A source of sound (1200 Hz) emits uniformly in all directions. An observer 2.99 m from the source measures a sound level of 41.1 dB. Calculate the average power output of the source.
To calculate the average power output of the source, we need to use the formula for sound intensity level in decibels.
The formula to calculate sound intensity level in decibels is:
L = 10 * log(I / I₀)
Where:
L is the sound level in decibels
I is the sound intensity (power per unit area)
I₀ is the reference intensity, which is defined as 10^(-12) watts per square meter (W/m²)
In this question, we are given the sound level (L = 41.1 dB) and the distance from the source to the observer (r = 2.99 m). To calculate the average power output of the source, we need to calculate the sound intensity I.
To find the sound intensity I, we can rearrange the formula for sound intensity level and solve for I:
I = I₀ * 10^(L / 10)
Given that I₀ = 10^(-12) W/m² and L = 41.1 dB, we can substitute these values into the formula:
I = (10^(-12)) * 10^(41.1 / 10)
Calculating this expression, we find:
I ≈ 1.258925412 × 10^(-9) W/m²
Now that we have the sound intensity I, we can calculate the power output P of the source using the equation:
P = I * A
Where:
P is the power output of the source
A is the surface area of the sphere centered at the source with a radius equal to the distance between the source and the observer
The surface area of a sphere A can be calculated using the formula:
A = 4πr²
Substituting the given distance r = 2.99 m into the formula, we get:
A = 4π * (2.99)^2 ≈ 112.319 m²
Finally, substitute the calculated values for I and A into the equation for power output:
P = (1.258925412 × 10^(-9)) * 112.319
Calculating this expression:
P ≈ 1.414 W
Therefore, the average power output of the source is approximately 1.414 watts.