<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lesson16intro" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">Introduction to Fourier Transforms and AM Radio</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson16_intro">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson16_intro" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
In previous lessons, we learned that <i class="itshape">dispersion</i> is a property of a medium that distorts waves as they travel. The dispersion of a medium can prohibit the transmission of information—by information, we mean wave pulses (typically square waves)—because dispersion makes waves with different wavenumbers travel at different speeds. Let's say the information that we want to send is described by the function [mathjaxinline]f(t)[/mathjaxinline]. </p><p>
In this lesson, we again return to our objective of learning how to send information in a non-distorted way. In pursuit of this objective, we first describe how to express a traveling wave pulse in terms of its frequency components by using a Fourier transform. Previously, we introduced Fourier transforms in terms of cosines and sines. Here, we introduce a more general formulation. </p><p>
The lesson this week becomes a little mathematically involved and, although it is important to develop a mathematical grounding in order to understand the derivations, let's not lose sight of our objective—to learn how to send a signal [mathjaxinline]f(t)[/mathjaxinline] in a non-distorted way, through a dispersive medium. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L24v1" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24v1: Transmission of Information by Wave Pulse</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v1" data-block-type="video" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L24v1: Transmission of Information by Wave Pulse</h3>
<div
id="video_L24v1"
class="video closed"
data-metadata='{"completionPercentage": 0.95, "ytMetadataEndpoint": "", "prioritizeHls": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v1/handler/transcript/available_translations", "generalSpeed": 1.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v1/handler/xmodule_handler/save_user_state", "savedVideoPosition": 0.0, "ytApiUrl": "https://www.youtube.com/iframe_api", "end": 0.0, "poster": null, "ytTestTimeout": 1500, "completionEnabled": false, "streams": "1.00:2B8xVBAyjKo", "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v1/handler/publish_completion", "showCaptions": "true", "start": 0.0, "sources": [], "autoplay": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v1/handler/transcript/translation/__lang__", "autoAdvance": false, "autohideHtml5": false, "lmsRootURL": "https://openlearninglibrary.mit.edu", "speed": null, "transcriptLanguage": "en", "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L24v1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L24v1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L24v1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/ff5c4695594c7fdd82295a97ecc4ae44/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L24v1.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex01" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q1: Dispersive vs. Non-dispersive Media</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_01" class="problems-wrapper" role="group"
aria-labelledby="lect_15_01-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="3.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_01-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01-problem-progress" tabindex="-1">
Phase Velocity, Group Velocity, and Dispersion Review
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part a)</b> Last week we learned the following definitions for phase velocity, [mathjaxinline]v_{p}[/mathjaxinline], and group velocity, [mathjaxinline]v_{g}[/mathjaxinline]: </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]v_{p}=\frac{\omega }{k}[/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]v_{g}=\frac{d\omega }{dk}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Calculate both [mathjaxinline]v_{p}[/mathjaxinline] and [mathjaxinline]v_{g}[/mathjaxinline] for the following dispersion relation: </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]\omega (k)=\omega _0\sin (k\alpha )[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Express your answers in terms of the variables given, which could include <code>k</code>, <code>omega_0</code> for [mathjaxinline]\omega _{0}[/mathjaxinline], or <code>alpha</code> for [mathjaxinline]\alpha[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]v_{p} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_15_01_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_15_01_2_1" id="input_lect_15_01_2_1" aria-describedby="status_lect_15_01_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_15_01_2_1"/>
<span class="status unanswered" id="status_lect_15_01_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_15_01_2_1" class="answer"/>
<div id="display_lect_15_01_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_15_01_2_1_dynamath" name="input_lect_15_01_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]v_{g} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_15_01_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_15_01_3_1" id="input_lect_15_01_3_1" aria-describedby="status_lect_15_01_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_15_01_3_1"/>
<span class="status unanswered" id="status_lect_15_01_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_15_01_3_1" class="answer"/>
<div id="display_lect_15_01_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_15_01_3_1_dynamath" name="input_lect_15_01_3_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<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>
<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>
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_15_01_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> Consider the limit [mathjaxinline]k\alpha \ll 1[/mathjaxinline]. Is the medium "non-dispersive" in this limit? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input ">
<select name="input_lect_15_01_4_1" id="input_lect_15_01_4_1" aria-describedby="status_lect_15_01_4_1">
<option value="option_lect_15_01_4_1_dummy_default">Select an option</option>
<option value="a) Yes"> a) Yes</option>
