A band playing music at a frequency `f` is moving towards a wall at a speed `v_(b)`. A motorist 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 motorist.

Explanation

Frequency received by motorist from band
`{:(rarr v_(o)=v_(m),,v_(s)=v_(b)rarr),(overset(*)(O),overset(v)rarr,overset(*)(S)):}`
`f_(1) = f((v-v_(o))/(v+v_(s))) = f((v-v_(m))/(v+v_(b)))` (i)
Frequency received by wall
`{:(rarr v_(s)=v_(b),,v_(o)=0 rarr),(overset(*)(S),overset(v)rarr,overset(*)(O)):}`
`f' = f((v-v_(o))/(v+v_(s))) = f((v)/(v+v_(b)))`
Now wall will act as source of frequency `f'` and motorist as observer.
Frequency received by motorist after reflection from wall
`{:(rarr v_(o)=v_(b),,rarr v_(s)=0),(overset(*)(O),overset(v)larr,overset(*)(S)):}`
`f_(2) = f'((v-v_(o))/(v+v_(s)))`
`= f((v)/(v+v_(b)) (v+v_(m))/(v)) = f((v+v_(m))/(v-v_(b)))`
Beta frequency heard by motorist
`f_(2) - f_(1) = f((v+v_(m))/(v-v_(b))) -f((v-v_(m))/(v+v_(b)))`
`= (f)/((v^(2)-v_(b)^(2)))[[(v + v_(m))(v+v_(b))],[-(v-v_(m))(v-v_(b))]]`
`= (f)/((v^(2)-v_(b)^(2))) xx [{v^(2) + v v_(b) + v_(m)v+v_(m)v_(b)} - {v^(2) -v v_(b)-v_(m)v+v_(m)v_(b)}]`
`= (f.2v(v_(b) + v_(m)))/((v^(2) - v_(b)^(2))) = (2f v(v_(b) + v_(m)))/((v^(2) - v_(b)^(2)))`