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<h2 class="hd hd-2 unit-title">Introduction to Reflection and Transmission</h2>
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<p>
Here, we continue the topic of traveling waves. </p><p>
As waves travel, they may encounter an object or move out of one material and into another. Thus, we consider how to treat waves that interact with a boundary. Specifically, we discuss the relation between <i class="itshape">incident</i>, <i class="itshape">reflected</i>, and <i class="itshape">transmitted</i> waves. </p><p>
In the following lesson, we will briefly discuss reflection and transmission coefficients, which are related to the interactions of waves with a boundary. </p>
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<h2 class="hd hd-2 unit-title">L17v1: Waves in Different Media - Reflection and Transmission</h2>
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<h3 class="hd hd-2">L17v1: Waves in different media - reflection and transmission</h3>
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<h2 class="hd hd-2 unit-title">L17Q1: Velocity of Waves in Different Media</h2>
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The wave velocity in a string is related to the properties of the string in the following way: </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=\sqrt {\dfrac {T}{\rho }}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]T[/mathjaxinline] is the tension and [mathjaxinline]\rho[/mathjaxinline] is the linear mass density. </p><p>
We can consider two strings connected to each other with different properties, so that the velocity of waves will be different in each string. </p><p>
Importantly, if the strings have different tensions, they must be connected to each other through a ring around a post, with the post exerting a horizontal force compensating for the difference in string tensions. Thus, the net horizontal force on the ring will be zero, and the strings will not move in the horizontal direction. We will make this assumption in all such cases moving forward. </p><center><img src="/assets/courseware/v1/9da5d9a963d8e73a2bb3a6710b37f722/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_pset_06_02_fig_1.svg" width="495"/></center><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
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Velocity of Waves in Different Media - part a
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<p><b class="bfseries">(Part a)</b> A wave travels from string 1 with mass per unit length [mathjaxinline]\rho _{1} = \rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{1}=T[/mathjaxinline], to string 2 with the same density [mathjaxinline]\rho _{2} = \rho _{L}[/mathjaxinline] but a higher tension [mathjaxinline]T_{2}=2T[/mathjaxinline]. In which string does the wave have higher velocity? </p>
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Velocity of Waves in Different Media - part b
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<p><b class="bfseries">(Part b)</b> A wave travels from string 1 with mass per unit length [mathjaxinline]\rho _{1} = 2\rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{1}=T[/mathjaxinline], to string 2 with [mathjaxinline]\rho _{2} = 4\rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{2}=2T[/mathjaxinline]. In what string does the wave have higher velocity? </p>
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Velocity of Waves in Different Media - part c
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<p><b class="bfseries">(Part c)</b> A wave travels from string 1 with mass per unit length [mathjaxinline]\rho _{1} = 5\rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{1}=T[/mathjaxinline], to string 2 with [mathjaxinline]\rho _{2} = 10\rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{2}=T[/mathjaxinline]. In what string does the wave have higher velocity? </p>
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<h2 class="hd hd-2 unit-title">L17Q2: Solving Transmission/Reflection of Wave at Boundary</h2>
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Boundary Conditions for Traveling Waves - part a
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<p><b class="bfseries">(Part a)</b> We stated two boundary conditions, which govern wave phenomena at boundaries. What is the physical meaning of each boundary condition? </p>
<p><b class="bfseries">(Part i)</b> [mathjaxinline]y_{L}(0^{-})=y_{R}(0^{+})[/mathjaxinline]: </p>
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<text> a) the sting doesn't move at the boundary</text>
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<text> b) the sum of the vertical forces due to the tensions in the two strings cancels at the boundary</text>
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<text> c) the string is continuous at the boundary</text>
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<text> d) the wave doesn't propagate through the boundary</text>
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<p><b class="bfseries">(Part ii)</b> [mathjaxinline]{T_{L}\dfrac {d}{dx}y_{L}(x)|}_{x=0}={T_{R}\dfrac {d}{dx}y_{R}(x)|}_{x=0}[/mathjaxinline]: </p>
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<text> a) the sting doesn't move at the boundary</text>
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<text> b) the sum of the vertical forces due to the tensions in the two strings cancels at the boundary</text>
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<text> c) the string is continuous at the boundary</text>
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<text> d) the wave doesn't propagate through the boundary</text>
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<h3 class="hd hd-3 problem-header" id="lect_11_05b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_11_05b-problem-progress" tabindex="-1">
Boundary Conditions for Traveling Waves - part b