<option value="b) No"> b) No</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_01_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_lect_15_01_4_1"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_01_solution_2"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Phase Velocity, Group Velocity, and Dispersion Review " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_01" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_01">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_01-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_01-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_01-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L24v2" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24v2: Pulse Function for a Non-dispersive Medium</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v2" data-block-type="video" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L24v2: Pulse Function for a Non-dispersive Medium</h3>
<div
id="video_L24v2"
class="video closed"
data-metadata='{"completionPercentage": 0.95, "ytMetadataEndpoint": "", "prioritizeHls": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v2/handler/transcript/available_translations", "generalSpeed": 1.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v2/handler/xmodule_handler/save_user_state", "savedVideoPosition": 0.0, "ytApiUrl": "https://www.youtube.com/iframe_api", "end": 0.0, "poster": null, "ytTestTimeout": 1500, "completionEnabled": false, "streams": "1.00:e39jfy_epfQ", "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v2/handler/publish_completion", "showCaptions": "true", "start": 0.0, "sources": [], "autoplay": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v2/handler/transcript/translation/__lang__", "autoAdvance": false, "autohideHtml5": false, "lmsRootURL": "https://openlearninglibrary.mit.edu", "speed": null, "transcriptLanguage": "en", "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L24v2"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L24v2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L24v2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/ac1eaea34706c14023913777e9f85adf/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L24v2.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex00_1" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q2: Definition of Functions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_1" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_00_1" class="problems-wrapper" role="group"
aria-labelledby="lect_15_00_1-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_1" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_1/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_00_1-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_1-problem-progress" tabindex="-1">
Definition of Functions
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_1-problem-progress"></div>
<div class="problem">
<div>
<p>
We have introduced two different functions, [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]\psi (x,t)[/mathjaxinline]. Select the options that correctly characterize each function. Select ALL that apply. </p>
<p><b class="bfseries">(Part a)</b> [mathjaxinline]f(t)[/mathjaxinline]: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_00_1_2_1">
<fieldset aria-describedby="status_lect_15_00_1_2_1">
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_2_1[]" id="input_lect_15_00_1_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_00_1_2_1-choice_0-label" for="input_lect_15_00_1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_2_1">
<text>a) a signal that we wish to transmit</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_2_1[]" id="input_lect_15_00_1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_00_1_2_1-choice_1-label" for="input_lect_15_00_1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_2_1">
<text>b) the time-dependent position of the end of the string</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_2_1[]" id="input_lect_15_00_1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_00_1_2_1-choice_2-label" for="input_lect_15_00_1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_2_1">
<text>c) the force on a traveling wave</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_2_1[]" id="input_lect_15_00_1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_15_00_1_2_1-choice_3-label" for="input_lect_15_00_1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_2_1">
<text>d) a traveling wave</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_2_1[]" id="input_lect_15_00_1_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="lect_15_00_1_2_1-choice_4-label" for="input_lect_15_00_1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_2_1">
<text>e) the position of the envelope of a traveling wave</text>
</label>
</div>
<span id="answer_lect_15_00_1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_00_1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_00_1_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> Definition of [mathjaxinline]\psi (x,t)[/mathjaxinline]: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_00_1_3_1">
<fieldset aria-describedby="status_lect_15_00_1_3_1">
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_3_1[]" id="input_lect_15_00_1_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_00_1_3_1-choice_0-label" for="input_lect_15_00_1_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_3_1">
<text>a) a signal that we wish to transmit</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_3_1[]" id="input_lect_15_00_1_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_00_1_3_1-choice_1-label" for="input_lect_15_00_1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_3_1">
<text>b) the time-dependent position of the end of the string</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_3_1[]" id="input_lect_15_00_1_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_00_1_3_1-choice_2-label" for="input_lect_15_00_1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_3_1">
<text>c) the force on a traveling wave</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_3_1[]" id="input_lect_15_00_1_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_15_00_1_3_1-choice_3-label" for="input_lect_15_00_1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_3_1">
<text>d) a traveling wave</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_1_3_1[]" id="input_lect_15_00_1_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="lect_15_00_1_3_1-choice_4-label" for="input_lect_15_00_1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_1_3_1">
<text>e) the position of the envelope of a traveling wave</text>
</label>
</div>
