3. A team of geophysicists is standing on the ground. Beneath their feet, at an

Question

3. A team of geophysicists is standing on the ground. Beneath their feet, at an unknown
distance, is the ceiling of a cavern. The floor of the cavern is a distance h below this ceiling. To
measure h, the team places microphones on the ground. At t = 0 s, a sound pulse is sent
straight downward through the ground and into the cavern. When this pulse reaches the ceiling
of the cavern, one part of it is reflected back toward the microphones, and a second part
continues downward, eventually to be reflected from the cavern floor. The sound reflected
from the cavern ceiling reaches the microphones at t = 0.0245 s, and the sound reflected from
the cavern floor arrives at t = 0.0437 s. The cavern is presumed to be filled with air at a
temperature of 9 °C. Assuming that air behaves like an ideal gas, what is the height h of the
cavern?

Explanation / Answer

Both sound pulses make the round trip through the rock over the cavern; so the round trip through the cavern itself takes 0.0437 - 0.0245 = 0.0192 s, and a one-way trip through the cavern takes half that or 0.0096 s.

The speed of sound in air is 331.3 + (1 + °C/273.15)1/2 if we assume it is an ideal gas, or

331.3 * (1 + 9/273.15)1/2 = 336.71 m/s

So the height of the cavern is (336.71 m/s)(0.0096 s) = 3.23 m

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