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We just derived a wave equation for the longitudinal displacement of gas molecules in a medium. We will also show that the wave equation will give us the velocity of waves in the medium. </p><p>
However, we are left with a choice. Different relations between pressure and density will yield different wave velocities. This gives us an opportunity to explore different relations (gas laws) to determine which gives the correct speed of sound that is observed. How very scientific! </p>
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<h2 class="hd hd-2 unit-title">L19v1: Degrees of Freedom for Ideal Gas and Speed of Sound</h2>
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<h2 class="hd hd-2 unit-title">L19Q1: Degrees of Freedom in Different Systems</h2>
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The adiabatic index [mathjaxinline]\gamma[/mathjaxinline] is related to the number of "degrees of freedom" of a gas 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]\gamma = \frac{\alpha + 1}{\alpha }[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]\alpha[/mathjaxinline] is equal to one half the number of quadratic terms in the Hamiltonian of the system. </p><p>
Since we are not studying Hamiltonian mechanics in this course, suffice it to say that each of the non-vibrational "degrees of freedom" contributes one quadratic term in the Hamiltonian, while each of the vibrational "degrees of freedom" contributes two quadratic terms to the Hamiltonian (equivalent to potential and kinetic energies). We will clarify what "degrees of freedom" are below. </p><p>
Degrees of freedom (DOF) are ways in which a molecule of gas can translate, rotate, and vibrate. For the following, let's just consider a simple linear molecule consisting of a "chain" of atoms: </p><ul class="itemize"><li><p><b class="bfseries">translational (DOF=3):</b> At any temperature, the molecule can move along all three axes of 3-dimensional space. </p></li><li><p><b class="bfseries">rotational (DOF=2):</b> At intermediate temperatures, the molecule can spin along two axes (excluding the axis along which the atoms are bound). Note, a single atom will not have rotational degrees of freedom, so this holds for systems with [mathjaxinline]N \geq 2[/mathjaxinline], where [mathjaxinline]N[/mathjaxinline] is the number of atoms in the molecule. </p></li><li><p><b class="bfseries">vibrational (DOF=3N-5):</b> At high temperatures, the atoms in the molecule vibrate. The DOF related to vibration can be thought of in the following way. Considering that each atom can move in three dimensions, there should be a TOTAL of [mathjaxinline]3N[/mathjaxinline] degrees of freedom (where [mathjaxinline]N[/mathjaxinline] is the number of atoms in the molecule). However, [mathjaxinline]3[/mathjaxinline] of those involve translation and [mathjaxinline]2[/mathjaxinline] involve rotation. So, the total remaining for vibration is [mathjaxinline]3N-5[/mathjaxinline]. </p></li></ul><p>
Since each vibrational degree of freedom contributes two quadratic terms to the Hamiltonian, we need to multiply the [mathjaxinline]3N-5[/mathjaxinline] by two when calculating [mathjaxinline]\alpha[/mathjaxinline] using one half the total number of quadratic modes: </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]\alpha =\dfrac {1}{2}\left(3 + 2 + 2(3N-5)\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Degrees of freedom in different systems - part a
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<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]\alpha[/mathjaxinline] for a monatomic gas at VERY LOW temperature? </p>
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<p style="display:inline">[mathjaxinline]\alpha =[/mathjaxinline]</p>
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Degrees of freedom in different systems - part b
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<p><b class="bfseries">(Part b)</b> What is [mathjaxinline]\alpha[/mathjaxinline] for a diatomic gas at VERY HIGH temperature? </p>
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<p style="display:inline">[mathjaxinline]\alpha =[/mathjaxinline]</p>
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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>
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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>
</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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<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">L19Q2: Speed of Sound and Pressure</h2>
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Speed of Sound and Pressure
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<p>
The speed of sound in a medium with [mathjaxinline]\gamma[/mathjaxinline], pressure [mathjaxinline]P_{0}[/mathjaxinline], and density [mathjaxinline]\rho[/mathjaxinline] is: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]v_{p}=\sqrt {\frac{\gamma P_{0}}{\rho }}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
At sea level and room temperature (T=[mathjaxinline]20^{\circ }[/mathjaxinline] C), the pressure of air is [mathjaxinline]P_{0}=101.33\times 10^{3}\, \mathrm{N/m^{2}}[/mathjaxinline] and the density is [mathjaxinline]\rho =1.2\, \mathrm{kg/m^{3}}[/mathjaxinline]. BUT, on top of Mount Everest (elevation 8.848 km), the pressure of air is approximately [mathjaxinline]0.30P_{0}[/mathjaxinline] and the air density is approximately [mathjaxinline]0.35\rho[/mathjaxinline]. </p>
<p>
Determine the speed of sound in air on top of Mount Everest, in terms of the speed of sound at sea level. </p>
<p>
<p style="display:inline">[mathjaxinline]v_{\mathrm{Everest}} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]v_{\mathrm{sea-level}}[/mathjaxinline] </p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
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<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">L19Q3: Speed of Sound in Other Media</h2>
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Speed of sound in other media
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<p>
Let's calculate the speed of sound in [mathjaxinline]\mathrm{CO_{2}}[/mathjaxinline] at room temperature. The pressure of the gas is [mathjaxinline]P_{0}=101.33\times 10^{3}\, \mathrm{N/m^{2}}[/mathjaxinline] and the density is [mathjaxinline]\rho =1.98\, \mathrm{kg/m^{3}}[/mathjaxinline]. Also, use what you know about the degrees of freedom of the system to calculate the speed of sound [mathjaxinline]v_{p}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]v_{p} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\mathrm{m/s}[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
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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>
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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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<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>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
</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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<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>
</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>
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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>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<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">L19v4: Misconception - Shape of a Wave Exiting a Tube</h2>
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<h2 class="hd hd-2 unit-title">L19v5: Summary and Boundary Conditions</h2>
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<h2 class="hd hd-2 unit-title">L19Q4: Visualizing Wave Propagation in Tubes</h2>
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Visualizing wave propagation in tubes
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Consider the following figures, which depict the normal modes of gasses in pipes with various combinations of open and closed ends. Each graph may represent [mathjaxinline]\psi (x)[/mathjaxinline] or [mathjaxinline]\psi _{p}(x)[/mathjaxinline]. For each of the following problems, you will match the normal mode to the situation that is described. </p>
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<p><b class="bfseries">(Part a)</b> Plot of [mathjaxinline]\psi _{p}(x)[/mathjaxinline], normal mode [mathjaxinline]m=2[/mathjaxinline], for a pipe with left end open and right end closed. </p>
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<p><b class="bfseries">(Part b)</b> Plot of [mathjaxinline]\psi (x)[/mathjaxinline], normal mode [mathjaxinline]m=1[/mathjaxinline], for a pipe with both ends open. </p>
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<p><b class="bfseries">(Part c)</b> Plot of [mathjaxinline]\psi _{p}(x)[/mathjaxinline], normal mode [mathjaxinline]m=3[/mathjaxinline], for a pipe with both ends closed. </p>
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