<span id="answer_lect_15_00_1_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_00_1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_00_1_solution_2"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Definition of Functions" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_00_1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_00_1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex00_0" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q3: Visualizing the Waveform [SIMULATION]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_00_0" class="problems-wrapper" role="group"
aria-labelledby="lect_15_00_0-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_00_0-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0-problem-progress" tabindex="-1">
Visualizing the Waveform
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider the graph below, which shows the function [mathjaxinline]f(t)[/mathjaxinline]. An instant in time, [mathjaxinline]t=t_{0}[/mathjaxinline] is shown as the vertical dashed line. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/7c9196b86345a982affe1ec7b1dabc8e/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_15_00_0_2.png" width="440"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
Which of the following plots below depicts the function [mathjaxinline]\psi (x,t=t_{0})[/mathjaxinline], the traveling wave produced by the function [mathjaxinline]f(t)[/mathjaxinline], shown as a function of position at the time [mathjaxinline]t_{0}[/mathjaxinline]? </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/d1a2a76f07cea875bacf95d97ddff526/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_15_00_0_1.png" width="880"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
<select name="input_lect_15_00_0_2_1" id="input_lect_15_00_0_2_1" aria-describedby="status_lect_15_00_0_2_1">
<option value="option_lect_15_00_0_2_1_dummy_default">Select an option</option>
<option value="a)"> a)</option>
<option value="b)"> b)</option>
<option value="c)"> c)</option>
<option value="d)"> d)</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_00_0_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_lect_15_00_0_2_1"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_00_0_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Visualizing the Waveform" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_00_0" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_00_0">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_0-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_0-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_0-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_15_00">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_15_00" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<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=non-dispersing_waveform_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/v1/8104719f7e27f3f75ada5695c53da6dd/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_non-dispersing_waveform_1.png"/></div><p/>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex01_1" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q4: Pulse Traveling to the Left</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01_02">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01_02" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_01_02" class="problems-wrapper" role="group"
aria-labelledby="lect_15_01_02-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01_02" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01_02/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_01_02-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01_02-problem-progress" tabindex="-1">
Pulse Going to the Left
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_01_02-problem-progress"></div>
<div class="problem">
<div>
<p>
We saw that if we shake our string as a function of time with [mathjaxinline]f(t)[/mathjaxinline], it creates a traveling wave in the string, [mathjaxinline]\psi (x,t)[/mathjaxinline]. In the lecture, we saw that a wave generated on the left travels to the right. What if instead we generate a wave on the right and let it travel to the left? </p>
<p>
What is the waveform for a leftward traveling wave (where positive [mathjaxinline]x[/mathjaxinline] points to the right)? </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/f21c2a6d135070f2e22d3a29d9d02c9c/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_15_01_02.svg" width="770"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_01_02_2_1">
<fieldset aria-describedby="status_lect_15_01_02_2_1">
<div class="field">
<input type="checkbox" name="input_lect_15_01_02_2_1[]" id="input_lect_15_01_02_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_01_02_2_1-choice_0-label" for="input_lect_15_01_02_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_01_02_2_1">
<text>a) [mathjaxinline]\psi (x,t) = f(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_01_02_2_1[]" id="input_lect_15_01_02_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_01_02_2_1-choice_1-label" for="input_lect_15_01_02_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_01_02_2_1">
<text>b) [mathjaxinline]\psi (x,t) = f(kx-\omega t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_01_02_2_1[]" id="input_lect_15_01_02_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_01_02_2_1-choice_2-label" for="input_lect_15_01_02_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_01_02_2_1">
<text>c) [mathjaxinline]\psi (x,t) = f(kx+\omega t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_01_02_2_1[]" id="input_lect_15_01_02_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_15_01_02_2_1-choice_3-label" for="input_lect_15_01_02_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_15_01_02_2_1">
<text>d) [mathjaxinline]\psi (x,t) = f(-kx+\omega t)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_15_01_02_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_01_02_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_01_02_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Pulse Going to the Left" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_01_02" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_01_02">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_01_02-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_01_02-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_01_02-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L24v3" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24v3: Set-up Fourier Transform for a Wave in a Dispersive Medium</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v3" data-block-type="video" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L24v3: Set-up Fourier Transform for a Wave in a Dispersive Medium</h3>