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<p><b class="bfseries">(Part b)</b> When an incoming wave encounters a boundary, we can express the result in terms of three traveling wave solutions:<br/>&#8195;&#8195;&#8195;1. &#8194;[mathjaxinline]f_{i}(x,t) \equiv f_{i}(-k_{1}x + \omega t)[/mathjaxinline] <br/>&#8195;&#8195;&#8195;2. &#8194;[mathjaxinline]f_{r}(x,t) \equiv f_{r}(+k_{1}x + \omega t)[/mathjaxinline] <br/>&#8195;&#8195;&#8195;3. &#8194;[mathjaxinline]f_{t}(x,t) \equiv f_{t}(-k_{2}x + \omega t)[/mathjaxinline] <br/></p>
<p><b class="bfseries">(Part i)</b> What do the indices [mathjaxinline]i[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline] represent? </p>
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<text> a) the two different media and the boundary</text>
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<text> b) the two different media only (there is no index for the boundary)</text>
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<text> c) the "incident", "reflected", and "transmitted" waves</text>
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<p><b class="bfseries">(Part ii)</b> What do the indices 1 and 2 represent? </p>
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<text> a) the two different media and the boundary</text>
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<input type="radio" name="input_lect_11_05b_3_1" id="input_lect_11_05b_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_11_05b_3_1-choice_2-label" for="input_lect_11_05b_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_11_05b_3_1">
<text> b) the two different media only (there is no index for the boundary)</text>
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<text> c) the "incident", "reflected", and "transmitted" waves</text>
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<p><b class="bfseries">(Part iii)</b> Select ALL the variables that must remain the same between the two different media. </p>
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<text>a) [mathjaxinline]k[/mathjaxinline]</text>
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<input type="checkbox" name="input_lect_11_05b_4_1[]" id="input_lect_11_05b_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_11_05b_4_1-choice_1-label" for="input_lect_11_05b_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_11_05b_4_1">
<text>b) [mathjaxinline]\omega[/mathjaxinline]</text>
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<input type="checkbox" name="input_lect_11_05b_4_1[]" id="input_lect_11_05b_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_11_05b_4_1-choice_2-label" for="input_lect_11_05b_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_11_05b_4_1">
<text>c) [mathjaxinline]T[/mathjaxinline]</text>
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<input type="checkbox" name="input_lect_11_05b_4_1[]" id="input_lect_11_05b_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_11_05b_4_1-choice_3-label" for="input_lect_11_05b_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_11_05b_4_1">
<text>d) [mathjaxinline]\rho[/mathjaxinline]</text>
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<input type="checkbox" name="input_lect_11_05b_4_1[]" id="input_lect_11_05b_4_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="lect_11_05b_4_1-choice_4-label" for="input_lect_11_05b_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_11_05b_4_1">
<text>e) [mathjaxinline]L[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L17v2: Reflection and Transmission Coefficients</h2>
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<h2 class="hd hd-2 unit-title">L17Q3: Transmission/Reflection at a Boundary</h2>
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Transmission/Reflection Coefficients - part a
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<b class="bfseries">Hint: instead of using the form of [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]T[/mathjaxinline] presented in lecture, one should derive a more general case when [mathjaxinline]T_{1} \neq T_{2}[/mathjaxinline] and [mathjaxinline]\rho _{1} \neq \rho _{2}[/mathjaxinline], assuming that the strings are connected through a post such that they do not move horizontally.</b>
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In each of the following situations, determine the reflection coefficient [mathjaxinline]R[/mathjaxinline], the transmission coefficient [mathjaxinline]T[/mathjaxinline] (denoted [mathjaxinline]\tau[/mathjaxinline] in the preceding video), and the sign of the reflected wave relative to the incident wave. </p>
<p><b class="bfseries">(Part a)</b> A wave travels from string 1 with mass per unit length [mathjaxinline]\rho _{1} = \rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{1}=T[/mathjaxinline], to string 2 with the same density [mathjaxinline]\rho _{2} = \rho _{L}[/mathjaxinline] but a higher tension [mathjaxinline]T_{2}=2T[/mathjaxinline]. Enter your answers for [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]T[/mathjaxinline] as pure numbers. </p>
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<p style="display:inline">[mathjaxinline]R =[/mathjaxinline] </p>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
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enter <code> 2+3*2 </code> for 8 </td>
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enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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Transmission/Reflection Coefficients - part b
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<p><b class="bfseries">(Part b)</b> A wave travels from string 1 with mass per unit length [mathjaxinline]\rho _{1} = 2\rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{1}=T[/mathjaxinline], to string 2 with [mathjaxinline]\rho _{2} = 4\rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{2}=2T[/mathjaxinline]. Enter your answers for [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]T[/mathjaxinline] as pure numbers. </p>
