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<h2 class="hd hd-2 unit-title">Introduction to 2D Waves</h2>
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Previously, we derived properties of waves propagating along one direction. To achieve this, we began by analyzing the behavior of coupled masses attached by strings, and determined the normal modes of oscillation for an infinite system. Similarly, this will be our starting point for analyzing 2D waves. </p><p>
For reference, let's consider where in the course we examined 1D waves before. </p><p>
In Lesson 11, we analyzed systems of infinite oscillators and concluded by deriving the eigenvectors for the translation symmetry matrix (L11v5), which were of the form [mathjaxinline]e^{ikx}[/mathjaxinline]. Adding time dependence to this solution yields solutions of the form [mathjaxinline]e^{i(kx-\omega t)}[/mathjaxinline]. </p><p>
In Lesson 12, starting with L12v2, we introduced the infinite mass-string system, which has the same eigenvectors of the translation symmetry matrix. Thus, we could apply our previous result. </p><p>
The infinite mass-string system has the following properties where [mathjaxinline]a[/mathjaxinline] is the distance between masses, [mathjaxinline]T[/mathjaxinline] is the tension of the string, and [mathjaxinline]m[/mathjaxinline] is the mass of an individual oscillator. </p><p>
Using the definition [mathjaxinline]\omega _{0}^{2}=\dfrac {T}{ma}[/mathjaxinline], in L12v4 we showed that the dispersion relation of this system is: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\omega (k)^{2}=4\omega _{0}^{2}\sin ^{2}\left( \dfrac {ka}{2}\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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<h2 class="hd hd-2 unit-title">L27v1: 2D Waves and Normal Modes of Square Plate</h2>
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<h2 class="hd hd-2 unit-title">L27Q1: Boundary Conditions for Chladni Plates</h2>
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Boundary Conditions for Chladni Plates
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In this problem, we will think about the 2D wave solutions and boundary conditions of square plates. We know that the general form of a 2D wave will involve the following combination of eigenvectors: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \psi (x,y) = A e^{i k_ x x}e^{i k_ y y}[/mathjaxinline]
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Taking the real part of the most general solution, we may either have [mathjaxinline]\psi (x,y) = A \sin (k_ x x)\sin (k_ y y)[/mathjaxinline] or [mathjaxinline]\psi (x,y) = A \cos (k_ x x)\cos (k_ y y)[/mathjaxinline] or another combination of sine and cosine, depending on the boundary conditions, which could be some combination of fixed and/or open. </p>
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Looking at some of the normal modes of a square Chladni plate driven at its center (shown below), what can you determine about the boundary conditions? </p>
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<text> c) The right and left sides are fixed while the top and bottom are open.</text>
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<text> d) The right and left sides are open while the top and bottom are fixed.</text>
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<text> e) There isn't enough information to know the boundary conditions.</text>
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Two Dimensional Mass+String Equation of Motion
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In one dimension, we saw that only the two nearest neighbors had non-zero contributions to the equation of motion for a single particle. In particular we had for beads on a string that: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle m \ddot{y_ j} = \frac{T}{a}(y_{j-1}-2y_ j+y_{j+1})[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]y[/mathjaxinline] is a displacement perpendicular to the line of the string. </p>
<p><b class="bfseries">(Part a)</b> Let's think about how this extends to a 2D system, for example identical beads connected by strings in a square grid </p>
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In this case, the string tensions and bead spacing can be different in the two directions and motion perpendicular to the strings will be in the [mathjaxinline]z[/mathjaxinline] direction. As before, only the nearest neighbors along the strings contribute to the forces on a given bead but now there are 4 of them. </p>
<p>
Denote the displacement in the [mathjaxinline]z[/mathjaxinline] direction of the central bead and its nearest neighbors in the following way:<br/>[mathjaxinline]zxy\equiv z(x_ i,y_ j)[/mathjaxinline]<br/>[mathjaxinline]zxpy\equiv z(x_{i+1},y_ j)[/mathjaxinline]<br/>[mathjaxinline]zxmy\equiv z(x_{i-1},y_ j)[/mathjaxinline]<br/>and similarly for [mathjaxinline]z(x_ i,y_{j+1})[/mathjaxinline] and [mathjaxinline]z(x_ i,y_{j-1})[/mathjaxinline]. Note that we do not need to consider the beads which are displaced from the central one in both [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
<p>
