Question

At a distance of $\mathbf{10}\mathbf{\text{\hspace{0.17em}km}}$, a$\mathbf{100}\mathbf{\text{\hspace{0.17em}Hz}}$horn, assumed to be an isotropic point source, is barely audible. At what distance would it begin to cause pain?

Short answer

The distance where the sound intensity begins to cause pain is $1\text{\hspace{0.17em}cm}$.

Step-by-step solution

The given data

1. Initial distance${\mathrm{r}}_2=10\text{\hspace{0.17em}kmor}10000\text{\hspace{0.17em}m}$
2. Frequency is$\mathrm{f}=100\text{\hspace{0.17em}Hz}$

Understanding the concept of sound intensity

We use the concept of sound intensity. Using the equation of sound intensity related to power and area, we can write the equations of intensities for ${\mathbf{r}}_{\mathbf{1}}$and${\mathbf{r}}_{\mathbf{2}}$. We are given,${\mathbf{r}}_{\mathbf{2}}$ so by taking the ratio, we find the distance${\mathbf{r}}_{\mathbf{1}}$.

Formulae:

The intensity of a wave,

$\mathbf{I}\mathbf{=}\frac{\mathbf{P}}{\mathbf{A}}$ …(i)

The surface area of the sphere,

$\mathbf{A}\mathbf{=}\mathbf{4}{\mathbf{\pi r}}^{\mathbf{2}}$ …(ii)

Calculation of the distance

We know sound waves travel spherically, so the area at which waves is focused is given using equation (ii).The power of the source is the same for any distance. Hence, using equation (i), we can write the intensity at surface of radius$r_1$and${\mathrm{r}}_2$as:

$\begin{array}{l}{\mathrm{I}}_1=\frac{\mathrm{P}}{4{\mathrm{\pi r}}_1^2}\\ {\mathrm{I}}_2=\frac{\mathrm{P}}{4{\mathrm{\pi r}}_2^2}\end{array}$

Dividing$I_2$by$I_1$, we get

$\begin{array}{c}\frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}=\frac{\left(\frac{\mathrm{P}}{4{\mathrm{\pi r}}_2^2}\right)}{\left(\frac{\mathrm{P}}{4{\mathrm{\pi r}}_1^2}\right)}\\ \frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}=\frac{4{\mathrm{\pi r}}_1^2}{4{\mathrm{\pi r}}_2^2}\end{array}$

$\frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}=\frac{{\mathrm{r}}_1^2}{{\mathrm{r}}_2^2}$ …(a)

We know that intensity ratio between barely audible and painful threshold is:

$\frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}={10}^{-12}$

Substitute all the value in the equation (a).

$\begin{array}{c}{10}^{-12}=\frac{{\left({\mathrm{r}}_1\right)}^2}{{\left(10000\text{\hspace{0.17em}m}\right)}^2}\\ {\mathrm{r}}_1^2={10}^{-12}\times {10}^8{\text{\hspace{0.17em}m}}^{\text{2}}\\ ={10}^{-4}{\text{\hspace{0.17em}m}}^{\text{2}}\\ {\mathrm{r}}_1=\sqrt{{10}^{-4}{\text{\hspace{0.17em}m}}^{\text{2}}}\\ ={10}^{-2}\text{\hspace{0.17em}m}\\ =0.01\text{\hspace{0.17em}m}\\ {\mathrm{r}}_1=1\text{\hspace{0.17em}cm}\end{array}$.

Hence, at $1\text{\hspace{0.17em}cm}$it would begin to cause pain.