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      <text>25.2.1.2 Wave Equation</text>
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      <text><![CDATA[<p>Thus we must be given <strong>initial conditions</strong> </p>
<p>\(\ u\ (x,0)\,=f(x)\qquad 0\le x\le L\)\(\qquad\)(initial position)</p>
<p> \(\dfrac{\partial u}{\partial t}(x,0)=g(x)\qquad 0\le x\le L\)\(\qquad\)(initial velocity)</p>
<p>where \(f(x)\) and \(g(x)\) are known.</p>
<br> These two initial conditions are in addition to the two <strong>boundary conditions</strong>
<p>\(\qquad u(0,t)=u(L,t)=0\) \(\qquad\) for \(t\ge 0\)</p> which indicate that the string is fixed at each end. In Example 2 discussed in Section 25.1 we had
<p>\(\qquad f(x)=u_0\ \sin\left(\dfrac{\pi x}{\ell}\right)\)</p>
<p> \(\qquad g(x)=0\ \ \ \ \) (string initially at rest).</p> The PDE (1) is the (undamped) wave equation. We will discuss solutions of it for various initial conditions later. More complicated forms of the wave equation would arise if some of the assumptions
were modified. For example:
<ol>

    <ol>
        <li>\(\dfrac{\partial^2u}{\partial t^2}=c^2\dfrac{\partial^2u}{\partial x^2}-g \qquad\) if the weight of the string was allowed for,</li>
        <li>\(\dfrac{\partial^2u}{\partial t^2}=c^2\dfrac{\partial^2u}{\partial x^2}-\alpha \dfrac{\partial u}{\partial t} \qquad\) if a damping force proportional to the velocity of the string (with damping constant \(\alpha\)) was included.</li>
    </ol>

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<p>Equation \((1)\) is referred to as the one-dimensional wave equation because only one space variable, \(x\), is present. The two-dimensional (undamped) wave equation is, in Cartesian coordinates, </p>

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            <td>\(\qquad \dfrac{\partial^2u}{\partial t^2}=c^2\left(\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}\right)\)</td>
            <td>\((2)\)</td>
        </tr>
    </tbody>
</table>

This arises for example when we model the transverse vibrations of a membrane. See the following Figure \((a)\) and \((b)\). Here \(u(x,y,t)\) is the definition of a point \((x,y)\) on the membrane at time \(t\). Again, a boundary condition must be specified: commonly
<p> \(\qquad u=0\qquad t\ge 0\)</p> on the boundary of the membrane, if this is fixed (clamped). Also initial conditions must be given
<p>\(\qquad u\ (x,y,0)=f(x,y)\qquad\) (initial position)</p>
<p>\(\qquad \dfrac{\partial u}{\partial t} (x,y,0)=g(x,y)\qquad\) (initial velocity)</p>
For a <strong>circular</strong> membrane, such as a drumhead, polar coordinates defined by \(x=r\cos\theta,\ y=r\sin\theta\) would be more convenient than Cartesian. In this case \((2)\) becomes
<p>\(\qquad \dfrac{\partial^2u}{\partial t^2}=c^2\left(\dfrac{\partial^2u}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial u}{\partial r}+\dfrac{1}{r^2}\dfrac{\partial^2u}{\partial\theta^2}\right)\ \qquad 0\le r\le R \) \(,\) \( 0\le \theta\le 2\pi\)</p> for a circular
membrane of radius \(R.\)
<p></p>
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    <img src="@@PLUGINFILE@@/workbook25fig2-1.svg" alt="(a) Image of a rectangular membrane, (b)Image of a circular membrane of radius R" width="461" height="143" class="img-responsive atto_image_button_text-bottom">
</figure>
<p></p>]]></text>
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