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<h2 class="hd hd-2 unit-title">Introduction to Wave Equation Solution</h2>
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<p>
We have seen how waves arise from single oscillators that are coupled together! </p><p>
In this lesson, we will work with the wave equation and discover its solution through a mathematical technique called "separation of variables." This will produce an equation that describes the amplitude of a wave at any position and time along the extent of the wave. </p><p>
We will subject this solution to boundary conditions to solve for the particular case of waves on a finite string—the string may be bound at both ends, open at both ends, or some combination. We address how to solve the normal modes of such systems. </p>
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<h2 class="hd hd-2 unit-title">L14v1: Review of the Wave Equation</h2>
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<h2 class="hd hd-2 unit-title">L14Q1: Continuous vs. Discrete Systems</h2>
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Continuous vs. Discrete Systems
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In the last lesson, we went from an infinite system of <i class="itshape">discrete</i> oscillators, to a <i class="itshape">continuous</i> system&#8212;both systems were infinitely long. We adopted a new approach to analyze the dynamics of the continuous system, and out came the wave equation! </p>
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Why couldn't we apply the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix to solve the equations of motion for the continuous infinite system? Select ALL that apply. </p>
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<text>a) We could have, but that approach is harder.</text>
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<text>b) We couldn't because, in the continuous limit, the positions of masses [mathjaxinline]j[/mathjaxinline] and [mathjaxinline]j+1[/mathjaxinline] are no longer defined</text>
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<text>c) We couldn't because, as the masses get closer together in the continuous limit, the matrix is less accurate</text>
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<h2 class="hd hd-2 unit-title">L14Q2: Separation of Variables</h2>
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Separation of Variables
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Previously, we started with an equation of the form: </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]\frac{\partial ^{2}}{\partial t^{2}}\psi _{m}(x,t) = v_{p}^{2}\frac{\partial ^{2}}{\partial x^{2}}\psi _{m}(x,t)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
and we assumed that the solution had the form [mathjaxinline]\psi _{m}(x,t)=A(x)B(t)[/mathjaxinline], which allowed us to "separate" the variables to get two equations: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\frac{1}{v_{p}^2 B(t)}\frac{\partial ^{2}}{\partial t^{2}}B(t) = -k_{m}^{2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
and </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\frac{1}{A(x)}\frac{\partial ^{2}}{\partial x^{2}}A(x) = -k_{m}^{2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
Now we will try to guess a solution to each equation. </p>
<p><b class="bfseries">(Part a)</b> Which of the following is a possible solution to the equation [mathjaxinline]\dfrac {1}{v_{p}^2 B(t)}\dfrac {\partial ^{2}}{\partial t^{2}}B(t) = -k_{m}^{2}[/mathjaxinline], for nonzero [mathjaxinline]k_{m}[/mathjaxinline]? Choose ALL that apply. </p>
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<text>a) [mathjaxinline]B(t)=B_{m}\left(\omega _{m}t + \beta _{m}\right)[/mathjaxinline]</text>
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<text>b) [mathjaxinline]B(t)=B_{m}/\left(\omega _{m}t + \beta _{m}\right)[/mathjaxinline]</text>
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<text>c) [mathjaxinline]B(t)=B_{m}\cos (\omega _{m}t + \beta _{m})[/mathjaxinline]</text>
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<text>d) [mathjaxinline]B(t)=B_{m}\sin (\omega _{m}t + \beta _{m})[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> Which of the following is a possible solution to the equation [mathjaxinline]\dfrac {1}{A(x)}\dfrac {\partial ^{2}}{\partial x^{2}}A(x) = -k_{m}^{2}[/mathjaxinline], for nonzero [mathjaxinline]k_{m}[/mathjaxinline]? Choose ALL that apply. </p>
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<text>a) [mathjaxinline]A(x)=A_{m}\left(k_{m}x + \alpha _{m}\right)[/mathjaxinline]</text>
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<text>b) [mathjaxinline]A(x)=A_{m}/\left(k_{m}x + \alpha _{m}\right)[/mathjaxinline]</text>
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<text>c) [mathjaxinline]A(x)=A_{m}\cos (k_{m}x + \alpha _{m})[/mathjaxinline]</text>
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<text>d) [mathjaxinline]A(x)=A_{m}\sin (k_{m}x + \alpha _{m})[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L14Q3: Validating the Solution</h2>
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Validating the Solution
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<p>
The solution to wave equation, for the [mathjaxinline]m^{\mathrm{th}}[/mathjaxinline] normal mode of the 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]\psi _{m}(x,t) = A_{m}\sin (\omega _{m}t + \beta _{m})\sin (k_{m}x + \alpha _{m})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]\omega _{m} = v_{p}k_{m}[/mathjaxinline]. </p>
<p>
Show that this solution satisfies the wave equations: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\frac{\partial ^{2}}{\partial t^{2}}\psi _{m}(x,t) = v_{p}^{2}\frac{\partial ^{2}}{\partial x^{2}}\psi _{m}(x,t)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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</table>
<p>
Calculate the following, using <code>A_m</code> for [mathjaxinline]A_{m}[/mathjaxinline], <code>omega_m</code> for [mathjaxinline]\omega _{m}[/mathjaxinline], <code>beta_m</code> for [mathjaxinline]\beta _{m}[/mathjaxinline], <code>k_m</code> for [mathjaxinline]k_{m}[/mathjaxinline], <code>alpha_m</code> for [mathjaxinline]\alpha _{m}[/mathjaxinline], <code>x</code>, and <code>t</code> in your answer, as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\frac{\partial ^{2}}{\partial t^{2}}\psi _{m}(x,t) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\frac{\partial ^{2}}{\partial x^{2}}\psi _{m}(x,t) =[/mathjaxinline] </p>
