Question

question 7 of 25
how does the intensity of a sound wave change if the distance from the source is decreased by a factor of 2?

a. the intensity decreases by a factor of 2.
b. the intensity increases by a factor of 4.
c. the intensity increases by a factor of 2.
d. the intensity decreases by a factor of 4.

Explanation:

Step1: Recall intensity - distance formula

The intensity $I$ of a sound wave is inversely proportional to the square of the distance $r$ from the source, i.e., $I\propto\frac{1}{r^{2}}$, or $I = \frac{k}{r^{2}}$ where $k$ is a constant.

Step2: Consider the change in distance

Let the initial distance be $r_1$ and the final distance be $r_2=\frac{r_1}{2}$. The initial intensity is $I_1=\frac{k}{r_1^{2}}$ and the final intensity is $I_2=\frac{k}{r_2^{2}}$. Substitute $r_2 = \frac{r_1}{2}$ into the formula for $I_2$: $I_2=\frac{k}{(\frac{r_1}{2})^{2}}=\frac{k}{\frac{r_1^{2}}{4}} = 4\times\frac{k}{r_1^{2}}$. Since $I_1=\frac{k}{r_1^{2}}$, we have $I_2 = 4I_1$.

Answer:

B. The intensity increases by a factor of 4.