<div
id="video_L24v3"
class="video closed"
data-metadata='{"completionPercentage": 0.95, "ytMetadataEndpoint": "", "prioritizeHls": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v3/handler/transcript/available_translations", "generalSpeed": 1.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v3/handler/xmodule_handler/save_user_state", "savedVideoPosition": 0.0, "ytApiUrl": "https://www.youtube.com/iframe_api", "end": 0.0, "poster": null, "ytTestTimeout": 1500, "completionEnabled": false, "streams": "1.00:MuSq4nQh1rQ", "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v3/handler/publish_completion", "showCaptions": "true", "start": 0.0, "sources": [], "autoplay": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v3/handler/transcript/translation/__lang__", "autoAdvance": false, "autohideHtml5": false, "lmsRootURL": "https://openlearninglibrary.mit.edu", "speed": null, "transcriptLanguage": "en", "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L24v3"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L24v3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L24v3">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/6a522cdb5e5de5714966dce1d8533964/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L24v3.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex00_0_0" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q5: Visualizing Dispersive Waveform [SIMULATION]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0_0">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0_0" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_00_0_0" class="problems-wrapper" role="group"
aria-labelledby="lect_15_00_0_0-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0_0" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0_0/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_00_0_0-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0_0-problem-progress" tabindex="-1">
Visualizing Dispersive Waveform
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_0_0-problem-progress"></div>
<div class="problem">
<div>
<p>
In a medium with dispersion, the traveling waveform [mathjaxinline]\psi (x,t)[/mathjaxinline] is not simply equivalent to the signal [mathjaxinline]f(t)[/mathjaxinline] traveling at a constant phase velocity. Consider the graphs shown below, which compare a traveling waveform moving to the right in a medium with and without dispersion (red dashed line vs. blue solid line). </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/49fc1eba4d2a812b238e76116896169a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_15_00_0_0_1.png" width="770"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/87c072305e32bb6ffa0dff36121a7f02/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_15_00_0_0_2.png" width="770"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/3c32b18c75397925e9dbb747f8d55d7e/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_15_00_0_0_3.png" width="770"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/8642d6bf71a5fcdbbb3ed1dbee184fe4/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_15_00_0_0_4.png" width="770"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
Select ALL statements that are generally true about traveling waveforms in dispersive media. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_00_0_0_2_1">
<fieldset aria-describedby="status_lect_15_00_0_0_2_1">
<div class="field">
<input type="checkbox" name="input_lect_15_00_0_0_2_1[]" id="input_lect_15_00_0_0_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_00_0_0_2_1-choice_0-label" for="input_lect_15_00_0_0_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_0_0_2_1">
<text>a) Waveforms disperse because each frequency component of the wave travels at a different velocity</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_0_0_2_1[]" id="input_lect_15_00_0_0_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_00_0_0_2_1-choice_1-label" for="input_lect_15_00_0_0_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_0_0_2_1">
<text>b) Waveforms disperse because the amplitude of each frequency component is affected differently</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_0_0_2_1[]" id="input_lect_15_00_0_0_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_00_0_0_2_1-choice_2-label" for="input_lect_15_00_0_0_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_0_0_2_1">
<text>c) Waveforms will <i class="itshape">spread out</i> when traveling left to right, but <i class="itshape">condense</i> when traveling right to left</text>
</label>
</div>
<span id="answer_lect_15_00_0_0_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_00_0_0_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_00_0_0_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Visualizing Dispersive Waveform" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_00_0_0" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_00_0_0">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_0_0-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_0_0-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_0_0-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_15_00_0">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_15_00_0" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<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=dispersing_waveform_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/v1/20700edbfe0d5c74320dbdcc59c7dbd2/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_dispersing_waveform_1.png"/></div><p/>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex00_2" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q6: The Meaning of Fourier Amplitude</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_2" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_00_2" class="problems-wrapper" role="group"
aria-labelledby="lect_15_00_2-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_2" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_2/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_00_2-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_2-problem-progress" tabindex="-1">
Definition of Functions
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_00_2-problem-progress"></div>
<div class="problem">
<div>
<p>
