Sound is emitted by a point source. You wish to compare the sound intensity and sound intensity level from this source at two different sites. If the distance to the second site is a factor of 6 greater than the distance to the first, determine the following.
(a) Determine the multiplicative factor by which the sound intensity decreases as you go from the first to the second site. (Assume the intensities at the first and second sites are I1 and I2, respectively.)
I1/I2 = ?
(b) Determine the additive amount by which the sound level intensity decreases as you go from the first to the second site. (Assume the sound level intensities at the first and second sites are β1 and β2, respectively.)
β1 − β2 = ? dB
In order to determine the multiplicative factor by which the sound intensity decreases as you go from the first site to the second site, you need to use the inverse square law for sound intensity. This law states that the sound intensity at a given distance from a point source is inversely proportional to the square of the distance from the source. Mathematically, it can be expressed as:
I1 / I2 = (r2 / r1)^2
In this equation, I1 and I2 represent the sound intensities at the first and second sites, respectively, and r1 and r2 are the distances from the point source to the first and second sites, respectively.
Given that the distance to the second site is a factor of 6 greater than the distance to the first site, we can write:
r2 = 6 * r1
Now we can substitute this relationship into the equation for the inverse square law:
I1 / I2 = (6 * r1 / r1)^2 = 36
Therefore, the multiplicative factor by which the sound intensity decreases as you go from the first site to the second site is 36 (I1/I2 = 36).
Moving on to part (b), to determine the additive amount by which the sound level intensity decreases as you go from the first site to the second site, you need to use the formula for sound level calculation. Sound level (β) is usually expressed in decibels (dB) and is given by:
β = 10 * log10(I / I0)
In this equation, β represents the sound level, I is the sound intensity, and I0 is the reference intensity.
Since we are comparing the sound level intensities at the first and second sites, we can subtract the two equations to find the additive amount:
β1 - β2 = 10 * log10(I1 / I0) - 10 * log10(I2 / I0)
Simplifying further:
β1 - β2 = 10 * (log10(I1) - log10(I2))
Since we already determined in part (a) that I1/I2 = 36, we can substitute this value into the equation:
β1 - β2 = 10 * (log10(36))
Using a calculator, we find that log10(36) ≈ 1.5563. Multiplying this by 10 gives us the final answer:
β1 - β2 = 15.563 dB
Therefore, the additive amount by which the sound level intensity decreases as you go from the first site to the second site is approximately 15.563 dB (β1 - β2 = 15.563 dB).