Question

You are standing at a distance D from an isotropic point source of sound. You walk$\mathbf{50}\mathbf{.0}\mathbf{}\mathbf{\text{m}}$ toward the source and observe that the intensity of the sound has doubled. Calculate the distanceD.

Short answer

The distance Dis $171\text{\hspace{0.17em}m}$.

Step-by-step solution

The given data

The value of the distance is $171\text{\hspace{0.17em}m}$.

Determine the concept of the intensity

Using the proportionality between the intensity of sound and distance, we can find the distance D.

Formula:

The intensity of the sound wave is inversely proportional to the square of distance or separation, $I\alpha \frac{1}{r^2}$ (i)

Calculate the distance

As given above we can say that

The distance betweenandis. Initially, the observer is at point D and then moves towardsthesource by 50.0m to reach at point d. At point d,the intensity of sound gets doubled.

So, we can write, the intensity and distance relation using equation (i) as:

$\begin{array}{c}\frac{I_{r=d}}{I_{r=D-d}}=\frac{{(D-d)}^2}{D^2}\\ =\frac{1}{2}\end{array}$

Solve by the quadratic formula:

$\begin{array}{c}\frac{D-d}{D}=\frac{1}{\sqrt{2}}\\ D=\frac{d}{1-\left(\frac{1}{\sqrt{2}}\right)}\\ D=170.77\\ D~171\text{\hspace{0.17em}m}\end{array}$

Hence, the value of distance, D is $171\text{\hspace{0.17em}m}$.