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<h2 class="hd hd-2 unit-title">Introduction to Dispersion, Phase Velocity, and Group Velocity</h2>
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Now, having introduced different examples of wave systems (e.g., strings, sound, EM waves), we return to discussing some general properties of waves. </p><p>
One general property is <i class="itshape">dispersion</i>, which is the dependence of frequency on wavenumber. We will show that dispersion causes waves with different frequencies to travel at different velocities in a medium. </p><p>
We also discuss ways to send information using EM waves. The property of dispersion will complicate information transmission, so we discuss the strategy of <i class="itshape">amplitude modulation</i> as a way to trasmit information with high fidelity. </p>
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<h2 class="hd hd-2 unit-title">L22v1: Sending a Square Pulse in a String with Linear Dispersion</h2>
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<h2 class="hd hd-2 unit-title">L22Q1: Dispersion of Waves Propagating on an Ideal String</h2>
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Dispersion of Waves Propagating on an Ideal String
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In the following problems, consider the regime where [mathjaxinline]|\alpha | k^{2} \leq 1[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Which of the following dispersion relations is "non-dispersive"? </p>
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<text> a) [mathjaxinline]\omega = vk[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\omega = vk\sqrt {1 + |\alpha | k^{2}}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\omega = vk\sqrt {1 - |\alpha | k^{2}}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> Which of the following dispersion relations has the smallest ratio [mathjaxinline]\frac{\omega }{k}[/mathjaxinline], for a given value [mathjaxinline]k[/mathjaxinline] (in the regime stated above)? </p>
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<text> b) [mathjaxinline]\omega = vk\sqrt {1 + |\alpha | k^{2}}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\omega = vk\sqrt {1 - |\alpha | k^{2}}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part c)</b> Which of the following dispersion relations has the largest ratio [mathjaxinline]\frac{\omega }{k}[/mathjaxinline], for a given value [mathjaxinline]k[/mathjaxinline] (in the regime stated above)? </p>
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<text> b) [mathjaxinline]\omega = vk\sqrt {1 + |\alpha | k^{2}}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\omega = vk\sqrt {1 - |\alpha | k^{2}}[/mathjaxinline]</text>
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<h3 class="hd hd-3 problem-header" id="lect_14_02a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_14_02a-problem-progress" tabindex="-1">
Phase Difference for Linear Dispersion
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<p><b class="bfseries">(Part a)</b> Consider a string that is non-dispersive: </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) = vk[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
What is the magnitude of the difference in phase, [mathjaxinline]|\Delta \phi |[/mathjaxinline], between two waves with wavenumbers [mathjaxinline]k_{1} \lt k_{2}[/mathjaxinline], after they travel past a fixed position for an amount of time [mathjaxinline]t[/mathjaxinline]. More specifically, the phase of each wave is measured at the position [mathjaxinline]x=0[/mathjaxinline], and they are initially in phase at [mathjaxinline]t=0[/mathjaxinline]. Determine the difference in phase at time [mathjaxinline]t[/mathjaxinline], still measured at [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
Express your answers in terms of <code>v</code>, <code>t</code>, <code>k_1</code> for [mathjaxinline]k_1[/mathjaxinline], and <code>k_2</code> for [mathjaxinline]k_2[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|\Delta \phi | =[/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> (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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<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>
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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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<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>
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<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>
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<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>
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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>
</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_14_02b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_14_02b-problem-progress" tabindex="-1">
Phase Difference for Non-Linear Dispersion
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<p><b class="bfseries">(Part b)</b> Now consider a string with the following dispersion relation, where [mathjaxinline]\alpha \gt 0[/mathjaxinline]: </p>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\omega (k) = vk\sqrt {1 + \alpha k^{2}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
With this non-linear dispersion relation, what is the difference in phase, [mathjaxinline]|\Delta \phi |[/mathjaxinline], between two waves with wavenumbers [mathjaxinline]k_{1} \lt k_{2}[/mathjaxinline], after they travel past a fixed position for an amount of time [mathjaxinline]t[/mathjaxinline]. </p>
<p>
Express your answers in terms of <code>v</code>, <code>t</code>, <code>alpha</code> for [mathjaxinline]\alpha[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_1[/mathjaxinline], and <code>k_2</code> for [mathjaxinline]k_2[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|\Delta \phi | =[/mathjaxinline] </p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<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>
</td>
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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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Limit of Phase Difference for Non-Linear Dispersion
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<p><b class="bfseries">(Part c)</b> Now, consider the case when the dispersion relation used in Part (b) has [mathjaxinline]\alpha \ll 1/(k^{2})[/mathjaxinline] so that [mathjaxinline]\alpha k^2 \ll 1[/mathjaxinline]. Determine an approximation for the phase difference that is caused SOLELY by the dispersive nature of the medium. That is, find [mathjaxinline]\Delta \phi _{\mathrm{dispersive}} - \Delta \phi _{\mathrm{non-dispersive}}[/mathjaxinline], where [mathjaxinline]\alpha =0[/mathjaxinline] in the non-dispersive case. Keep only the lowest order term in [mathjaxinline]\alpha[/mathjaxinline]. </p>
<p>
Express your answers in terms of <code>v</code>, <code>t</code>, <code>alpha</code> for [mathjaxinline]\alpha[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_1[/mathjaxinline], and <code>k_2</code> for [mathjaxinline]k_2[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]| \Delta \phi _{\mathrm{dispersive}} | - | \Delta \phi _{\mathrm{non-dispersive}} |[/mathjaxinline] </p>
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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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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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<code>abs, ln, sqrt</code>
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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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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
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<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>
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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>
</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">L22Q3: Speed of Waves at Different Frequencies I</h2>
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Speed of Waves at Different Frequencies I
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<p>
Consider two waves with the SAME wavenumber [mathjaxinline]k[/mathjaxinline] traveling in two different strings. String 1 is non-dispersive, but string 2 is dispersive&#8212;their dispersion relations are: </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 _{1}(k)=vk[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<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]\omega _{2}(k) = vk\sqrt {1 + \alpha k^{2}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]\alpha \gt 0[/mathjaxinline]. </p>
<p>
Let string 1 have length [mathjaxinline]L_{1}[/mathjaxinline] and string 2 have length [mathjaxinline]L_{2}[/mathjaxinline]. How much longer should string 2 be than string 1 such that the waves take the same amount of time to travel down each string. Find the ratio [mathjaxinline]L_{2}/L_{1}[/mathjaxinline], and express your answer in terms of <code>k</code> and <code>alpha</code> for [mathjaxinline]\alpha[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]L_{2}/L_{1} =[/mathjaxinline] </p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
</td>
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<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>
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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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<h2 class="hd hd-2 unit-title">L22Q4: Dispersing Waves</h2>
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Dispersing Waves
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We have seen that a dispersion relationship like the one shown in red in the figure below causes waves to disperse as they travel because the different frequencies travel at different speeds. </p>
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What if, instead, our dispersion relationship looks the red line below? </p>
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In this case, will you still see the wave spread out as it travels? </p>
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