Question

A SONAR system fixed in a submarine operates at a frequency $ 40.0kHz $ . An enemy submarine moves towards the sonar with a speed of 360 km/h. What is the frequency of sound reflected by the submarine? Take the speed of sound in water to be 1450 m/s.

Answer 1

Hint : The frequency reflected by the enemy submarine is the sound frequency perceived by that submarine. The sound perceived can be calculated from Doppler Effect.
Formula used: In this solution we will be using the following formulae;
 $ f' = \dfrac{{v + {v_L}}}{v} $ where $ f' $ is the frequency of sound as perceived by an observer moving toward a source of sound wave, $ v $ is the velocity of sound in the medium of travel and $ {v_L} $ is the velocity of the observer, and $ f $ is the frequency of sound emitted by the source.

Complete step by step answer
The SONAR system emits sound at frequency 40 Hz. To calculate the reflected sound frequency, we shall note that the reflected frequency from the enemy ship would be the frequency observed by the enemy ship. This can be calculated from the Doppler equation as given below.
 $ f' = \dfrac{{v + {v_L}}}{v}f $ where $ f' $ is the frequency of sound as perceived by an observer moving toward a source of sound wave, $ v $ is the velocity of sound in the medium of travel and $ {v_L} $ is the velocity of the observer.
Hence, by inserting known values to formula, we have
 $ f' = \dfrac{{1450 + 100}}{{1450}}40 $ (since 360 km/h is equal to 100 m/s).
Hence, by computation, we have that
 $ f' = 42.759Hz \approx 43Hz $ .

Note
As seen, the application of the Doppler effect is used in the military for the detection of enemy ships and submarines. However, instead of finding the frequency of reflection from the speed, it’s the other way around. Sensors are placed on the submarines, which detect the frequency of reflected sound waves from other ships and an algorithm according to the Doppler effect formula is built into the computer system to estimate the velocity of the ship.