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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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<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">L14Q4: Spatio-Temporal Behavior of Wave Solution [WITH SIMULATION]</h2>
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Spatio-temporal Behavior of Wave Solution
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The solution to wave equation, for the [mathjaxinline]m^{\mathrm{th}}[/mathjaxinline] normal mode of the 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]\psi _{m}(x,t) = A_{m}\sin (\omega _{m}t + \beta _{m})\sin (k_{m}x + \alpha _{m})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
where [mathjaxinline]\omega _{m} = v_{p}k_{m}[/mathjaxinline]. </p>
<p>
In this problem we want to understand the interplay between position and time. For simplicity, let's consider a normal mode of the following form (with [mathjaxinline]A_{0}=1[/mathjaxinline]): </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]A(x,t)=A_{0}\cos (\omega _{m}t)\cos (k_{m}x)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
The graph below shows the solution [mathjaxinline]A(x,t_{0})[/mathjaxinline] at time [mathjaxinline]t_{0}=0[/mathjaxinline], as a function of position [mathjaxinline]x[/mathjaxinline]. The red point on the plot indicates the amplitude at a particular position [mathjaxinline]x=x_{0}[/mathjaxinline]. </p>
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Which of the following graphs represents [mathjaxinline]A(x_{0},t)[/mathjaxinline]&#8212;that is, the amplitude of the wave at the position [mathjaxinline]x=x_{0}[/mathjaxinline] (shown in the plot above), as a function of time [mathjaxinline]t[/mathjaxinline]? </p>
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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=wave_equation_solution_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/v1/5b1d13885fe77240fa6a0efcd2ec3a9c/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_wave_equation_solution_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L14Q5: Unknown Constants and Initial/Boundary Conditions</h2>
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Dependent Constants
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The solution to wave equation, for the [mathjaxinline]m^{\mathrm{th}}[/mathjaxinline] normal mode of the 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]\psi _{m}(x,t) = A_{m}\sin (\omega _{m}t + \beta _{m})\sin (k_{m}x + \alpha _{m})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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Where there are five unknown parameters, which must be determined: [mathjaxinline]A_{m}[/mathjaxinline], [mathjaxinline]\omega _{m}[/mathjaxinline], [mathjaxinline]\beta _{m}[/mathjaxinline], [mathjaxinline]k_{m}[/mathjaxinline], and [mathjaxinline]\alpha _{m}[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> There are only four independent parameters (two solved by initial conditions, and two solved by boundary conditions), which means one of the parameters depends explicitly on one of the other parameters. Which two parameters depend each other? </p>
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<text>a) [mathjaxinline]A_{m}[/mathjaxinline]</text>
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<text>b) [mathjaxinline]\omega _{m}[/mathjaxinline]</text>
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<text>c) [mathjaxinline]\beta _{m}[/mathjaxinline]</text>
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<text>d) [mathjaxinline]k_{m}[/mathjaxinline]</text>
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<text>e) [mathjaxinline]\alpha _{m}[/mathjaxinline]</text>
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Unknown Constants and Initial Conditions I
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<p><b class="bfseries">(Part b)</b> Which parameters of the solution must be solved using <i class="itshape">initial</i> conditions? Check ALL that apply. </p>
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<text>a) [mathjaxinline]A_{m}[/mathjaxinline]</text>
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<text>b) [mathjaxinline]\omega _{m}[/mathjaxinline]</text>
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<text>c) [mathjaxinline]\beta _{m}[/mathjaxinline]</text>
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<text>d) [mathjaxinline]k_{m}[/mathjaxinline]</text>
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<text>e) [mathjaxinline]\alpha _{m}[/mathjaxinline]</text>
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Unknown Constants and Boundary Conditions II
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<p><b class="bfseries">(Part c)</b> Which parameters of the solution must be solved using <i class="itshape">boundary</i> conditions? Check ALL that apply. </p>
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<text>a) [mathjaxinline]A_{m}[/mathjaxinline]</text>
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<text>b) [mathjaxinline]\omega _{m}[/mathjaxinline]</text>
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<text>c) [mathjaxinline]\beta _{m}[/mathjaxinline]</text>
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<text>d) [mathjaxinline]k_{m}[/mathjaxinline]</text>
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<text>e) [mathjaxinline]\alpha _{m}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L14Q6: Practice with Boundary Conditions I</h2>
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Boundary Condition Practice - part a
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Consider the following descriptions of systems consisting of strings of length [mathjaxinline]L[/mathjaxinline], each with different boundary conditions. For each case, write the restrictions on the wave solution required to satisfy the given boundary conditions. Recall that the general solution to the wave equation for such systems 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]\psi _{m}(x,t) = A_{m}\sin (\omega _{m}t + \beta _{m})\sin (k_{m}x + \alpha _{m})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Note, only TWO boundary conditions can be used, since these systems are one-dimensional strings with two end-points&#8212;if a boundary condition cannot be used, write <code>NA</code>. </p>
<p><b class="bfseries">(Part a)</b> A string with two fixed ends. </p>
