Notebook

Initializing the program

To initialize the visualization, you may need to click "Run all initialization cells" above (see button location in figure).

You should see the visualization directly below this line, before the next section.

In [1]:

#This code was authored by Alex Shvonski, Copyright 2020 MIT All Rights Reserved.

%matplotlib notebook
import ipywidgets as widgets
from IPython.display import display, clear_output
from ipywidgets import interact, interactive, interactive_output, fixed, IntSlider, HBox, Layout, Output, VBox
import numpy as np
import matplotlib.pyplot as plt
from math import pi

In [2]:

#Define functions
###############################################
###############################################
def amplitude_x(x,t0):
    result = np.sin(x - t0)
    return result

def amplitude_t(x0,t):
    result = np.sin(x0 - t)
    return result

In [3]:

#Define plot
###############################################
###############################################
fig, ax = plt.subplots(1, 2, figsize=(9.5, 5))
plt.subplots_adjust(left=0.05, bottom=None, right=1, top=None, wspace=None, hspace=1.)
x = np.linspace(0., 3.*pi, 1000)
t = np.linspace(0., 3.*pi, 1000)
x0=0.
t0=0.

#############################
#plot1
y_max = 2.
line_1, = ax[0].plot(x, amplitude_x(x,t0),
        'b-', lw=2)

line_2, = ax[0].plot([x0,x0], [-y_max,y_max],
        'r--', lw=1, label='$x_0$')

line_3, = ax[0].plot(x0, amplitude_x(x0,t0),
        'ro', lw=1, label='$A(x_0,t_0)$')

#plot params
ax[0].set_title('Amplitude: $A(x,t=t_0)$', fontsize=16)
ax[0].set_xlabel('$x$', fontsize=16)
ax[0].set_xlim(min(x),max(x))
ax[0].set_ylim(-y_max,y_max)
ax[0].grid(True)
ax[0].legend(loc=1, fontsize=16)
plt.setp(ax[0].get_xticklabels(), fontsize=14)
plt.setp(ax[0].get_yticklabels(), fontsize=14)

#############################
#plot2
line_4, = ax[1].plot(t, amplitude_t(x0,t),
        'r-', lw=2)

line_5, = ax[1].plot([t0,t0], [-y_max,y_max],
        'b--', lw=1, label='$t_0$')

line_6, = ax[1].plot(t0, amplitude_t(x0,t0),
        'bo', lw=1, label='$A(x_0,t_0)$')

#plot params
ax[1].set_title('Amplitude: $A(x=x_0,t)$', fontsize=16)
ax[1].set_xlabel('$t$', fontsize=16)
ax[1].set_xlim(min(x),max(x))
ax[1].set_ylim(-y_max,y_max)
ax[1].grid(True)
ax[1].legend(loc=1, fontsize=16)
plt.setp(ax[1].get_xticklabels(), fontsize=14)
plt.setp(ax[1].get_yticklabels(), fontsize=14)


#Define plot updater
###############################################
###############################################
def update(x0=2.*pi, t0=0.):
    #plot1
    #############################
    line_1.set_ydata(amplitude_x(x,t0))
    line_2.set_xdata([x0,x0])
    line_3.set_xdata(x0)
    line_3.set_ydata(amplitude_x(x0,t0))
    
    #plot2
    #############################
    line_4.set_ydata(amplitude_t(x0,t))
    line_5.set_xdata([t0,t0])
    line_6.set_xdata(t0)
    line_6.set_ydata(amplitude_t(x0,t0))
    fig.canvas.draw_idle()
    return


#Define control elements
###############################################
###############################################
s1=widgets.FloatSlider(
    min=0.,
    max=3.*pi,
    step=0.2,
    value=0.,
    layout=Layout(width='500px'),
    description='$x_0$',
    style = {'description_width': 'initial'})

s2=widgets.FloatSlider(
    min=0.,
    max=3.*pi,
    step=0.2,
    value=0.,
    layout=Layout(width='500px'),
    description='$t_0$',
    style = {'description_width': 'initial'})


#Connect controls to plot
###############################################
###############################################
out = interactive_output(update, {'x0': s1, 't0': s2})


#Set layout
###############################################
###############################################
box_layout = Layout(display='flex', flex_flow='column', justify_content='space-between', align_items='center')


#Display output
###############################################
###############################################
display(VBox([s1, s2], layout=box_layout))

VBox(children=(FloatSlider(value=0.0, description='$x_0$', layout=Layout(width='500px'), max=9.42477796076938,…

About the Visualization

These plots show traveling waves of the form $A(x,t)=A\sin(kx - \omega t)$. The left plot shows the solution at a fixed time $t=t_{0}$ (as a function of position $x$) and the right plot shows the solution at a fixed position $x=x_{0}$ (as a function of time $t$).

Plot 1 (left):

(blue line) $A(x,t)=A\sin(kx - \omega t)$ is plotted as a funciton of $x$, at a fixed time $t_{0}$
(red dashed line) the position $x_{0}$ is shown
(red dot) the amplitude $A(x_{0},t_{0})$ is shown
the graph changes when $t_{0}$ is varied

Plot 2 (right):

(red line) $A(x,t)=A\sin(kx - \omega t)$ is plotted as a function of $t$, at a fixed position $x_{0}$
(blue dashed line) the position $t_{0}$ is shown
(blue dot) the amplitude $A(x_{0},t_{0})$ is shown
the graph changes when $x_{0}$ is varied

Sliders:

$x_{0}$: vary the position at which the amplitude is "observed"; this will change the vertical dashed line in Plot 1 and the shape of the wave in Plot 2.
$t_{0}$: vary the time at which the amplitude is "observed"; this will change the vertical dashed line in Plot 2 and the shape of the wave in Plot 1.

Exploration

Consider the following questions and possible actions:

How does the amplitude change as a funciton of time, for fixed position?
How does the amplitude change as a funciton of position, for fixed time?

Viewing the Code

You are encouraged to click the button below to view the source code. You can alter the code and rerun it within this notebook, or download the notebook itself and run the code locally on your own machine.

In [4]:

#The code in this block includes content from StackOverFlow User: harshil (CC BY-SA 4.0)
#and can be found here: https://stackoverflow.com/questions/27934885/how-to-hide-code-from-cells-in-ipython-notebook-visualized-with-nbviewer
###############################################
###############################################
#Enable hidden code
from IPython.display import HTML

HTML('''<script>
code_show=true; 
function code_toggle() {
 if (code_show){
 $('div.input').hide();
 } else {
 $('div.input').show();
 }
 code_show = !code_show
} 
$( document ).ready(code_toggle);
</script>
<form action="javascript:code_toggle()"><input type="submit" value="Click here to toggle on/off the raw code."></form>''')

Out[4]: