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<h2 class="hd hd-2 unit-title">Introduction Sound Waves</h2>
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Next, we introduce <i class="itshape">longitudinal</i> waves, i.e., waves where the oscillation is in the same direction as the velocity of the waves—the primary example is <i class="itshape">sound</i>. In order to derive a wave equation, we consider how the pressure and density of molecules in an air column are related,. </p>
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<h2 class="hd hd-2 unit-title">L18v1: Longitudinal Waves</h2>
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<h3 class="hd hd-2">L18v1: Longitudinal Waves</h3>
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<h2 class="hd hd-2 unit-title">L18Q1:Sound Wave Definitions</h2>
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Sound Wave Definitions
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Choose the correct definition for the following functions. </p>
<p><b class="bfseries">(Part a)</b> [mathjaxinline]\psi (x)[/mathjaxinline] </p>
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<text> a) the amplitude of the sound wave</text>
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<text> b) the average position of a volume element of air</text>
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<text> c) the position of the boundary of a volume element of air</text>
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<text> d) the pressure of a volume element of air</text>
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<text> e) the change in pressure of a volume element of air, relative to ambient pressure</text>
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<p><b class="bfseries">(Part b)</b> [mathjaxinline]\psi _{p}(x)[/mathjaxinline] </p>
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<text> a) the amplitude of the sound wave</text>
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<text> b) the average position of a volume element of air</text>
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<text> c) the position of the boundary of a volume element of air</text>
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<text> d) the pressure of a volume element of air</text>
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<text> e) the change in pressure of a volume element of air, relative to ambient pressure</text>
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<h2 class="hd hd-2 unit-title">L18Q2: What is Pressure?</h2>
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What is Pressure?
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All of the waves that we have encountered so far in this course are composed of moving particles, or infinitesimal units of a continuous medium, which respond to restoring forces. After all, we learned that restoring forces produce oscillatory motion in the very first lesson! </p>
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Sound waves, consisting of air molecules, are no different! Differences in pressure are the restoring forces that arise from the displacement of air molecules. What is the microscopic origin of pressure? </p>
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<text> a) the speed of molecules</text>
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<text> b) the momentum of molecules</text>
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<text> c) the force of molecules on each other</text>
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<h2 class="hd hd-2 unit-title">L18v2: Longitudinal Waves - Pressure and Ideal Gas Law</h2>
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<h2 class="hd hd-2 unit-title">Sound Waves and Pressure: Review I</h2>
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A difference in pressure will cause displacement of gas, by Newton's second law. The relation between pressure difference, [mathjaxinline]\Delta P[/mathjaxinline], and change in volume, [mathjaxinline]\Delta V[/mathjaxinline], will enable us to relate the functions [mathjaxinline]\psi (x)[/mathjaxinline] and [mathjaxinline]\psi _{p}(x)[/mathjaxinline]. This is the objective of our derivation. </p><p>
So far, we began by deriving expressions for [mathjaxinline]\Delta V[/mathjaxinline] and [mathjaxinline]\Delta P[/mathjaxinline]: </p><table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000003"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta V[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = A\left( \psi (x + \Delta x) - \psi (x) \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000004"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
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[mathjaxinline]\displaystyle = A\frac{\partial }{\partial x} \psi (x) \Delta x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
having used the definition [mathjaxinline]\dfrac {\partial }{\partial x} \psi (x) = \dfrac {\psi (x + \Delta x) - \psi (x)}{\Delta x}[/mathjaxinline]. </p><p>
Similarly, we noted </p><table id="a0000000005" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000006"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta P[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\psi _{p}(x + \Delta x) + \psi _{p}(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000007"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\frac{\partial }{\partial x}\psi _{p}(x) \Delta x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
