Chapter 15, Problem 83A

To determine

To calculate: The frequency received on the ship that was reflected from the whale.

Answer to Problem 83A

The frequency received on the ship that was reflected from the whaleis $\underset{\_}{\text{22}\text{.4 kHz}}$ .

Explanation of Solution

Given:

Frequency of sonar used by the ship = $22.5\text{ kHz}$

Speed of sound in seawater = $\text{1533 m/s}$

Velocity of the whale = $\text{4}\text{.15 m/s}$

The ship is at rest.

Formula used:

The Doppler’s effect equation is given by:

  $f_d=f_s\left(\frac{v-v_d}{v-v_s}\right)$

Where $f_s$ is frequency of the source, $v$ is the speed of sound, $v_d$ is velocity of the detector and $v_s$ is velocity of the source.

Calculation:

The frequency received by the whale from the ship is calculated as:

$\begin{array}{l}{f}_d=f_s\left(\frac{v-v_d}{v-v_s}\right)\\ \text{ =22}\text{.5}\times {\text{10}}^3\text{ Hz}\left(\frac{1533\text{ m/s}-4.15\text{ m/s}}{1533\text{ m/s}-0\text{ m/s}}\right)\\ \text{ =22}\text{.5}\times {\text{10}}^3\text{ Hz}\times \frac{1528.85\text{ m/s}}{1533\text{ m/s}}\\ \text{ =22439}\text{.0 Hz}\end{array}$

The frequency received by the ship from the whale is calculated as:

$\begin{array}{l}{f}_d=f_s\left(\frac{v-v_d}{v-v_s}\right)\\ \text{ =22439 Hz}\left(\frac{1533\text{ m/s}-0\text{ m/s}}{1533\text{ m/s}+4.15\text{ m/s}}\right)\\ \text{ =22439 Hz}\times \frac{1533\text{ m/s}}{1537.15\text{ m/s}}\\ \text{ =22378}\text{.4 Hz}\\ \text{ =22}\text{.4 kHz}\end{array}$

Conclusion:

The frequency received on the ship that was reflected from the whaleis $\underset{\_}{\text{22}\text{.4 kHz}}$ .