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 (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. is the average power generated by the horn?
To find the average power generated by the horn, we can use the concept of sound intensity and the decibel scale.
(a) We are given that the horn sound can be heard at a level of 59 dB by an observer 14 km away. The sound level in decibels is related to the sound intensity by the equation:
dB = 10 * log10(I/I0)
where I is the sound intensity and I0 is the reference intensity (usually taken as the threshold of hearing, which is 10^(-12) W/m^2). Rearranging the equation, we can solve for the sound intensity:
I = I0 * 10^(dB/10)
In this case, the sound level is 59 dB and the distance from the observer to the train is 14 km (or 14,000 m). Let's calculate the sound intensity:
I = (10^(-12) W/m^2) * 10^(59/10)
I ≈ 1.26 W/m^2
(b) Now, assuming that the observer is waiting at an intersection 46 m from the train, we can use the inverse square law to calculate the new sound intensity. The inverse square law states that the intensity of sound decreases inversely with the square of the distance from the source. Therefore, the new sound intensity can be found using the equation:
I2 = I1 * (r1/r2)^2
where I1 is the initial sound intensity, I2 is the new sound intensity, r1 is the initial distance, and r2 is the new distance.
In this case, I1 is the sound intensity calculated in part (a) as 1.26 W/m^2, r1 is 14,000 m, and r2 is 46 m. Let's calculate the new sound intensity:
I2 = (1.26 W/m^2) * (14,000 m / 46 m)^2
I2 ≈ 3.06 × 10^6 W/m^2
So, the intensity level of the horn's sound observed by someone waiting at an intersection 46 m from the train is approximately 3.06 × 10^6 W/m^2.
To find the average power generated by the horn, we can assume that the sound is radiated equally in all directions, so the power is spread over the surface area of a sphere with a radius equal to the distance from the observer to the source. The power can be calculated using the equation:
Power = Intensity * Area
where Intensity is the sound intensity and Area is the surface area of the sphere, given by 4πr^2.
In this case, the sound intensity is the value calculated in part (b) as 3.06 × 10^6 W/m^2 and the distance from the observer to the train is 46 m. Let's calculate the average power:
Power = (3.06 × 10^6 W/m^2) * 4π(46 m)^2
Power ≈ 8.6 × 10^9 W
Therefore, the average power generated by the horn is approximately 8.6 × 10^9 watts.