How Are Lighthouses Powered
how are lighthouses powered
Sound Level Question?
A boat sounds a foghorn which is heard by a lighthouse operator at a sound level of 54 dB. (a) The lighthouse operator knows that the sound power of the boat's horn is 4.5 W. How far away is the boat? Assume that the horn is a sound source that radiates equally in all directions, and that the fog does not absorb any sound. (b) For someone standing on the boat, 7.5 meters from the horn, what is the sound level?
(a)
Sound intensity level in dB associated to sound intensity I is defined as
L = 10∙log₁₀ ( I/I₀ )
where I₀ = 10⁻¹² W/m² is the threshold of hearing
So the sound intensity experienced by the light house operator is:
I = I₀∙10^(L/10) = 10⁻¹² W/m² ∙10^(54/10) = 2.51×10⁻⁷W/m²
Sound waves propagate uniformly in all directions as spherical wave. That means at distance R to the source the sound energy emitted by the source is uniformly distributed over the surface of a sphere of radius R. That means the intensity at distance R to a source of power P is:
I = P/ (4∙π∙R²)
Solve for R
R = √[ P/ (4∙π∙I)]
= √[ 4.5W / (4∙π∙2.51×10⁻⁷W/m²)]
= 1194m
(b)
I = P/ (4∙π∙R²) = 4.5W/ (4∙π∙7.5m²) = 6.36×10⁻³W/m²
=>
L = 10∙log₁₀ ( 6.36×10⁻³W/m² / 10⁻¹² W/m² ) = 98dB
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