A train sounds its horn as it approaches an intersection. The horn can just be heard at a level of 59 dB by an observer 14 km away.
(a) What is the average power generated by the horn? (b) What intensity level of the horns sound is observed by someone waiting at an intersection 46 m from the train? Treat the horn as a point source and neglect any absorption of sound by the air.
To determine the average power generated by the train's horn, we can use the relationship between power, intensity, and distance.
(a) Average Power generated by the horn:
We are given the sound level in decibels (dB) and the distance from the observer to the source.
1. Convert the sound level from dB to intensity level (IL):
IL = 10^((dB - 120) / 10)
Where 120 is the reference intensity level of 0 dB.
2. Calculate the intensity (I) of the sound:
I = IL / (4πr^2)
Where r is the distance from the source to the observer.
3. Use the formula for power (P) in terms of intensity:
P = I * A
Where A is the area through which the sound is spreading.
Since we are treating the horn as a point source, the area can be considered as a sphere with a radius equal to the distance of the observer from the horn.
So, the average power generated by the horn is:
P = I * (4πr^2)
Substituting the given values:
- IL = 59 dB (sound level)
- r = 14 km = 14,000 m
Calculating the average power generated by the horn:
1. Convert the sound level from dB to intensity level (IL):
IL = 10^((59 - 120) / 10) = 10^(-6.1) ≈ 7.943 x 10^(-7) W/m^2
2. Calculate the intensity (I) of the sound:
I = (7.943 x 10^(-7)) / (4π(14,000)^2) ≈ 9.802 x 10^(-13) W/m^2
3. Calculate the average power generated by the horn:
P = (9.802 x 10^(-13)) * (4π(14,000)^2) ≈ 2.53 W (approximately)
Therefore, the average power generated by the horn is approximately 2.53 Watts.
(b) Intensity level observed by someone waiting at the intersection:
To find the intensity level observed by someone waiting at the intersection, we need to calculate the intensity (I) of the sound at their location and convert it back to dB.
1. Calculate the intensity (I) of the sound at the intersection:
I = P / A
Where P is the average power generated by the horn, and A is the area through which the sound is spreading.
Since we are treating the horn as a point source, the area can be considered as a sphere with a radius equal to the distance of the intersection from the train.
So, the intensity of the sound at the intersection is:
I = P / (4πr^2)
Substituting the given values:
- P = 2.53 W (average power generated by the horn)
- r = 46 m
Calculating the intensity of the sound at the intersection:
I = (2.53) / (4π(46)^2) ≈ 2.05 x 10^(-5) W/m^2
2. Calculate the intensity level (IL) in decibels:
IL = 10 * log10(I / I_ref)
Where I_ref is the reference intensity level of 0 dB.
IL = 10 * log10((2.05 x 10^(-5)) / (10^(-12))) ≈ 85 dB
Therefore, the intensity level of the horn's sound observed by someone waiting at the intersection is approximately 85 dB.