Physics Sound Level, Sound Power and Distance?

Ju Yeon sits in a park where people come to fly small radio-controlled airplanes. She can hear down to a sound level of 20 dB. If a plane emits sound with a power of 2.0e-6 W, how far away will the plane be when she is just able to hear it?|||Sound intensity level in dB is defined as


L = 10∙log₁₀( I / I₀)


with I₀ = 1.0e-12W/m²





So the minimum intensity she can hear is


I = I₀ ∙ 10^( L/10)


= 1.0e-12W/m² ∙ 10^( 20/10)


= 1.0e-10W/m²





Sound waves propagate as spherical waves. Therefore at distance r to the source the sound power is uniformly distributed over the surface of a sphere of radius r. That means the sound intensity at distance r to the source is:


I = P/A = P / (4∙π∙r²)





Solve for r:


r = √[ P/(4∙π∙I₀²) ]


= √[ 2.0e-06W /(4∙π∙1.0e-10) ]


= 39.89....m


≈ 40m