The sound intensity (I) is related to the amplitude (A) and the distance (d) from the source by the inverse square law. According to the inverse square law, the intensity is inversely proportional to the square of the distance. Therefore, we can express the relationship between the sound intensity (I₁) at distance d and the sound intensity (I₂) at distance 0.5D as follows:
I₁ / I₂ = (d / 0.5D)²
Since the source emits waves of amplitude 2A instead of A, the amplitude factor affects the intensity as the square of the amplitude. Therefore, we can further modify the equation as:
I₁ / I₂ = (d / 0.5D)² * (2A / A)²
Simplifying this equation gives us:
I₁ / I₂ = 4 * (d / 0.5D)²
Now, we can substitute the value of d / 0.5D = 1 (since we are comparing the intensity at the same distance) and simplify further:
I₁ / I₂ = 4
Therefore, the sound intensity at a distance of 0.5D from the source when the source emits waves of amplitude 2A is four times greater (4I) compared to the sound intensity (I) at a distance d from the source when the source emits waves of amplitude A.