A car moves with a speed of `54 kmh^-1` towards a cliff. The horn of the car emits sound of frequency 400 Hz at a speed of `335 m s^-1`. (a) Find the wavelength of the sound emitted by the horn in front of the car. (b) Find the wavelength of the wave reflected from the cliff (c) What frequency does a person sitting in the car hear for the reflected sound wave ? (d) How many beats does he hear in 10 seconds between the sound coming directly from the horn and that coming after the reflection ?

Explanation

`{:(rarr v_(s) underset(=15 m//s)(= 54km//hr),,rarr v_(o) = 0),(overset(*)(S)" "overset(v)rarr,,overset(*)(O)),(,,):}`
(a) Apparent frequency in front of car
`f' = f((v)/(v - v_(s))) = 400((335)/(335 - 15)) = 400 xx (335)/(320)`
`lambda' = (v)/(f') = (335 xx 320)/(400 xx 335) = 0.8 m`
(As source moves, wavelength changes)
(b) Frequency received by cliff is same `= 400 xx (335)/(320)` now cliff acts as source.
Wavelength of reflected wave
`= (335 xx 320)/(400 xx 335) = 0.8 m`
(As source is stationary, wavelength remains same)
(c ) Frequency of reflected sound
`{:(rarr v_(s) =15 m//s,,rarr v_(s) = 0),(overset(*)(O)" "overset(v)rarr,,overset(*)(S)^(f')):}`
`f_(r) = f'((v + v_(0))/(v)) = (400 xx 335)/(320) ((335 + 15)/(335))`
`= 437.5 Hz`
OR
`f_(r) = f((v + u)/(v - u)) = 400 ((335 + 15)/(335)) = 437.5 Hz`
(d) Beat frequency `= f_(r) - f = 437.5 - 400 = 37.5` but beat frequency `gt 10`, hence no beats are heard.