What is the equation of motion for a bead at position [mathjaxinline](x_ i,y_ j)[/mathjaxinline]? Write your answer in terms of <code>T_H</code> for [mathjaxinline]T_ H[/mathjaxinline],<code>T_V</code> for [mathjaxinline]T_ V[/mathjaxinline], <code>a_H</code> for [mathjaxinline]a_ H[/mathjaxinline],<code>a_V</code> for [mathjaxinline]a_ V[/mathjaxinline], <code>zxy</code>, <code>zxpy</code>, <code>zxmy</code>, <code>zxyp</code>, and <code>zxym</code>, as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]m \ddot{z}(x_ i,y_ j) =[/mathjaxinline] </p>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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Conditions for Two Dimensional Mass+String Motion
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<p><b class="bfseries">(Part b)</b> What assumptions must be made in order for this equation of motion to be valid? Select ALL that are true. </p>
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Two Dimensional Mass+String Eigenfrequencies
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<p><b class="bfseries">(Part c)</b> Using the eigenvectors [mathjaxinline]z(x_{b},y_{c})=B_{b}C_{c}e^{i(\omega t + \phi )}[/mathjaxinline] where [mathjaxinline]B_{b}=e^{ibk_{x}a_{H}}[/mathjaxinline] and [mathjaxinline]C_{c}=e^{ick_{y}a_{V}}[/mathjaxinline] are the amplitudes of the [mathjaxinline]b^{\mathrm{th}}[/mathjaxinline] and [mathjaxinline]c^{\mathrm{th}}[/mathjaxinline] bead in the [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] directions, respectively, solve for [mathjaxinline]\omega ^{2}[/mathjaxinline]. Express your answer in terms of <code>m</code>, <code>T_H</code> for [mathjaxinline]T_ H[/mathjaxinline],<code>T_V</code> for [mathjaxinline]T_ V[/mathjaxinline], <code>a_H</code> for [mathjaxinline]a_ H[/mathjaxinline],<code>a_V</code> for [mathjaxinline]a_ V[/mathjaxinline], <code>k_x</code> for [mathjaxinline]k_{x}[/mathjaxinline], and <code>k_y</code> for [mathjaxinline]k_{y}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\omega ^{2} =[/mathjaxinline] </p>
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<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L27Q3: 2D Dispersion Relation [WITH SIMULATION]</h2>
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Parameter Space of kx,ky Values
</h3>
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<p>
The dispersion relation of the 2D mesh system is: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\omega ^{2}(k_{x},k_{y}) = \frac{4T_ H}{ma_ H}\sin ^{2}\left(\frac{k_{x}a_{H}}{2}\right) + \frac{4T_ V}{ma_ V}\sin ^{2}\left(\frac{k_{y}a_{V}}{2}\right)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
The plots presented in the lesson video are "slices" through the surface defined by [mathjaxinline]\omega (k_{x},k_{y})[/mathjaxinline]&#8212;the intersection between the slice at fixed [mathjaxinline]\omega[/mathjaxinline] gives the dispersion plot in k-space. Two examples are shown below: </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
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<img src="/assets/courseware/v1/5b4e7a1355912c6464f988610e8bf1b2/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_17_04_1.png" width="770"/>
</div>
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<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
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<img src="/assets/courseware/v1/7c45cb25bf296fae686248679977ead2/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_17_04_2.png" width="770"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
The shape of the dispersion plots in k-space are, of course, governed by the parameters of the system, [mathjaxinline]T_{H}[/mathjaxinline], [mathjaxinline]T_{V}[/mathjaxinline], [mathjaxinline]a_{H}[/mathjaxinline], [mathjaxinline]a_{V}[/mathjaxinline], and [mathjaxinline]m[/mathjaxinline]. Consider how the following plots depend on these parameters. </p>
<p><b class="bfseries">(Part a)</b> Which relation between parameters will generate the plot below? </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/033898ff570cca21cc616b17c69ab954/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_17_04_a.png" width="440"/>
</div>
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<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_17_04_2_1">
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<input type="radio" name="input_lect_17_04_2_1" id="input_lect_17_04_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_17_04_2_1-choice_1-label" for="input_lect_17_04_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_2_1">
<text> a) [mathjaxinline]T_{H} \lt T_{V}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_17_04_2_1" id="input_lect_17_04_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_17_04_2_1-choice_2-label" for="input_lect_17_04_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_2_1">
<text> b) [mathjaxinline]T_{H} \gt T_{V}[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="radio" name="input_lect_17_04_2_1" id="input_lect_17_04_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_17_04_2_1-choice_3-label" for="input_lect_17_04_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_2_1">
