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 (in $Hz$) of sound reflected by the submarine? Take the speed of sound in water to be $1450\; m s ^{-1}$
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Operating frequency of the $SONAR$ system, $v=40 \,kHz$
Speed of the enemy submarine, $v_{ e }=360\, km / h =100 \,m / s$
Speed of sound in water, $v=1450 \,m / s$
The source is at rest and the observer (enemy submarine) is moving toward it. Hence, the apparen frequency ( $v^{\prime}$ ) received and reflected by the submarine is given by the relation:
$v^{\prime}=\left(\frac{v+v_{e}}{v}\right) \,v$
$=\left(\frac{1450+100}{1450}\right) \times 40=42.76\, kHz$
The frequency $\left(v^{\prime \prime}\right)$ received by the enemy submarine is given by the relation:
$v^{\prime \prime}=\left(\frac{v}{v+v_{s}}\right) v^{\prime}$
Where, $v_{ s }=100 \,m / s$
$\therefore v^{\prime \prime}=\left(\frac{1450}{1450-100}\right) \times 42.76=45.93 \,kHz$
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