To search for a fossilized dinosaur embedded in rock, paleontologists can use sound waves to produce a

Question:

To search for a fossilized dinosaur embedded in rock, paleontologists can use sound waves to produce a computer image of the dinosaur. The image then guides the paleontologist as they dig the dinosaur out of the rock. (The technique is shown in the opening scenes of the movie Jurassic Park.) The basic idea of the detection technique is that a strong pulse of sound is emitted by a source (a seismic gun) at ground level and then detected by hydrophones that lie at evenly spaced depths in a bore hole drilled into the ground. The source and one hydrophone are shown in Figure. If the sound wave travels from the source to the hydrophone through only rock as in Figure it travels at a known speed V and takes a certain time T. If instead, it travels through a fossilized bone along the way, it takes slightly more time because it travels more slowly in the bone than in the rock. By measuring the difference Δt between the expected and measured travel times, the distance d traveled in the bone can be determined. After this procedure is repeated for many locations of the source and hydrophones, a computer can transform the many computed distances d into an image of the fossil.

(a) Let the speed of sound through fossilized bone be V ?? ΔV, where ΔV is small relative to V. Show that the distance d is given by d ?? V2 Δt/ ΔV.

(b) For V = 5000 m/s and ΔV = 200 m/s, what typical value of Δt can be expected if the sound passes along the diameter of a leg bone of an adult T rex? (Estimate the bone's diameter.)