An interstate highway has been built though a poor neighborhood in a city. In the afternoon, the sound level in a rented room is 114.0 dB as 172 cars pass outside the window every minute. Late at night, the traffic flow is only 7 cars per minute.
(b) What is the sound intensity at night? (c) What is the average late-night sound level?
To find the sound intensity at night, we need to determine the sound intensity level (SIL) first using the given information. The formula to calculate SIL is:
SIL = 10 * log10(I/I0)
where I is the sound intensity in watts per square meter (W/m^2) and I0 is the reference sound intensity (usually taken to be 10^-12 W/m^2).
Since the sound level is given in decibels (dB), we can use the formula to find the sound intensity:
SIL = 114.0 dB
SIL = 10 * log10(I/I0)
114.0 = 10 * log10(I/I0)
Dividing both sides by 10:
11.4 = log10(I/I0)
Now, let's find the sound intensity at night when the traffic flow is only 7 cars per minute.
(b) To calculate the sound intensity at night, we need to know the relationship between the traffic flow and the sound intensity. Let's assume that the traffic flow is directly proportional to the sound intensity.
When there are 172 cars passing outside the window every minute, the sound intensity is 114 dB.
Let's assign this value to I1 and the corresponding traffic flow of 172 cars/minute to F1.
I1 = 114 dB
F1 = 172 cars/minute
Now, when the traffic flow is only 7 cars per minute, we need to find the corresponding sound intensity (I2). Let's assign this value to I2 and the traffic flow of 7 cars/minute to F2.
F2 = 7 cars/minute
We can use the following equation to find the sound intensity at night (I2):
(I2/I1) = (F2/F1)
Substituting the values we have:
(I2/114) = (7/172)
Multiplying both sides by 114:
I2 = (7/172) * 114
Calculating this:
I2 ≈ 4.688 W/m^2
So, the sound intensity at night is approximately 4.688 W/m^2.
(c) To find the average late-night sound level, we can use the formula:
SIL = 10 * log10(I/I0)
Using the sound intensity value we found at night (I2 ≈ 4.688 W/m^2) and the reference sound intensity (I0 = 10^-12 W/m^2), we can calculate the average late-night sound level:
SIL = 10 * log10(4.688/10^-12)
Calculating this:
SIL ≈ 10 * log10(4.688 * 10^12)
SIL ≈ 10 * log10(4.688) + 10 * log10(10^12)
SIL ≈ 10 * 0.6702 + 10 * 12
SIL ≈ 6.702 + 120
SIL ≈ 126.702 dB
Therefore, the average late-night sound level is approximately 126.702 dB.
To find the answers to questions (b) and (c), we need to understand the relationship between sound intensity and sound level, as well as how it is affected by the number of cars passing by.
Sound intensity (I) is a measure of the amount of sound energy passing through a unit area per unit time. It is usually measured in watts per square meter (W/m²).
Sound level (L) is a logarithmic measure of the sound intensity relative to a reference intensity. It is measured in decibels (dB). The formula for sound level is:
L = 10 * log10(I / I₀)
Where L is the sound level, I is the sound intensity, and I₀ is the reference intensity (usually the threshold of human hearing, which is 1.0 × 10^(-12) W/m²).
Given that the sound level in the afternoon is 114.0 dB and there are 172 cars passing outside the window every minute, we can find the sound intensity (I) during that time.
First, let's calculate the sound intensity during the afternoon:
L = 10 * log10(I / I₀)
114 = 10 * log10(I / I₀)
Divide both sides by 10:
11.4 = log10(I / I₀)
Now, let's find the intensity ratio (I / I₀):
I / I₀ = 10^(11.4)
Using a calculator, we find that the intensity ratio is approximately 2.51 × 10^(11).
Now that we know the intensity ratio during the afternoon, we can use it to find the sound intensity at night when there are only 7 cars passing per minute.
We are given that the traffic flow at night is 7 cars per minute. We can assume that the sound intensity is directly proportional to the number of cars passing by (assuming no other factors affecting the sound level change). Therefore, we can calculate the sound intensity at night using the proportion:
(car flow at night) / (car flow in the afternoon) = (sound intensity at night) / (sound intensity in the afternoon)
Replacing the values:
7 / 172 = (sound intensity at night) / (2.51 × 10^(11))
Let's solve for the sound intensity at night:
(sound intensity at night) = (7 / 172) * (2.51 × 10^(11))
Using a calculator, we find that the sound intensity at night is approximately 1.02 × 10^9 W/m².
Finally, to calculate the average late-night sound level, we use the sound intensity during that time:
L = 10 * log10(sound intensity / I₀)
L = 10 * log10(1.02 × 10^9 / 1 × 10^(-12))
Using a calculator, we find that the late-night sound level is approximately 190.4 dB.