Question

At a distance of \(20.0 \mathrm{~m}\) from a sound source, the intensity of the sound is \(60.0 \mathrm{~dB}\). What is the intensity (in dB) at a point \(2.00 \mathrm{~m}\) from the source? Assume that the sound radiates equally in all directions from the source.

Short answer

Answer: Follow the solution steps provided, using the formula for sound intensity level and the inverse square law to find the intensity at the new distance. Calculate the sound intensity level (in dB) at 2 m from the source, following the steps outlined in the solution.

Step-by-step solution

Recall the formula for sound intensity level (in dB)

Sound intensity level (L) in decibels (dB) is given by the formula, $$L = 10 \log{\frac{I}{I_0}}$$ Where, \(I\) is the intensity of sound, \(I_0\) is the reference intensity (usually taken as \(10^{-12} \mathrm{W/m^2}\)).

Find the intensity at the initial distance

To find the intensity at the initial distance, use the formula for sound intensity level and solve for \(I\): $$60 = 10 \log{\frac{I}{10^{-12}}}$$ Next, divide by 10 and take the antilog (10 raised to the power of) both sides: $$I = 10^{60/10} \times 10^{-12}$$

Apply the inverse square law to find the intensity at the new distance

The inverse square law states that the intensity (I) of sound is inversely proportional to the square of the distance (d) from the source, i.e., $$I \propto \frac{1}{d^2}$$ Let's denote the initial distance as \(d_1\) (20 m) and the new distance as \(d_2\) (2 m). To find the intensity at the new distance \(I_2\), we can write the equation: $$\frac{I_1}{I_2} = \frac{d_2^2}{d_1^2}$$ We know the value of \(I_1\), and the distances \(d_2\), and \(d_1\). Solve for \(I_2\), $$I_2 = \frac{d_1^2}{d_2^2} \times I_1$$

Find the sound intensity level (in dB) at the new distance

Now that we have the new intensity (I_2), use the formula to find the sound intensity level at the new distance: $$L_2 = 10 \log{\frac{I_2}{I_0}}$$ The result will give us the intensity level at 2.00 m from the source. By completing these steps, the intensity (in dB) at 2.00 m from the source can be found.