A boat is travelling in a driver in river with a speed `10 m//s` along the stream flowing with a speed `2m//s`. From this, boat, a sounjd transmitter is lowered into the river through a right support. The wavelength of the sound emitted from the transmitter inside the water is `14.45 mm`. Assume that attention of sound in water and air is negligible.
(a) What will be frequency detected by receiver kept inside downstream?
(b) The transmitter and the receiver are now pulled up into air. The air is blowing with a speed `5 m//s` in the direction opposite the river stream. Determine the frequency of the sound detected by the receiver. (Temperature of the air and water `= 20^(@)C`, Density of river water `= 10^(3) kg//m^(3)`. Bulk modulus of the water `= 2.088 xx 10^(6) Pa`, Gas constant `R = 8.31 J//mol-K`, Mean molecular mass of air `= 28.8 xx 10^(-3) kg//mol`, `C_(P)//C_(V)` for air `= 1.4`).

Explanation

(a) Speed of boat `v_(s) = 10 m//s`
Stream speed `u = 2 m//s`
Wavelength of sound in water `lambda = 14.45 mm`
`= 14.45 xx 10^(-3) m`
Speed of sound in water
`v = sqrt((B)/(rho)) = sqrt((2.088 xx 10^(9))/(10^(3))) = 1445 m//s`
Frequency of wave inside water
`f = (v)/(lambda) = (1445)/(14.45 xx 10^(-3)) = 10^(5) Hz`
`{:(rarrv_(s),,,rarrv_(o)=0,),(overset(.)S,rarru,,overset(.)O,),(,rarrv,,,):}`
`f' = f((v+u-v_(o))/(v+u-v_(s))) =10^(5) ((1445 + 2-0)/(1445+2-10))`
`= 1.007 xx 10^(5) Hz`
(b) Speed of sound in air
`v = sqrt((gamma RT)/(M_(0))) = sqrt((1.4 xx 8.3 xx (273 + 20))/(28.8 xx 10^(-3))) ~= 344 m//s`
`{:(rarrv_(s)=10m//s,,,rarrv_(o)=0,),(overset(.)S,rarrw,,overset(.)O,),(,rarrv,,,):}`
`f' = f((v+w-v_(o))/(v+w-v_(s))) = 10^(5)((344+5-0)/(344+5-10))`
`=1.03 xx 10^(5) Hz`