<text> c) [mathjaxinline]a_{H} \lt a_{V}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_17_04_2_1" id="input_lect_17_04_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_17_04_2_1-choice_4-label" for="input_lect_17_04_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_2_1">
<text> d) [mathjaxinline]a_{H} \gt a_{V}[/mathjaxinline]</text>
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</p>
<p><b class="bfseries">(Part b)</b> Which relation between parameters will generate the plot below? </p>
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<div align="center">
<img src="/assets/courseware/v1/398b5d79464bf732a073634cec37c8da/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_17_04_b.png" width="440"/>
</div>
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<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_17_04_3_1">
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<input type="radio" name="input_lect_17_04_3_1" id="input_lect_17_04_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_17_04_3_1-choice_1-label" for="input_lect_17_04_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_3_1">
<text> a) [mathjaxinline]T_{H} \lt T_{V}[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_lect_17_04_3_1" id="input_lect_17_04_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_17_04_3_1-choice_2-label" for="input_lect_17_04_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_3_1">
<text> b) [mathjaxinline]T_{H} \gt T_{V}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_17_04_3_1" id="input_lect_17_04_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_17_04_3_1-choice_3-label" for="input_lect_17_04_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_3_1">
<text> c) [mathjaxinline]a_{H} \lt a_{V}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_17_04_3_1" id="input_lect_17_04_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_17_04_3_1-choice_4-label" for="input_lect_17_04_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_17_04_3_1">
<text> d) [mathjaxinline]a_{H} \gt a_{V}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_17_04_3_1"/>
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</p>
<p><b class="bfseries">(Part c)</b> Which relation between parameters will generate the plot below? </p>
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<text> a) [mathjaxinline]T_{H} \lt T_{V}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]T_{H} \gt T_{V}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]a_{H} \lt a_{V}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]a_{H} \gt a_{V}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part d)</b> Which relation between parameters will generate the plot below? </p>
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<text> b) [mathjaxinline]T_{H} \gt T_{V}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]a_{H} \lt a_{V}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]a_{H} \gt a_{V}[/mathjaxinline]</text>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=2D_dispersion_no_boundary_2.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/dbc85affd39f653837f0f055b6930332/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_2D_dispersion_no_boundary_2.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L27v4: Boundary Conditions for a Finite 2D System</h2>
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<h2 class="hd hd-2 unit-title">L27Q4: Visualizing Normal Modes of 2D Array of Masses [WITH SIMULATION]</h2>
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Visualizing Normal Modes of 2D Array of Masses - I
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<p>
Consider a 3X3 array of masses connected by strings, with fixed boundary conditions. The following plots show several normal modes of the system. For each plot, identify the index of the normal mode that is shown&#8212;i.e., the [mathjaxinline](n_{x},n_{y})[/mathjaxinline] values? </p>
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<b class="bfseries">(Part a)</b>
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<p style="display:inline">[mathjaxinline]n_{x} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]n_{y} =[/mathjaxinline] </p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Visualizing Normal Modes of 2D Array of Masses - II
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<b class="bfseries">(Part b)</b>
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<p style="display:inline">[mathjaxinline]n_{x} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]n_{y} =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Visualizing Normal Modes of 2D Array of Masses - III
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<b class="bfseries">(Part c)</b>
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<p style="display:inline">[mathjaxinline]n_{x} =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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enter <code> 2+3*2 </code> for 8 </td>
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enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=2D_rank_freq_spatial_modes_with_boundary_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/c32c884587cdc88bf4a311186ef5aa8a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_2D_rank_freq_spatial_modes_with_boundary_1.png"/></div><p/>
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