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<h2 class="hd hd-2 unit-title">Introduction to Infinite Mass-String System</h2>
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Now we look at the case of an infinite number of masses attached to STRINGS (last time was <i class="itshape">springs</i>). First we carry out an analysis similar to the previous lesson, by writing the equation of motion for adjacent masses and finding the normal mode frequencies of the system. As we are now familiar with, we utilize the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix to carry out this calculation. Note, even though we are dealing with STRINGS (not springs), and there is no spring constant in the system, we still use the terminology "[mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix." </p><p>
For an infinite array of oscillators, the normal mode frequencies are continuous! </p>
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<h2 class="hd hd-2 unit-title">L12v1: Review of Symmetry Matrices and Normal Mode Eigenvector Problems</h2>
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<h2 class="hd hd-2 unit-title">L12Q1: Amplitude of Infinite System</h2>
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Amplitude Plot
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An infinite system of masses on springs has translational symmetry, and the amplitude of the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] spring is given by: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]A_{j}=A_{0}e^{ijka}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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Which of the following plots represents the value of [mathjaxinline]A_{j}[/mathjaxinline] in the complex plane as [mathjaxinline]j[/mathjaxinline] changes? Note that the various plots may correspond to different values of [mathjaxinline]A_0[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">L12Q2: Wavenumber and Wavelength</h2>
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Wavenumber and Wavelength
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We showed that a solution to the infinite oscillator problem with translational symmetry is [mathjaxinline]\beta =e^{ika}[/mathjaxinline] (where [mathjaxinline]a[/mathjaxinline] is the separation of the masses) for which the complex amplitude of the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] mass in the system was found to be [mathjaxinline]A_{j}=A_0 e^{ijka}[/mathjaxinline]. We've had some practice with this before, and we showed that this has the form of a wave in real space. </p>
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Express the wavenumber as a function of the wavelength&#8212;that is, find [mathjaxinline]k(\lambda )[/mathjaxinline]. To answer this, it may be helpful to remember that the location of the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] mass in the system is [mathjaxinline]x=ja[/mathjaxinline]. Express your answer in term of any of the variables <code>j</code>, <code>a</code>, or <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]k =[/mathjaxinline] </p>
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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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<td class="formulainput">enter (english) name of letter</td>
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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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<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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<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">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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Consider the force diagram for the [mathjaxinline](j-1)^{\mathrm{st}}[/mathjaxinline], [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline], and [mathjaxinline](j+1)^{\mathrm{st}}[/mathjaxinline] masses in an infinite system of masses connected by strings. </p>
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In the limit where the vertical displacements are very small compared to the separation of the masses&#8212;i.e. where [mathjaxinline]\theta _{1},\, \theta _{2} \ll 1[/mathjaxinline]&#8212;what are the expressions for [mathjaxinline]\theta _{1}[/mathjaxinline] and [mathjaxinline]\theta _{2}[/mathjaxinline] in terms of the parameters of the system? </p>
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<text> a) [mathjaxinline]\theta _{1}=\frac{y_{j}+y_{j-1}}{a}[/mathjaxinline], [mathjaxinline]\theta _{2}=\frac{y_{j}+y_{j+1}}{a}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\theta _{1}=\frac{y_{j}+y_{j+1}}{a}[/mathjaxinline], [mathjaxinline]\theta _{2}=\frac{y_{j}+y_{j-1}}{a}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\theta _{1}=\frac{y_{j}-y_{j-1}}{a}[/mathjaxinline], [mathjaxinline]\theta _{2}=\frac{y_{j}-y_{j+1}}{a}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\theta _{1}=\frac{y_{j}-y_{j+1}}{a}[/mathjaxinline], [mathjaxinline]\theta _{2}=\frac{y_{j}-y_{j-1}}{a}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L12Q4: Dispersion Relation</h2>
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Dispersion Relation - part a
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The "dispersion relation" for a system relates the frequency of waves to the wavelength of those waves. More to the point, the function [mathjaxinline]\omega (k)[/mathjaxinline] describes how the frequency [mathjaxinline]\omega[/mathjaxinline] depends on the wavevector [mathjaxinline]k[/mathjaxinline]. </p>
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The dispersion relation of the system of oscillators discussed in the previous lecture segment 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\omega _{0}\sin {\frac{ka}{2}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part a)</b> Which of the following plots represents [mathjaxinline]\omega (k)[/mathjaxinline]? Note that in making these plots, a value of [mathjaxinline]2\omega _0=1.0[/mathjaxinline] was used. </p>
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Dispersion Relation - part b
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<p><b class="bfseries">(Part b)</b> Under certain conditions, the above dispersion relation reduces to the following form: [mathjaxinline]\omega (k) = \omega _{0}ka[/mathjaxinline]. Under what condition will this occur? </p>
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<text> d) when [mathjaxinline]\lambda =\frac{2\pi }{k}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L12Q5: Shortest and Longest Wavelengths and Wavenumbers</h2>
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Shortest Wavelength
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Once again, consider the system of infinitely many oscillators discussed in the previous lecture segment. One of the infinitely many normal modes of the system is depicted below: </p>
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<p><b class="bfseries">(Part a)</b> What is the <i class="itshape">wavelength</i> of the <i class="itshape">shortest</i> wave that can propagate through the system, with nonzero amplitude? Express your answer using <code>a</code> or use <code>infty</code> for [mathjaxinline]\infty[/mathjaxinline] if necessary. </p>
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<p style="display:inline">[mathjaxinline]\lambda _{\mathrm{shortest}}=[/mathjaxinline] </p>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<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>
</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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<h3 class="hd hd-3 problem-header" id="lect_09_03_2b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_03_2b-problem-progress" tabindex="-1">
Longest Wavelength
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<p><b class="bfseries">(Part b)</b> What is the <i class="itshape">wavelength</i> of the <i class="itshape">longest</i> wave that can propagate through the system, with nonzero amplitude? Express your answer using <code>a</code> or use <code>infty</code> for [mathjaxinline]\infty[/mathjaxinline] if necessary. </p>
<p>
<p style="display:inline">[mathjaxinline]\lambda _{\mathrm{longest}}=[/mathjaxinline] </p>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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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"><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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<h3 class="hd hd-3 problem-header" id="lect_09_03_2c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_03_2c-problem-progress" tabindex="-1">
Shortest Wavenumber
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<p><b class="bfseries">(Part c)</b> What is the <i class="itshape">wavenumber</i> of the <i class="itshape">shortest</i> wave that can propagate through the system, with nonzero amplitude? Express your answer using <code>a</code>, <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline], or use <code>infty</code> for [mathjaxinline]\infty[/mathjaxinline] if necessary. </p>
<p>
<p style="display:inline">[mathjaxinline]k_{\mathrm{shortest}}=[/mathjaxinline] </p>
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<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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<h3 class="hd hd-3 problem-header" id="lect_09_03_2d-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_03_2d-problem-progress" tabindex="-1">
Longest Wavenumber
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<p><b class="bfseries">(Part d)</b> What is the <i class="itshape">wavenumber</i> of the <i class="itshape">longest</i> wave that can propagate through the system, with nonzero amplitude? Express your answer using <code>a</code>, <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline], or use <code>infty</code> for [mathjaxinline]\infty[/mathjaxinline] if necessary. </p>
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<p style="display:inline">[mathjaxinline]k_{\mathrm{longest}}=[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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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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