having used the definition [mathjaxinline]\dfrac {\partial }{\partial x} \psi _{p}(x) = \dfrac {\psi _{p}(x + \Delta x) - \psi _{p}(x)}{\Delta x}[/mathjaxinline]. </p><p>
It is essential to note that [mathjaxinline]\Delta P[/mathjaxinline] is a difference in pressure between two boundaries, located at [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]x+\Delta x[/mathjaxinline], whereas [mathjaxinline]\psi _{p}[/mathjaxinline] is the change in pressure at a given location relative to ambient pressure. </p>
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<h2 class="hd hd-2 unit-title">L18Q3: Understanding Different Gas Laws: Part I</h2>
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Pressure and Volume in Ideal Gas Law - part a
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Now, let's explore two relations between pressure and volume, with the ultimate goal of relating [mathjaxinline]\psi (x)[/mathjaxinline] and [mathjaxinline]\psi _{p}(x)[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> One relation between [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]V[/mathjaxinline] is given by the <b class="bfseries">ideal gas law</b>, which (for constant temperature) states that </p>
<table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]PV=C[/mathjax]</td>
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where [mathjaxinline]C[/mathjaxinline] is a constant related to the temperature and the number of gas molecules. Using the ideal gas law, determine the change in pressure [mathjaxinline]\psi _{p}[/mathjaxinline] resulting from a change in volume [mathjaxinline]\Delta V[/mathjaxinline]. To do this, define the initial pressure and volume of the system as [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]V[/mathjaxinline], and the final pressure and volume of the system as [mathjaxinline]P + \psi _{p}[/mathjaxinline] and [mathjaxinline]V + \Delta V[/mathjaxinline]. Solve for [mathjaxinline]\psi _{p}[/mathjaxinline] in terms of <code>P</code>, <code>V</code>, and <code>DeltaV</code> for [mathjaxinline]\Delta V[/mathjaxinline]. (Your answer as shown in the grey box will look more correct if you put <code>DeltaV</code> in parentheses, but not doing this will not affect the answer checking.) </p>
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<p style="display:inline">[mathjaxinline]\psi _{p} =[/mathjaxinline] </p>
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Pressure and Volume in Adiabatic Gas Law - part b
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<p><b class="bfseries">(Part b)</b> Another relation between [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]V[/mathjaxinline] occurs for an <b class="bfseries">adiabatic process</b>, where </p>
<table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]PV^{\gamma }=C[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
and, again, [mathjaxinline]C[/mathjaxinline] and [mathjaxinline]\gamma[/mathjaxinline] are constants. Using this relation, determine the change in pressure [mathjaxinline]\psi _{p}[/mathjaxinline] resulting from a change in volume [mathjaxinline]\Delta V[/mathjaxinline]. As before, define the initial pressure and volume of the system as [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]V[/mathjaxinline], and the final pressure and volume of the system as [mathjaxinline]P + \psi _{p}[/mathjaxinline] and [mathjaxinline]V + \Delta V[/mathjaxinline]. Solve for [mathjaxinline]\psi _{p}[/mathjaxinline] in terms of <code>P</code>, <code>V</code>, <code>DeltaV</code> for [mathjaxinline]\Delta V[/mathjaxinline], and <code>gamma</code> for [mathjaxinline]\gamma[/mathjaxinline]. </p>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</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>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<code>e, pi</code>
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enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</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>
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<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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<p style="display:inline">[mathjaxinline]\psi _{p} =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L18Q4: Understanding Different Gas Laws: Part II</h2>
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Pressure in Small Volume Limit - part a
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Using your results for the ideal gas law in the previous problem, derive expressions for [mathjaxinline]\psi _{p}[/mathjaxinline] in the limit [mathjaxinline]\Delta V \ll V[/mathjaxinline]. Express your answer in terms of <code>P</code>, <code>V</code>, and <code>DeltaV</code> for [mathjaxinline]\Delta V[/mathjaxinline]. (Your answer as shown in the grey box will look more correct if you put <code>DeltaV</code> in parentheses, but not doing this will not affect the answer checking.) </p>
<p><b class="bfseries">(Part a)</b> Using the relation given by the <i class="itshape">ideal gas law</i>: [mathjaxinline]PV=C[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]\psi _{p} =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Pressure in Small Volume Limit - part b
