There is evidence that elephants communicate via infrasound, generating rumbling vocalizations as low as 14 Hz that can travel up to 10 km. The intensity level of these sounds can reach 105 dB, measured a distance of 4.5 m from the source. Determine the intensity level of the infrasound 10 km from the source, assuming the sound energy radiates uniformly in all directions.
To determine the intensity level of the infrasound 10 km from the source, we can use the inverse square law for sound propagation. According to the inverse square law, the intensity of sound decreases with the square of the distance from the source.
Given:
- Intensity level at a distance of 4.5 m from the source: 105 dB
- Distance of 10 km (or 10,000 m) from the source
First, we need to convert the intensity level from decibels (dB) to intensity in watts per square meter (W/m²), as intensity is typically measured in watts per square meter of area.
The formula to convert intensity level from decibels to intensity is:
I = 10^(L/10)
Where:
- I is the intensity in W/m²
- L is the intensity level in decibels
Using this formula, we can calculate the intensity at a distance of 4.5 m from the source:
I1 = 10^(105/10)
= 10^10.5
= 316227.766
Next, we need to calculate the intensity at a distance of 10 km (10,000 m) from the source using the inverse square law. The formula for the inverse square law is:
I2 = I1 / (r1/r2)^2
Where:
- I2 is the intensity at the new distance (10,000 m)
- I1 is the intensity at the initial distance (4.5 m)
- r1 is the initial distance (4.5 m)
- r2 is the new distance (10,000 m)
Calculating the intensity at 10 km:
I2 = I1 / (r1/r2)^2
= 316227.766 / (4.5/10000)^2
= 316227.766 / (0.00045)^2
= 316227.766 / 0.000002025
= 156078431373.588 W/m²
Lastly, to convert the intensity back to an intensity level in decibels, we can use the formula:
L = 10 * log10(I/I0)
Where:
- L is the intensity level in decibels
- I is the intensity in W/m²
- I0 is the reference intensity (typically 10^(-12) W/m²)
Calculating the intensity level at 10 km:
L = 10 * log10(I/I0)
= 10 * log10(156078431373.588/10^(-12))
= 10 * log10(156078431373588000000000000)
= 10 * 25.193
= 251.93 dB
Therefore, the estimated intensity level of the infrasound 10 km from the source is approximately 251.93 dB.