Select ALL options that correctly characterize [mathjaxinline]c(\omega )[/mathjaxinline]. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_00_2_2_1">
<fieldset aria-describedby="status_lect_15_00_2_2_1">
<div class="field">
<input type="checkbox" name="input_lect_15_00_2_2_1[]" id="input_lect_15_00_2_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_00_2_2_1-choice_0-label" for="input_lect_15_00_2_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_2_2_1">
<text>a) the frequency of the signal [mathjaxinline]f(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_2_2_1[]" id="input_lect_15_00_2_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_00_2_2_1-choice_1-label" for="input_lect_15_00_2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_2_2_1">
<text>b) the amplitude of the signal [mathjaxinline]f(t)[/mathjaxinline] in real space</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_2_2_1[]" id="input_lect_15_00_2_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_00_2_2_1-choice_2-label" for="input_lect_15_00_2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_2_2_1">
<text>c) the amplitude of the signal [mathjaxinline]f(t)[/mathjaxinline] in frequency space</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_00_2_2_1[]" id="input_lect_15_00_2_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_15_00_2_2_1-choice_3-label" for="input_lect_15_00_2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_15_00_2_2_1">
<text>d) the magnitude of the "contribution" to [mathjaxinline]f(t)[/mathjaxinline] from frequency component [mathjaxinline]\omega[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_15_00_2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_00_2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_00_2_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Definition of Functions" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_00_2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_00_2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_00_2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex02" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q7: Properties of Fourier Transform</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_02">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_02" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_02" class="problems-wrapper" role="group"
aria-labelledby="lect_15_02-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_02" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_02/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="3.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_02-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_02-problem-progress" tabindex="-1">
Identification of Traveling Waves
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_02-problem-progress"></div>
<div class="problem">
<div>
<p>
We have defined [mathjaxinline]f(t)[/mathjaxinline] as: </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]f(t)=\int _{-\infty }^{\infty } d\omega \, c(\omega ) e^{-i\omega t}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Use a change of variables to solve each of the following integrals. Consider a non-dispersive medium, where [mathjaxinline]k=\omega /v_{p}[/mathjaxinline] </p>
<p><b class="bfseries">(Part a)</b> [mathjaxinline]\int _{-\infty }^{\infty } d\omega \, c(\omega ) e^{-i\omega t - i k x}=[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_02_2_1">
<fieldset aria-describedby="status_lect_15_02_2_1">
<div class="field">
<input type="checkbox" name="input_lect_15_02_2_1[]" id="input_lect_15_02_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_02_2_1-choice_0-label" for="input_lect_15_02_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_2_1">
<text>a) [mathjaxinline]f(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_2_1[]" id="input_lect_15_02_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_02_2_1-choice_1-label" for="input_lect_15_02_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_2_1">
<text>b) [mathjaxinline]f(t-x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_2_1[]" id="input_lect_15_02_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_02_2_1-choice_2-label" for="input_lect_15_02_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_2_1">
<text>c) [mathjaxinline]f(-t-x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_2_1[]" id="input_lect_15_02_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_15_02_2_1-choice_3-label" for="input_lect_15_02_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_2_1">
<text>d) [mathjaxinline]f(-t+x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_2_1[]" id="input_lect_15_02_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="lect_15_02_2_1-choice_4-label" for="input_lect_15_02_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_2_1">
<text>e) [mathjaxinline]f(t+x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_15_02_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_02_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_02_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> [mathjaxinline]\int _{-\infty }^{\infty } d\omega \, c(\omega ) e^{+i\omega t - i k x}=[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_02_3_1">
<fieldset aria-describedby="status_lect_15_02_3_1">
<div class="field">
<input type="checkbox" name="input_lect_15_02_3_1[]" id="input_lect_15_02_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_02_3_1-choice_0-label" for="input_lect_15_02_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_3_1">
<text>a) [mathjaxinline]f(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_3_1[]" id="input_lect_15_02_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_02_3_1-choice_1-label" for="input_lect_15_02_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_3_1">
<text>b) [mathjaxinline]f(t-x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_3_1[]" id="input_lect_15_02_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_02_3_1-choice_2-label" for="input_lect_15_02_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_3_1">
<text>c) [mathjaxinline]f(-t-x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_3_1[]" id="input_lect_15_02_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_15_02_3_1-choice_3-label" for="input_lect_15_02_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_3_1">
<text>d) [mathjaxinline]f(-t+x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_02_3_1[]" id="input_lect_15_02_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="lect_15_02_3_1-choice_4-label" for="input_lect_15_02_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_15_02_3_1">
