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      <text>14.2.1.3 Engineering example</text>
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      <text><![CDATA[<hr>
<h4 class="HELM_example">Engineering Example</h4><br><br>
<h5>Sonic boom</h5><br>
<h6>Introduction</h6>

<p>Impulsive signals are described by their peak amplitudes and their duration. Another quantity of interest is the total energy of the impulse. The effect of a blast wave from an explosion on structures, for example, is related to its total energy. This
    Example looks at the calculation of the energy on a sonic boom. Sonic booms are caused when an aircraft travels faster than the speed of sound in air. An idealized sonic-boom pressure waveform is shown in the figure below where the instantaneous sound
    pressure \(p(t)\) is plotted versus time \(t\). This wave type is often called an N-wave because it resembles the shape of the letter N. The energy in a sound wave is proportional to the square of the sound pressure.</p>
<img src="@@PLUGINFILE@@/14-2-fig-6-Sonic.svg" alt=" An idealized sonic-boom pressure waveform (an N-wave)." width="389" height="125" class="img-responsive atto_image_button_text-bottom"><br><br>

    <h6>Problem in words</h6>

    <p>Calculate the energy in an ideal N-wave sonic boom in terms of its peak pressure, its duration and the density and sound speed in air.</p><br>

    <h6>Mathematical statement of problem</h6>

    <p>Represent the positive peak pressure by \(P_0\) and the duration by \(T\). The total acoustic energy \(E\) carried across unit area normal to the sonic-boom wave front during time \(T\) is defined by \[E ~=~ &lt;\! p(t)^2\!&gt; T /\rho c\qquad \text{(1)}\]
        where \(\rho\) is the air density, \(c\) the speed of sound and the time average of \([p(t)]^2\) is \[ &lt;\!p(t)^2\!&gt; ~=~ \frac{1}{T}\int_0^T p(t)^2\,\mathrm{d}t\qquad \text{(2)}\]</p>

    <ol class="HELM_parts">
        <li>Find an appropriate expression for \(p(t)\).</li>
        <li>Hence show that \(E\) can be expressed in terms of \(P_0,\ T,\ \rho\) and \(c\) as \(\displaystyle E ~=~\frac{TP_0^2}{3\rho c}\).</li>
    </ol><br>

    <h6>Mathematical analysis</h6><br>

    <ol class="HELM_parts">
        <li>
            <p>The interval of integration needed to compute (2) is \([0, T].\) Therefore it is necessary to find an expression for \(p(t)\) only in this interval. The figure above shows that, in this interval, the dependence of the sound pressure \(p\)
                on the variable \(t\) is linear, i.e. \(p(t) = a t + b.\)</p>
            <p>From the figure above also \(p(0) = P_0\) and \(p(T) = - P_0\). The constants \(a\) and \(b\) are determined from these conditions.</p>
            <p>At \(t = 0,\ a \times 0 + b = P_0\) implies that \(b = P_0\).</p>
            <p>At \(t = T,\ a \times T + b = - P_0\) implies that \(a = -2P_0/T.\)</p>
            <p>Consequently, the sound pressure in the interval \([0, T]\) may be written \(\displaystyle p(t) = \frac{-2P_0}{T}\ t + P_0.\)</p>
        </li>
        <li>This expression for \(p(t)\) may be used to compute the integral (2) \[\begin{aligned} \frac{1}{T}\int_0^T p(t)^2 \,\mathrm{d}t&amp;~=\frac{1}{T}\int_0^T\ \left(\frac{-2P_0}{T}\ t+P_0\right)^2 \,\mathrm{d}t =\frac{1}{T}\ \int_0^T\ \left(\frac{4P_0^2}{T^2}\ t^2-\frac{4P_0^2}{T}\
            t+P_0^2\right) \,\mathrm{d}t\\ \\ &amp;~=\frac{1}{T}\left[\frac{4P_0^2}{3T^2}\ t^3-\frac{2P_0^2}{T}\ t^2+P_0^2 t\right]_0^T\\ \\ &amp;~=\frac{P_0^2}{T}\left(\frac{4}{3T^2}\ T^3-\frac{2}{T}\ T^2+T\right)-0=P_0^2/3.\end{aligned}\] Hence, from Equation
            (1), the total acoustic energy \(E\) carried across unit area normal to the sonic-boom wave front during time \(T\) is \(E = \dfrac{TP_0^2}{3\rho c}.\)</li>
    </ol><br>

    <h6>Interpretation</h6>

    <p>The energy in an N-wave is given by a third of the sound intensity corresponding to the peak pressure multiplied by the duration.</p>]]></text>
<file name="14-2-fig-6-Sonic.svg" path="/" encoding="base64">[encoded data, 25864 characters]</file>
<file name="14-2-fig-6-sonicboom.svg" path="/" encoding="base64">[encoded data, 27620 characters]</file>
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