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<p><b class="bfseries">(Part b)</b> Using the relation for an <i class="itshape">adiabatic process</i>: [mathjaxinline]PV^{\gamma }=C[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\psi _{p} =[/mathjaxinline] </p>
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enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L18v3: Derivation of Wave Equation for Pressure in a Tube</h2>
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<h2 class="hd hd-2 unit-title">L18Q5: Relationship Between Displacement and Pressure</h2>
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Relationship Between Displacement and Pressure - part a
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<p>
We have derived the following relation between [mathjaxinline]\psi (x)[/mathjaxinline] and [mathjaxinline]\psi _{p}(x)[/mathjaxinline]: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\psi _{p}(x) = -\gamma P_{0} \frac{\partial }{\partial x} \psi (x)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
where [mathjaxinline]\gamma[/mathjaxinline] and [mathjaxinline]P_{0}[/mathjaxinline] are constants. </p>
<p><b class="bfseries">(Part a)</b> Consider the following plot, which depicts the displacement of air as a function of position, [mathjaxinline]\psi (x)[/mathjaxinline], in a pipe of length [mathjaxinline]L[/mathjaxinline]. </p>
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Identify the graph below that correctly shows [mathjaxinline]\psi _{p}(x)[/mathjaxinline]. (Units on the vertical scale are normalized to 1.) </p>
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Relationship Between Displacement and Pressure - part b
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<p><b class="bfseries">(Part b)</b> Consider the following plot, which depicts the change in pressure as a function of position, [mathjaxinline]\psi _{p}(x)[/mathjaxinline], in a pipe of length [mathjaxinline]L[/mathjaxinline]. </p>
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Identify the graph below that correctly shows [mathjaxinline]\psi (x)[/mathjaxinline]. (Units on the vertical scale are normalized to 1.) </p>
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<h2 class="hd hd-2 unit-title">Sound Waves and Pressure: Review II</h2>
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<p>
Let's recap what we have done. Previously, we had derived the relation [mathjaxinline]\psi (x)[/mathjaxinline] and [mathjaxinline]\psi _{p}(x)[/mathjaxinline] (where [mathjaxinline]\gamma[/mathjaxinline] and [mathjaxinline]P_{0}[/mathjaxinline] are constants): </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\psi _{p}(x) = -\gamma P_{0} \frac{\partial }{\partial x} \psi (x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
To carry out the derivation and obtain an equation of motion (i.e., a relation between [mathjaxinline]\ddot{\psi }(x)[/mathjaxinline] and [mathjaxinline]\psi (x)[/mathjaxinline]), we must now use a force equation: </p><table id="a0000000003" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000004"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle F[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =A\Delta P[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000005"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -A \frac{\partial }{\partial x} \psi _{p}(x) \Delta x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000006"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = A\gamma P_{0} \Delta x \frac{\partial ^{2}}{\partial x^{2}} \psi (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
where we used [mathjaxinline]\Delta P = -\dfrac {\partial }{\partial x} \psi _{p}(x) \Delta x[/mathjaxinline], and the relation [mathjaxinline]\psi _{p}(x) = -\gamma P_{0} \dfrac {\partial }{\partial x} \psi (x)[/mathjaxinline]. </p><p>
Of course, Newton's 2nd law relates force and acceleration, therefore, using [mathjaxinline]\Delta m = \rho A \Delta x[/mathjaxinline], the sum of the forces is also equal to: </p><table id="a0000000007" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000008"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle F[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\Delta m \ddot{\psi }(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \rho A \Delta x \ddot{\psi }(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
Now, we are in a position to combine everything! </p><table id="a0000000010" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000011"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \rho A \Delta x \ddot{\psi }(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = A\gamma P_{0} \Delta x \frac{\partial ^{2}}{\partial x^{2}} \psi (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000012"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \ddot{\psi }(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{\gamma P_{0}}{\rho } \frac{\partial ^{2}}{\partial x^{2}} \psi (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
This is the equation of motion that governs the displacement of the gas. It is a wave equation! </p>
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