Question

One way to monitor global warming is to measure the average temperature of the ocean. Researchers are doing this by measuring the time it takes sound pulses to travel underwater over large distances. At a depth of 1000 m, where ocean temperatures hold steady near 4∘C, the average sound speed is 1480m/s. It's known from laboratory measurements that the sound speed increases 4.0m/s for every 1.0∘C increase in temperature. In one experiment, where sounds generated near California are detected in the South Pacific, the sound waves travel 8500 km . If the smallest time change that can be reliably detected is 1.0 s, what is the smallest change in average temperature that can be measured?

Answer 1

Solution: The speed of sound in water could be written as

v (T) = 1480 + 4 (T − 4◦C)

For the given distance, at 4◦C, the travel time is

t(4◦C) = (8500 × 10^3 m)/ (1480 m/s) = 5743.243s

At a temperature of 5◦C, the travel time would be

t(5◦C) = (8500 × 10^3 m)/ (1484 m/s) = 5727.762s

So near this temperature, one degree C corresponds to a ∆t of(5743.243-5727.762) = 15.481 s.
So if we can measure a ∆t of 1.0 s, then we can see a temperature difference of

∆T = 1s/15.481s = 0.645deg C