Many airports have noise ordinances restricting the maximum sound intensity that an aircraft may produce when it takes off. At one California airport the maximum allowable sound intensity level is 98.5 dB as measured by a microphone at the end of the 1740-m-long runway. A certain airliner produces a sound intensity level of 100.0 dB on the ground when it flies over at an altitude of 100 m. On takeoff, this airliner rolls for 1200 m along the runway before leaving the ground, at which point it climbs at a 15º angle. Does this airliner violate the noise ordinance? You can ignore any effects due to reflection of the sound waves from the ground.
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The altitude of the plane when it passes over the end of the runway is (1740 m – 1200 m)tan 15º = 145 m, and so the sound intensity is 1/(1.45) 2 of what the intensity would be at 100 m. The intensity level is then
100.0 dB – 10 × log[(1.45) 2 ] = 96.8 dB,
so the airliner is not in violation of the ordinance.
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