Let W and W' be the two parallel walls and the bat is flying towards the wall W. The bat will hear two sounds, one from the reflection of the sound produced by it from the front wall W and other from the reflection from the hind wall W'.
The frequency of sound produced by the bat, ν= 4.5x10⁴ Hz. For the front wall W, the apparent frequency coming to it ν'=330ν/(330-6) Hz, {Since source approaching the observer} =330ν/324 Hz
This apparent frequency is reflected and for the bat now the wall W is source and observer (bat) moving. The apparent frequency for the bat is now ν₁ =(V+u)ν'/V
→ν₁ ={(330+6)/330}ν'
→ν₁ = 336ν'/330 =(336/330)*(330/324)ν
→ν₁ = 336ν/324
→ν₁ = 336*4.5x10⁴/324 Hz
→ν₁ = 1.037*4.5x10⁴ Hz
→ν₁ = 4.67x10⁴ Hz
At the back wall W', the coming sound has an apparent frequency ν" = 330ν/(330+6) {The source is receding, observer stationary} ν" = 330ν/336
When it is reflected, it is the source for the bat.
This source is stationary and the observer receding. The apparent frequency ν₂ =(V-u)ν"/V
→ν₂ = (330-6)ν"/330 Hz
→ν₂ =324ν"/330 Hz
→ν₂ =(324/330)(330/336)ν Hz
→ν₂ =(324/336)*4.5x10⁴ Hz
→ν₂ =0.964*4.5x10⁴ Hz
→ν₂ =4.34x10⁴ Hz
The beat frequency = Difference between the two frequencies =ν₁-ν₂ = 336ν/324 - 324ν/336
=(336²-324²)ν/(336*324)
=(336+324)(336-324)ν/(336*324)
=660*12ν/(336*324)
=110*2ν/(56*54)
=55ν/(28*27) =0.0727*4.5x10⁴ Hz
=0.327x10⁴ Hz
=3270 Hz
