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 �� = 5.84 dB, the sound level at your listener must also increase. What separation rf is then required?
To raise the received level of your voice by 5.84 dB, the received intensity of your voice must be higher by a factor
10^(5.84/10)= 10^0.584 = 3.837
To acheieve this increase, the distance must decrease by a factor (because of the inverse square law of inteneity)
x = sqrt(3.837) = 1.959
That makes the new distance
1.65/1.959 = 0.842 m
For a review of the subject, see
http://en.wikipedia.org/wiki/Decibel
To model the situation, we can use the inverse square law for sound propagation. According to the inverse square law, the sound intensity decreases as the square of the distance from the source increases.
The sound intensity at the initial separation (ri) can be calculated using the formula:
Ii = P / (4πri²)
where Ii is the initial sound intensity, P is the power of the source, and ri is the initial separation.
Given that the background noise increases by 5.84 dB, we can convert this to a ratio of sound intensities using the formula:
dB ratio = 10 log10(I2 / I1)
where I2 is the final sound intensity and I1 is the initial sound intensity.
Converting the background noise increase to a ratio, we have:
dB ratio = 10 log10(I2 / Ii) = 5.84 dB
Now, we can rearrange the equation to solve for the final sound intensity (I2) in terms of the initial sound intensity (Ii):
I2 = Ii * 10^(dB ratio / 10)
Next, we can use the inverse square law to find the new separation (rf) required, given the final sound intensity (I2) and the power (P) of the source:
I2 = P / (4πrf²)
Rearranging the equation, we have:
rf² = P / (4πI2)
Finally, we can solve for the new separation (rf):
rf = √(P / (4πI2))
Substituting the given values, we have:
ri = 1.65 m
dB ratio = 5.84 dB
Now we can substitute these values into the formulas and calculate the new separation (rf).
To model the situation described, we can use the inverse square law for sound intensity. According to this law, the sound intensity decreases with the square of the distance from the source.
Let's assume that the initial sound level at the listener position is L1 and the sound level after increasing the separation is L2.
The sound level is typically measured in decibels (dB) and is given by the equation:
L = 10 * log10(I/I0)
Where L is the sound level in decibels, I is the sound intensity, and I0 is the reference intensity (typically the threshold of human hearing, which is 1 x 10^(-12) W/m^2).
Given that the background noise increases by 5.84 dB, we can write:
L2 = L1 + 5.84
Since we are dealing with a point source of fixed power P, the sound intensity at a distance r1 (initial separation) is given by:
I1 = P / (4 * pi * r1^2)
Similarly, the sound intensity at a distance r2 (required separation) is given by:
I2 = P / (4 * pi * r2^2)
Using the inverse square law, we know that:
I2 = I1 * (r1^2 / r2^2)
Substituting the expressions for I1 and I2:
P / (4 * pi * r2^2) = P / (4 * pi * r1^2) * (r1^2 / r2^2)
Canceling similar terms:
1 / r2^2 = 1 / (r1^2) * (r1^2 / r2^2)
Simplifying:
1 / r2^2 = 1 / r1^2
Taking the square root of both sides:
1 / r2 = 1 / r1
Rearranging the equation:
r2 = r1
Therefore, the required separation, rf, is equal to the initial separation, ri, which is 1.65 m.