A band playing music at a frequency f is moving towards a wall at a speed v$_{b}$. A car is following the…
A band playing music at a frequency f is moving towards a wall at a speed v$_{b}$. A car is following the band with a speed v$_{m}$. If v is the speed of sound, obtain an expression for the beat frequency heard by the driver sitting in the car.
Solution
Sol. Here → vo = vm vs = vb →
Frequency received by driver from band,
$f_1 = f\left(\frac{v - vo}{v + vs}\right) = f\left(\frac{v - vm}{v + vb}\right) \qquad \ldots (i)$
[figure showing car (speed v$_{m}$), band (observer speed vo) and wall]
Here, → vs = vb vo = 0 →
$f' = f\left(\frac{v - vo}{v + vs}\right) = f\left(\frac{v}{v + vb}\right)$
Now, wall will act as source of frequency $f'$ and driver as observer.
Again, → vo = vm vs = vb →
Frequency received by driver after reflection from wall,
$f_2 = f'\left(\frac{v - vo}{v + vs}\right) = f\left(\frac{v}{v + vb}\right)\left(\frac{v + vm}{v}\right) = f\left(\frac{v + vm}{v - vb}\right)$
Beat frequency heard by driver,
$f_2 - f_1 = f\left(\frac{v + vm}{v - vb}\right) - f\left(\frac{v - vm}{v + vb}\right)$
$= \dfrac{f}{(v^2 - v_b^2)}\big[(v + v_m)(v + v_b) - (v - v_m)(v - v_b)\big]$
$= \dfrac{f}{(v^2 - v_b^2)}\{[v^2 + vv_b + v_mv + v_mv_b] - [v^2 - vv_b - v_mv + v_mv_b]\}$
$= \dfrac{f\cdot 2v (v_b + v_m)}{(v^2 - v_b^2)}$
$= \dfrac{2fv (v_b + v_m)}{(v^2 - v_b^2)}$
Answer: $\dfrac{2fv(v_b+v_m)}{v^2-v_b^2}$