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

To compare the sound intensity and sound intensity level at two different sites, we need to understand the relationship between sound intensity and distance.

The sound intensity (I) of a point source decreases with the square of the distance (r) according to the inverse square law. Mathematically, this relationship can be expressed as:

I ∝ 1/r^2

Now, let's solve the given problem step by step.

(a) Determine the multiplicative factor by which the sound intensity decreases as you go from the first to the second site. (I1/I2 = ?)

Given that the distance to the second site is a factor of 6 greater than the distance to the first site, we can express it as:

r2 = 6 * r1

Using the inverse square law relationship, we can compare the sound intensities as follows:

I1/I2 = (1/r1^2) / (1/r2^2)

Substituting r2 = 6 * r1:

I1/I2 = (1/r1^2) / (1/(6*r1)^2)

Simplifying the equation:

I1/I2 = r1^2 / (36 * r1^2)

The common term "r1^2" cancels out:

I1/I2 = 1/36

Therefore, the multiplicative factor by which the sound intensity decreases as you go from the first to the second site is 1/36.

(b) Determine the additive amount by which the sound level intensity decreases as you go from the first to the second site. (β1 − β2 = ? dB)

The sound level intensity (β) is measured using the decibel (dB) scale. The relationship between sound intensity (I) and sound level intensity (β) is given by:

β = 10 * log10(I/I₀)

where I₀ is the reference intensity level.

To find the difference in sound level intensities, we can subtract the sound level intensities at the two sites:

β1 - β2 = 10 * log10(I1/I₀) - 10 * log10(I2/I₀)

Using the relation I1/I2 = 1/36 from part (a), we can substitute it into the equation:

β1 - β2 = 10 * log10((1/36) * I₀/I₀)

Simplifying the equation:

β1 - β2 = 10 * log10(1/36)

Taking the logarithm of 1/36 (which is equivalent to the negative logarithm of 36):

β1 - β2 = 10 * (-log10(36))

Using the property of logarithms:

β1 - β2 = -10 * log10(36)

Calculating log10(36) ≈ 1.556 the equation becomes:

β1 - β2 ≈ -10 * 1.556

Therefore, the additive amount by which the sound level intensity decreases as you go from the first to the second site is approximately -15.56 dB.