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<p style="display:inline">[mathjaxinline]R =[/mathjaxinline] </p>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><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>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h3 class="hd hd-3 problem-header" id="lect_11_06c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_11_06c-problem-progress" tabindex="-1">
Transmission/Reflection Coefficients - part c
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<p><b class="bfseries">(Part c)</b> A wave travels from string 1 with mass per unit length [mathjaxinline]\rho _{1} = 5\rho _{L}[/mathjaxinline] and tension [mathjaxinline]T_{1}=T[/mathjaxinline], to string 2 with a higher density [mathjaxinline]\rho _{2} = 10\rho _{L}[/mathjaxinline] but the same tension [mathjaxinline]T_{2}=T[/mathjaxinline]. Enter your answers for [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]T[/mathjaxinline] as pure numbers. </p>
<p>
<p style="display:inline">[mathjaxinline]R =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]T =[/mathjaxinline] </p>
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<p style="display:inline">Sign of reflected wave:</p>
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<option value="a) same sign as incident wave"> a) same sign as incident wave</option>
<option value="b) flipped sign relative to incident wave"> b) flipped sign relative to incident wave</option>
<option value="c) there is no reflected wave"> c) there is no reflected wave</option>
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<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L17Q4: Properties of Transmission/Reflection</h2>
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Properties of Transmission/Reflection - part a
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Limits of [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]T[/mathjaxinline] as [mathjaxinline]\rho \rightarrow 0[/mathjaxinline] and [mathjaxinline]\rho \rightarrow \infty[/mathjaxinline]. </p>
<p>
Consider a wave in string 1 that is incident on string 2, <i class="itshape">where the tensions of the strings are equal.</i> What are the limits of the reflection and transmission coefficients, [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]T[/mathjaxinline], when the linear mass density of string 2 is either zero or infinite? </p>
<p><b class="bfseries">(Part a)</b> Determine the following values when [mathjaxinline]\rho _{2} \to \infty[/mathjaxinline]: </p>
<p>
<p style="display:inline">[mathjaxinline]R =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]T =[/mathjaxinline] </p>
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<p style="display:inline">Sign of reflected wave:</p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Properties of Transmission/Reflection - part b
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<p><b class="bfseries">(Part b)</b> Determine the following values when [mathjaxinline]\rho _{2} \to 0[/mathjaxinline]: </p>
<p>
<p style="display:inline">[mathjaxinline]R =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]T =[/mathjaxinline] </p>
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<p style="display:inline">Sign of reflected wave:</p>
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<option value="a) same sign as incident wave"> a) same sign as incident wave</option>
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<option value="c) there is no reflected wave"> c) there is no reflected wave</option>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
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<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L17Q5: Reflection at a Boundary: Fixed vs. Open [WITH SIMULATION]</h2>
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Reflection at a Boundary: Fixed vs. Open
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In the figures below, a wave is incident on a boundary from the left, and there is no string to the right of the boundary. The behavior of the total waveform (purple line) can be modeled as the superposition of an incident wave, [mathjaxinline]f_{i}(x,t)[/mathjaxinline] (red dot-dashed line), and a reflected wave that is counter-propagating, [mathjaxinline]f_{r}(x,t)[/mathjaxinline] (blue-dashed line). In each case, plots (a), (b), and (c) show the situation before, during, and after the wave hits the boundary, respectively. </p>
<p>
Depending on the boundary condition (open vs. fixed), the counter-propagating wave should be "flipped" with respect to the incident wave. For each of the following series of plots, indicate whether the boundary is "open" or "fixed." </p>
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<h4 onclick="hideshow(this);" style="margin: 0px">More on waves in the "non-existent" string<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">The text says that there is "no string to the right of the boundary" but the plots show a wave there. This is not a "real" wave, but rather a visualization trick to help understand what happens at the boundary. By projecting the incident wave forward into the "non-string" region, or extrapolating the reflected wave back, one can understand what happens at the boundary by thinking of the situation as two waves passing each other. </div>
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<p><b class="bfseries">(Part a)</b> Series 1: indicate whether the boundary is "open" or "fixed." </p>
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<p><b class="bfseries">(Part a)</b> Series 2: indicate whether the boundary is "open" or "fixed." </p>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=reflection_boundary_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/70225e7425d578aef2228cd82d3c6a63/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_reflection_boundary_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L17v5: Reflection of Waves at Boundary Between Different Media I [DEMO]</h2>
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<h2 class="hd hd-2 unit-title">L17v6: Reflection of Waves at Boundary Between Different Media II [DEMO]</h2>
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