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      <text>25.2.1.1 Wave Equation</text>
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      <text><![CDATA[<p><span style="font-size: 0.9375rem;"></span></p>
<h3>Wave Equation</h3>
The simplest situation to give rise to the one-dimensional wave equation is the motion of a stretched string - specifically the <strong style="font-size: 0.9375rem;">transverse vibrations</strong><span style="font-size: 0.9375rem;"> of a string such as the string of a musical instrument. Assume that a string is placed along the \(x\)-axis, is stretched and then fixed at ends \(x=0\) and \(x=L\); it is then deflected and at some instant, which we call \(t=0\), is released and allowed to vibrate. The quantity of interest is the deflection \(u\) of the string at any point \(x\), \(0\le x\le L\), and at any time \(t&gt;0\). We write \(u=u(x,t)\). The following figure shows a possible displacement of the string at a fixed time \(t\).</span><br>

<p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/workbook25fig1-1.svg" alt="Image of a wave along a string" width="381" height="97" class="img-responsive atto_image_button_text-bottom">
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</figure>
<p></p>

<p>Subject to various assumptions \(\cdots\)</p>
<ol>
    <li> damping forces such as air resistance are negligible </li>
    <li> the weight of the string is negligible </li>
    <li> the tension in the string is tangential to the curve of the string at any point </li>
    <li> the string performs small transverse oscillations i.e. every particle of the string moves strictly vertically and such that its deflection and slope as every point on the string is small. </li>
</ol>
<p>\(\cdots\) it can be shown, by applying Newton’s law of motion to a small segment of the string, that \(u\) satisfies the PDE </p>

<table width="100%">
    <tbody>
        <tr>
            <td>\(\qquad \dfrac{\partial^2u}{\partial t^2}= c^2\dfrac{\partial^2u}{\partial x^2}\)</td>
            <td>\((1)\)</td>
        </tr>
    </tbody>
</table>

where \(c^2=\dfrac{T}{\rho}\), \(\rho\) being the mass per unit length of the string and \(T\) being the (constant) horizontal component of the tension in the string. To determine \(u(x,t)\) uniquely, we must also know
<p></p>
<ol>
    <li> the initial definition of the string at the time \(t=0\) at which it is released</li>
    <li> the initial velocity of the string. </li>
</ol>]]></text>
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