<text>e) [mathjaxinline]f(t+x/v_{p})[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_15_02_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_02_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_02_solution_2"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Identification of Traveling Waves" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_02" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_02">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_02-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_02-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_02-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L24v4" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24v4: Delta Function to Extract the Fourier Transform Weights</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v4">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L24v4" data-block-type="video" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L24v4: Delta Function to Extract the Fourier Transform Weights</h3>
<div
id="video_L24v4"
class="video closed"
data-metadata='{"completionPercentage": 0.95, "ytMetadataEndpoint": "", "prioritizeHls": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v4/handler/transcript/available_translations", "generalSpeed": 1.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v4/handler/xmodule_handler/save_user_state", "savedVideoPosition": 0.0, "ytApiUrl": "https://www.youtube.com/iframe_api", "end": 0.0, "poster": null, "ytTestTimeout": 1500, "completionEnabled": false, "streams": "1.00:sUxZ5hvPmIM", "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v4/handler/publish_completion", "showCaptions": "true", "start": 0.0, "sources": [], "autoplay": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L24v4/handler/transcript/translation/__lang__", "autoAdvance": false, "autohideHtml5": false, "lmsRootURL": "https://openlearninglibrary.mit.edu", "speed": null, "transcriptLanguage": "en", "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L24v4"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L24v4">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L24v4">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/e6a65aaab6a425ec5d009f15242a08e3/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L24v4.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-review_delta_func" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">Review of Delta Functions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@review_delta_func">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@review_delta_func" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><h2>DEFINITIONS</h2></p><p>
A Dirac delta function is defined as the following piecewise function: </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]\delta (x)= \begin{cases} \infty & x=0 \\ 0 & x \neq 0 \\ \end{cases}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
The delta function can also be defined through the following Fourier integral: </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]\delta (\omega -\omega ^{\prime }) = \frac{1}{2\pi }\int _{-\infty }^{\infty } dt\, e^{i(\omega - \omega ^{\prime })t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h2>PROPERTIES</h2></p><p>
The delta function has the convenient property that its integral is equal to 1: </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]\int _{-\infty }^{\infty } dx \, \delta (x) = 1[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Additionally, the <i class="itshape">convolution</i> of the delta-function with another function has the following result: </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{-\infty }^{\infty } dx \, \delta (x-\alpha )f(x) = f(\alpha )[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Finally, the following relation may be useful to know: </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]\delta (\alpha x) = \frac{1}{|\alpha |}\delta (x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
A special case of the above relation is: </p><table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\delta (-x) = \delta (x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex03_1" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q8: Properties of Delta Functions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03_1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03_1" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_03_1" class="problems-wrapper" role="group"
aria-labelledby="lect_15_03_1-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03_1" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03_1/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_03_1-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03_1-problem-progress" tabindex="-1">
Properties of delta functions
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03_1-problem-progress"></div>
<div class="problem">
<div>
<p>
Solve each of the following integrals, which explore the properties of delta functions. </p>
<p><b class="bfseries">(Part a)</b> [mathjaxinline]\int _{-\infty }^{\infty }dx\, x^{2} \delta (x-2)=[/mathjaxinline] </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_15_03_1_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_15_03_1_2_1" id="input_lect_15_03_1_2_1" aria-describedby="status_lect_15_03_1_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_15_03_1_2_1"/>
<span class="status unanswered" id="status_lect_15_03_1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_15_03_1_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_03_1_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> [mathjaxinline]\int _{-\infty }^{\infty }dx\, \sin (x) \delta (x+\pi /2)=[/mathjaxinline] </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_15_03_1_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_15_03_1_3_1" id="input_lect_15_03_1_3_1" aria-describedby="status_lect_15_03_1_3_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_15_03_1_3_1"/>
<span class="status unanswered" id="status_lect_15_03_1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_15_03_1_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_03_1_solution_2"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part c)</b> [mathjaxinline]\frac{1}{2\pi }\int _{-\infty }^{\infty }dx\, \int _{-\infty }^{\infty }dk\, \cos (x) e^{ik(x-\pi /2)} =[/mathjaxinline] </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_lect_15_03_1_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_15_03_1_4_1" id="input_lect_15_03_1_4_1" aria-describedby="status_lect_15_03_1_4_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_15_03_1_4_1"/>
