Party hearing. As the number of people at a party increases, you must raise your voice for a listener to hear you against the background noise of the other partygoers. However, once you reach the level of yelling, the only way you can be heard is if you move closer to your listener, into the listener's “personal space.” Model the situation by replacing you with an isotropic point source of fixed power P and replacing your listener with a point that absorbs part of your sound waves. These points are initially separated by ri = 1.65 m. If the background noise increases by Äâ = 5.84 dB, the sound level at your listener must also increase. What separation rf is then required?

Question

Party hearing. As the number of people at a party increases, you must raise your voice for a listener to hear you against the background noise of the other partygoers. However, once you reach the level of yelling, the only way you can be heard is if you move closer to your listener, into the listener's “personal space.” Model the situation by replacing you with an isotropic point source of fixed power P and replacing your listener with a point that absorbs part of your sound waves. These points are initially separated by ri = 1.65 m. If the background noise increases by Äâ = 5.84 dB, the sound level at your listener must also increase. What separation rf is then required?

Answer 1

To solve this problem, we can use the concept of sound intensity and the inverse square law, which states that sound intensity decreases with the square of the distance from the source.

Let's denote the initial intensity at the listener position as Ii, and the final intensity required as If. We know that the initial separation between the isotropic point source and the listener, ri, is 1.65 m.

By definition, the quantity Δβ (ΔdB) represents the change in sound level or intensity between the initial and final situations. In this case, Δβ is given as 5.84 dB. We also know that the sound power of the source is fixed at P.

The relationship between sound intensity and sound power is given by:

I = P / (4πr²)

Where I is the sound intensity, P is the sound power, and r is the distance from the source.

To compare two intensity levels, we can use the formula:

Δβ = 10 log10(If / Ii)

Substituting the expressions for sound intensity into this equation:

Δβ = 10 log10((P / (4πr_f²)) / (P / (4πr_i²)))
Δβ = 10 log10(r_i² / r_f²)
Δβ = 20 log10(r_i / r_f)

Now we can solve for the final separation, rf. Rearranging the equation:

Δβ / 20 = log10(r_i / r_f)
10^(Δβ / 20) = r_i / r_f
r_f = r_i / 10^(Δβ / 20)

Plugging in the values:

ri = 1.65 m
Δβ = 5.84 dB

r_f = 1.65 / 10^(5.84 / 20)
r_f ≈ 0.929 m

Therefore, the required separation, rf, is approximately 0.929 meters.