A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy submarine moves towards the SONAR with a speed of 360 km h–1. What is the frequency of sound reflected by the submarine? Take the speed of sound in water to be 1450 ms–1.
True frequency of SONAR system ν = 40.0 kHz
Speed of sound in water v = 1450 m s-1
Speed of enemy submarine towards SONAR v0 = 360 km h-1 = 360 × (5/18) m s-1 = 100 m s-1
As the source is at rest and observer is in motion towards the source, apparent frequency received by observer (enemy submarine) will be:
ν' = (v + v 0 )/v × ν
= (1450 + 100)/1450 × 40.0 kHz
= 42.76 kHz
Then the frequency is reflected by enemy submarine (which now behaves as source, moving with velocity vs = −100 m s-1) and is observed by SONAR. Hence, frequency as felt by SONAR system:
ν'' = v/(v − vs) × ν'
= 1450/(1450 − 100) × 42.76 kHz
= 45.9 kHz