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Mechanical Properties Of Fluids

Question
CBSEENPH11020246

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 m s–1.

Solution
Operating frequency of the SONAR system, ν = 40 kHz 
Speed of the enemy submarine, ve = 360 km/h
                                                      = 100 m/s 
Speed of sound in water, = 1450 m/s 
The source is at rest and the observer (enemy submarine) is moving toward it. Hence, the apparent frequency (v') received and reflected by the submarine is given by the relation,

Frequency space of space the space SONAR space system comma space
straight nu space equals space 40 space kHz space equals space 40 space cross times space 10 cubed space Hz

The space enrgy space submarine space first space acts space as space an space observer
which space is space approaching space the space source left parenthesis SONAR right parenthesis space with
straight a space speed space of colon

straight nu subscript straight o space space equals space 360 space km divided by hr space equals space 360 cross times 5 over 18 space straight m divided by straight s
equals space 100 space straight m divided by straight s

Speed space of space sound space in space water comma space straight nu space equals space 1450 space straight m divided by straight s

Apparent space frequency space of space sound space waves space is comma space

straight nu to the power of apostrophe space equals open parentheses space fraction numerator straight nu space plus space straight nu subscript straight o over denominator straight nu end fraction close parentheses straight nu space

space space space space equals space fraction numerator 1450 space plus space 100 space over denominator 1450 end fraction cross times space 4 space cross times space 10 to the power of 4 space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis space

This space frequency space is space the space frequency space of space sound space received
by space enemy space submarine. space

Now comma space the space enemy space submarine space will space reflect
the space waves space of space frequency space straight nu to the power of apostrophe space anf space thus space acts space as space straight a space
source space which space is space moving space towards space the space observer left parenthesis SONAR right parenthesis space
with space straight a space speed space given space by comma space

straight nu subscript straight s space equals space 100 space straight m divided by straight s

Therefore comma space

Apparent space frequency space is space given space by comma space

straight nu to the power of apostrophe subscript straight a space equals space open parentheses fraction numerator straight nu over denominator straight nu space minus space straight nu subscript straight s end fraction close parentheses space cross times space straight nu to the power of apostrophe space

space space space space space equals space fraction numerator 1450 over denominator 1450 space minus space 100 end fraction cross times fraction numerator 1450 space plus space 100 space over denominator 1450 end fraction space cross times space 4 space cross times space 10 to the power of 4 space

space space space space equals space 1550 over 1350 space cross times space 4 space cross times space 10 to the power of 4 space

space space space space equals space 45.92 space cross times space 10 cubed space Hz
space
space space space space equals space 45.92 space KHz