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 = 1.90 m. If the background noise increases by Δβ = 4.90 dB, the sound level at your listener must also increase. What separation rf is then required?

Answer 1

To model the situation, we consider the decrease in sound intensity as we move away from the source. Using the inverse square law, we can calculate the required separation between the source and listener.

Step-by-step explanation:

To model the situation, we need to consider the decrease in sound intensity as we move away from the source. According to the inverse square law, the sound intensity decreases with the square of the distance from the source. So, when the listener is at a distance of ri from the source, the sound intensity is given by:

Ii = P / (4πri^2)

When the separation between the source and listener increases to rf, the sound intensity will decrease to:

If = P / (4πrf^2)

Since the sound level is given by L = 10log(I/I0), where I0 is the reference intensity, we can calculate the change in sound level:

ΔL = 10log(If/Ii)

Using the given information that Δβ = 4.90 dB, we can solve for the required separation rf.

Answer 2

Step-by-step explanation:

If the background noise increases by Δβ = 4.90 dB , the separation must decrease so that intensity of sound to be listened should increase by the same level .

If I₁ be the intensity of previous sound and I₂ be the intensity of sound at nearer distance

I₂ / I₁ = 4.9 dB = 10°⁴⁹

= 3.09

If r₁ be the previous separation and r₂ be the new separation

r₁² / r₂² = 3.09 ( intensity of sound decreases at square of distance )

r₁ / r₂ = 1.758

r₂ = r₁ / 1.758

= 1.9 / 1.758

= 1.08 m

This is separation required.