2 Engineering Example 2

2.1 Sonic boom

Introduction

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 Figure 6 where the instantaneous sound pressure $p\left(t\right)$ 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.

Figure 6 :

Problem in words

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.

Mathematical statement of problem

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

$E=<p{\left(t\right)}^2>T∕\rho c$ (1)

where $\rho$ is the air density, $c$ the speed of sound and the time average of ${\left[p\left(t\right)\right]}^2$ is

$<p{\left(t\right)}^2>=\frac{1}{T}{\int \; <!--nolimits\; -->}_0^{T}p{\left(t\right)}^{2}dt$ (2)

1. Find an appropriate expression for $p\left(t\right)$ .
2. Hence show that $E$ can be expressed in terms of $P_0,T,\rho$ and $c$ as    $E=\frac{T{P}_0^2}{3\rho c}.$

Mathematical analysis

1. The interval of integration needed to compute (2) is $\left[0,T\right].$ Therefore it is necessary to find an expression for $p\left(t\right)$ only in this interval. Figure 6 shows that, in this interval, the dependence of the sound pressure $p$ on the variable $t$ is linear, i.e. $p\left(t\right)=at+b.$

   From Figure 6 also $p\left(0\right)=P_0$ and $p\left(T\right)=-P_0$ . The constants $a$ and $b$ are determined from these conditions.

   At $t=0,a\times 0+b=P_0$ implies that $b=P_0$ .

   At $t=T,a\times T+b=-P_0$ implies that $a=-2P_0∕T.$

   Consequently, the sound pressure in the interval $\left[0,T\right]$ may be written    $p\left(t\right)=\frac{-2P_0}{T}t+P_0.$

2. This expression for $p\left(t\right)$ may be used to compute the integral (2) $$\begin{array}{rcll}\frac{1}{T}{\int \; <!--nolimits\; -->}_0^{T}p{\left(t\right)}^{2}dt& =& \frac{1}{T}{\int \; <!--nolimits\; -->}_0^T{\left(\frac{-2P_0}{T}t+P_0\right)}^{2}dt=\frac{1}{T}{\int \; <!--nolimits\; -->}_0^T\left(\frac{4P_0^2}{T^2}{t}^2-\frac{4P_0^2}{T}t+P_0^2\right)dt& \\ & & & \\ & =& \frac{1}{T}{\left[\frac{4P_0^2}{3T^2}{t}^3-\frac{2P_0^2}{T}{t}^2+P_0^{2}t\right]}_0^T& \\ & & & \\ & =& \frac{P_0^2}{T}\left(\frac{4}{3T^2}{T}^3-\frac{2}{T}{T}^2+T\right)-0=P_0^2∕3.& \end{array}$$

   Hence, from Equation (1), the total acoustic energy $E$ carried across unit area normal to the sonic-boom wave front during time $T$ is $E=\frac{T{P}_0^2}{3\rho c}.$

Interpretation

The energy in an N-wave is given by a third of the sound intensity corresponding to the peak pressure multiplied by the duration.

Exercises

1. Calculate the mean value of the given functions across the specified interval.
   1. $f\left(t\right)=1+t$ across $\left[0,2\right]$
   2. $f\left(x\right)=2x-1$ across $\left[-1,1\right]$
   3. $f\left(t\right)=t^2$ across $\left[0,1\right]$
   4. $f\left(t\right)=t^2$ across $\left[0,2\right]$
   5. $f\left(z\right)=z^2+z$ across $\left[1,3\right]$
2. Calculate the mean value of the given functions over the specified interval.
   1. $f\left(x\right)=x^3$ across $\left[1,3\right]$
   2. $f\left(x\right)=\frac{1}{x}$ across $\left[1,2\right]$
   3. $f\left(t\right)=\sqrt{t}$ across $\left[0,2\right]$
   4. $f\left(z\right)=z^3-1$ across $\left[-1,1\right]$
3. Calculate the mean value of the following:
   1. $f\left(t\right)=sint$ across $\left[0,\frac{\pi }{2}\right]$
   2. $f\left(t\right)=sint$ across $\left[0,\pi \right]$
   3. $f\left(t\right)=sin\omega t$ across $\left[0,\pi \right]$
   4. $f\left(t\right)=cost$ across $\left[0,\frac{\pi }{2}\right]$
   5. $f\left(t\right)=cost$ across $\left[0,\pi \right]$
   6. $f\left(t\right)=cos\omega t$ across $\left[0,\pi \right]$
   7. $f\left(t\right)=sin\omega t+cos\omega t$ across $\left[0,1\right]$
4. Calculate the mean value of the following functions:
   1. $f\left(t\right)=\sqrt{t+1}$ across $\left[0,3\right]$
   2. $f\left(t\right)=e^t$ across $\left[-1,1\right]$
   3. $f\left(t\right)=1+e^t$ across $\left[-1,1\right]$

Answer