How does the intensity of a sound wave change if the distance from the source is reduced by a factor of 3?
A. The intensity increases by a factor of 9.
B. The intensity decreases by a factor of 9.
C. The intensity decreases by a factor of 3.
D. The intensity increases by a factor of 3.
Answer (1)
The intensity of a sound wave is inversely proportional to the square of the distance from the source.
The distance from the source is reduced by a factor of 3: d 2 = 3 d 1 .
The ratio of the final intensity to the initial intensity is I 1 I 2 = d 2 2 d 1 2 = 9 .
Therefore, the intensity increases by a factor of 9: T h e in t e n s i t y in cre a ses b y a f a c t oro f 9.
Explanation
Understanding the Relationship Let's analyze the relationship between the intensity of a sound wave and the distance from its source. The intensity of a sound wave is inversely proportional to the square of the distance from the source. This means if you move closer to the source, the intensity increases, and if you move farther away, the intensity decreases.
Setting up the Equations We are given that the distance from the source is reduced by a factor of 3. Let's denote the initial distance as d 1 and the final distance as d 2 . Then, we have:
d 2 = 3 d 1
Let I 1 be the initial intensity and I 2 be the final intensity. Since intensity is inversely proportional to the square of the distance, we can write:
I 1 = d 1 2 k
I 2 = d 2 2 k
where k is a constant of proportionality.
Calculating the Change in Intensity Now, we want to find how the intensity changes, so we need to find the ratio I 1 I 2 . Substituting the expressions for I 1 and I 2 , we get:
I 1 I 2 = d 1 2 k d 2 2 k = d 2 2 d 1 2
Since d 2 = 3 d 1 , we can substitute this into the equation:
I 1 I 2 = ( 3 d 1 ) 2 d 1 2 = 9 d 1 2 d 1 2 = 9
This means that I 2 = 9 I 1 .
Conclusion Therefore, the intensity increases by a factor of 9.
Examples
Imagine you're at a concert. If you move three times closer to the speakers, the sound intensity you experience increases by a factor of 9. This is because sound intensity decreases with the square of the distance. Understanding this relationship helps you appreciate how sound levels change dramatically with even small changes in distance, which is crucial for sound engineering and event planning.
