='paramp')
    ax1.legend()

div_container = document.createElement('div') # now we call some HTML from python to add it to the page
div_container.innerHTML = '''
<br>
<br>
<h> Generator Ready!
</h>
<p>
Let's simulate the driven harmonic oscillator.
</p>
<br>
Signal Frequency \(\omega_s\):
<input id='omega_s' value='1' type="number">
<br><br>
Parameteric Pump Frequency \(\omega_p\):
<input id='omega_0' value='1' type="number">
<br><br>
Resonance Frequency \(\omega_0\):
<input id='omega_p' value='1' type="number">
<br><br>
<button onclick='pyodide.globals.generate_plot_img()'>Plot</button>
<br>
<img id="fig" />'''

div_container2 = document.createElement('div') # now we call some HTML from python to add it to the page
div_container2.innerHTML = '''
<br>
<br>
<p>
Here's another simulation.
</p>
<br>
Parameter 1
<input id='A' value='1' type="number">
<br><br>
Parameter 2
<input id='B' value='1' type="number">
<br><br>
Parameter 3
<input id='c' value='1' type="number">
'''


document.getElementById("container").appendChild(div_container)
document.getElementById("container").appendChild(div_container2)

def generate_plot_img():
	# get values from inputs
	print('1')
	omega_s = int(document.getElementById('omega_s').value)
	omega_0 = int(document.getElementById('omega_0').value)  # generate an interval
	omega_p = int(document.getElementById('omega_p').value)  # generate an interval
	print('2')
	plot_paramp(omega_s, omega_p, omega_0)
	print('3')
	buf = io.BytesIO()
	# copy the plot into the buffer
	plt.savefig(buf, format='png')
	buf.seek(0)
	print('4')
	# encode the image as Base64 string
	img_str = 'data:image/png;base64,' + base64.b64encode(buf.read()).decode('UTF-8')  # show the image
	img_tag = document.getElementById('fig')
	img_tag.src = img_str
	buf.close()
	print('5')
`
//end of python code

	function evaluatePython() {
		console.log(pyodide.runPythonAsync(code));
    }

    // init pyodide
    languagePluginLoader.then(() => { evaluatePython()});


	</script>
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