<span class="status unanswered" id="status_lect_15_03_1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_15_03_1_4_1" class="answer"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<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>
<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>
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_15_03_1_solution_3"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Properties of delta functions" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_03_1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_03_1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_03_1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_03_1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_03_1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_15_ex03" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">L24Q9: Inverting the Derivation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-request-token="888b113cedc011eeb92f16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03" data-block-type="problem" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_15_03" class="problems-wrapper" role="group"
aria-labelledby="lect_15_03-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_15_03-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03-problem-progress" tabindex="-1">
Inverting the Derivation
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_15_03-problem-progress"></div>
<div class="problem">
<div>
<p>
In the lesson, we used the properties of delta functions to prove the following: </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]c(\omega ) = \frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, f(t) e^{i\omega t}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Now, let's invert these steps to show that the following is also true (this was our previous expression for [mathjaxinline]f(t)[/mathjaxinline]): </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]\int _{-\infty }^{\infty }d\omega \, c(\omega ) e^{-i\omega t} = f(t)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
First, let's plug the definition of [mathjaxinline]c(\omega )[/mathjaxinline] into the equation above, and collect exponential terms. </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int _{-\infty }^{\infty }d\omega \, c(\omega ) e^{-i\omega t}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{-\infty }^{\infty }d\omega \, \left[\frac{1}{2\pi }\int _{-\infty }^{\infty }dt^{\prime } f(t^{\prime }) e^{i\omega t^{\prime }}\right] e^{-i\omega t}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
<tr id="a0000000006">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\int _{-\infty }^{\infty }d\omega \, \frac{1}{2\pi }\int _{-\infty }^{\infty }dt^{\prime } f(t^{\prime }) e^{-i\omega (t-t^{\prime })}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Select the option that shows a legitimate next step. Select ALL that apply. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_03_2_1">
<fieldset aria-describedby="status_lect_15_03_2_1">
<div class="field">
<input type="checkbox" name="input_lect_15_03_2_1[]" id="input_lect_15_03_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_15_03_2_1-choice_0-label" for="input_lect_15_03_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_15_03_2_1">
<text>a) [mathjaxinline]\frac{1}{2\pi } \int _{-\infty }^{\infty }d\omega \, e^{-i\omega (t-t^{\prime })} \int _{-\infty }^{\infty }dt^{\prime } f(t^{\prime })[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_03_2_1[]" id="input_lect_15_03_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_15_03_2_1-choice_1-label" for="input_lect_15_03_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_03_2_1">
<text>b) [mathjaxinline]\frac{1}{2\pi } \int _{-\infty }^{\infty }d\omega \, \int _{-\infty }^{\infty }dt^{\prime } f(t^{\prime }) e^{-i\omega (t-t^{\prime })}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_15_03_2_1[]" id="input_lect_15_03_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_15_03_2_1-choice_2-label" for="input_lect_15_03_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_03_2_1">
<text>c) [mathjaxinline]\int _{-\infty }^{\infty }dt^{\prime } f(t^{\prime }) \frac{1}{2\pi }\int _{-\infty }^{\infty }d\omega \, e^{-i\omega (t-t^{\prime })}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_15_03_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_03_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> What delta function substitution is necessary to complete the derivation? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_15_03_3_1">
<fieldset aria-describedby="status_lect_15_03_3_1">
<div class="field">
<input type="radio" name="input_lect_15_03_3_1" id="input_lect_15_03_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_15_03_3_1-choice_1-label" for="input_lect_15_03_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_15_03_3_1">
<text> a) [mathjaxinline]\delta (\omega -\omega ^{\prime })[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_15_03_3_1" id="input_lect_15_03_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_15_03_3_1-choice_2-label" for="input_lect_15_03_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_15_03_3_1">
<text> b) [mathjaxinline]\delta (\omega +\omega ^{\prime })[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_15_03_3_1" id="input_lect_15_03_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_15_03_3_1-choice_3-label" for="input_lect_15_03_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_15_03_3_1">
<text> c) [mathjaxinline]\delta (t-t^{\prime })[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_15_03_3_1" id="input_lect_15_03_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_15_03_3_1-choice_4-label" for="input_lect_15_03_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_15_03_3_1">
<text> d) [mathjaxinline]\delta (t+t^{\prime })[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_15_03_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_15_03_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_15_03_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Inverting the Derivation" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_15_03" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_15_03">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_03-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_03-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_15_03-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>