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 4C, the average sound speed is 1480 m/s. It’s known from laboratory measurements that the sound speed increases 4.0 m/s for every 1.0C increase in temperature. In one experiment, where sounds generated near California are detected in the South Pacific, the sound waves travel 8000 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?

Short answer

The researchers can measure a temperature change of$0.{07}^{o}C.$

Step-by-step solution

Given data

The average sound speed is, $v=1480m/s$

From California to the South Pacific, the wave travels a distance of$d=8000km=8\times {10}^{6}m$

determine the temperature decrease 

From California to the South Pacific, the time taken by the wave is,

$$\begin{gathered} t=\frac{d}{v} \\ t=\frac{8\times {10}^{6}m}{1480m/s} \\ t=5405.4s \end{gathered}$$

Therefore, the wave speed has decreased to,

role="math" localid="1649961885901" $$\begin{gathered} v\text{'}=\frac{d}{t} \\ v\text{'}=\frac{8\times {10}^{6}m}{5405.4s} \\ v\text{'}=1480.28m/s \end{gathered}$$

For an increase in velocity of $4m/s$, the temperature changes $1^{o}C$. Thus, for a velocity increase by $\left(1480-1480.28\right)m/s=0.28m/s$, the increase in temperature should be,

$$\begin{gathered} T=\frac{0.28m/s}{4m/s}\times \left(1^{o}C\right) \\ T=0.{07}^{o}C \end{gathered}$$