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<p style="display:inline">[mathjaxinline]\psi (x=0) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\psi (x=L) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\frac{d}{dx}\psi (x=0) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\frac{d}{dx}\psi (x=L) =[/mathjaxinline] </p>
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Boundary Condition Practice - part b
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<p><b class="bfseries">(Part b)</b> A string with left end ([mathjaxinline]x=0[/mathjaxinline]) open and right end ([mathjaxinline]x=L[/mathjaxinline]) fixed. </p>
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<p style="display:inline">[mathjaxinline]\psi (x=0) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\psi (x=L) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\frac{d}{dx}\psi (x=0) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\frac{d}{dx}\psi (x=L) =[/mathjaxinline] </p>
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Boundary Condition Practice - part c
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<p><b class="bfseries">(Part c)</b> A string with two ends free to move. </p>
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<p style="display:inline">[mathjaxinline]\psi (x=0) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\psi (x=L) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\frac{d}{dx}\psi (x=0) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\frac{d}{dx}\psi (x=L) =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L14Q7: Practice with Boundary Conditions II</h2>
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Finding Parameters from Boundary Conditions
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Consider the following descriptions of systems consisting of strings of length [mathjaxinline]L[/mathjaxinline], each with different boundary conditions. Recall that the general solution to the wave equation for such systems 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]\psi _{m}(x,t) = A_{m}\sin (\omega _{m}t + \beta _{m})\sin (k_{m}x + \alpha _{m})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
For each case below, determine the values [mathjaxinline]\alpha _{m}[/mathjaxinline] and [mathjaxinline]k_{m}[/mathjaxinline] that give oscillatory solutions, based on the boundary conditions. Your answer could possibly depend on the index <code>m</code> (where [mathjaxinline]m=1,2,3\ldots[/mathjaxinline], with [mathjaxinline]m=1[/mathjaxinline] corresponding to the lowest oscillatory normal mode of the system), the length <code>L</code>, and/or numerical factors. If your answer includes [mathjaxinline]\pi[/mathjaxinline], use <code>pi</code>. </p>
<p><b class="bfseries">Note 1:</b> There may be cases where [mathjaxinline]k_ m=0[/mathjaxinline] is a mathematically valid solution. However, this corresponds to an infinite wavelength (i.e., a flat line) and is therefore not an oscillatory normal mode. </p>
<p><b class="bfseries">Note 2:</b> As we have seen, the requirement in all cases will be that some trig function is equal to zero. An equation of this form has multiple answers since adding an integer multiple of [mathjaxinline]\pi[/mathjaxinline] to any solution also gives a correct solution. <b class="bfseries">You should enter the smallest value of [mathjaxinline]\alpha _ m[/mathjaxinline] and a value of [mathjaxinline]k_ m[/mathjaxinline] that is valid for all [mathjaxinline]m[/mathjaxinline], where [mathjaxinline]m=1[/mathjaxinline] gives the lowest oscillatory mode.</b> </p>
<p><b class="bfseries">(Part a)</b> A string with two fixed ends. </p>
<p>
<p style="display:inline">[mathjaxinline]\alpha _{m}=[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
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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>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
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<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]
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<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
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>
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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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Practice with boundary conditions I
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<p><b class="bfseries">(Part b)</b> A string with left end ([mathjaxinline]x=0[/mathjaxinline]) open and right end ([mathjaxinline]x=L[/mathjaxinline]) fixed. </p>
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<p style="display:inline">[mathjaxinline]\alpha _{m}=[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
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<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>
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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>
<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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Practice with boundary conditions II
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<p><b class="bfseries">(Part c)</b> A string with two ends free to move. </p>
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<p style="display:inline">[mathjaxinline]\alpha _{m}=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]k_{m}=[/mathjaxinline] </p>
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<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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<h2 class="hd hd-2 unit-title">L14Q8: Normal Modes of a String</h2>
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Normal Modes of a String
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Consider the following figures, which depict the normal modes of different systems (where the lowest oscillatory mode is defined at [mathjaxinline]m=1[/mathjaxinline]). For each of the following problems, you will match the normal mode to the situation that is described. </p>
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<p><b class="bfseries">(Part a)</b> The [mathjaxinline]m=2[/mathjaxinline] normal mode of a string with one fixed end and one moving end. </p>
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<p><b class="bfseries">(Part b)</b> The [mathjaxinline]m=1[/mathjaxinline] normal mode of a string with two moving ends. </p>
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<p><b class="bfseries">(Part c)</b> The [mathjaxinline]m=3[/mathjaxinline] normal mode of a string with two fixed ends. </p>
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