Question

A boat is travelling in a river with a speed $10m/s$ along the stream flowing with a speed $2m/s$. From this boat, a sound transmitter is lowered into the river through a rigid support. The wavelength off the sound emitted from the transmitter inside the water is $14.45mm$. Assume that attenuation of sound in water and air is negligible.
(a) What will be the frequency detected by a receiver kept inside the river downstream?
(b) The transmitter and the receiver are now pulled up into air. The air is blowing with a speed $5m/s$ in the direction opposite the river stream. Determine the frequency of the sound detected by the receiver.(Temperature of the air and water $={20}^{o}C$; Density of river water $={10}^3$; Bulk modulus of the water $=2.088\times {10}^{9}Pa$; Gas constant $R=8.31j/mol-k$; Mean molecular mass of air $=28.8\times {10}^{-3}kg/mol;C_p/C_v$ for air $=1.4$)

Answer 1

speed of second in water

$=(k/e{)}^{1/2}$

$={\left(\frac{2.088\times {10}^9}{{10}^3}\right)}^{1/2}$

$=1445m/s$

$\therefore V_{water}=\frac{v}{\lambda }=\frac{1445}{14.45}$

$=100Hz$

speed of second in air

$\sqrt{\frac{\gamma RT}{M_0}}$

$=\sqrt{\frac{1.4\times 8.31\times 293}{28.8\times {10}^{-3}}}$

$=344m/s$

$v_{air}\approx 4.2$

$a)v^{\prime }=\frac{(v+v_{water})}{(v+v_{water})-v_s}v$

$=\frac{1445+2}{1445+2-10}\times 100$

$=100.696Hz$

$b)v^{\prime }=\frac{(v+v_{air})}{(v+v_{air})+v_s}{v}_{air}$

$=\frac{344+5}{344+5-10}\times 4.2$

$=\frac{349}{339}$

$=4.324Hz$