In a large room, a person receives direct sound waves from a source 120 meters away from him. He also receives waves from the same source which reach him, being reflected from the 25 meter high ceiling at a point halfway between them. The two waves interfere constuctively for wavelength of

Options

A. `20,20//3`etc
B. `10,5,2.5`etc
C. `10,20,30`etc
D. `15,25,35`etc

Answer

A

Explanation

Let S be source of sound and P the person or listener The waves from S reach point P directly following the path SMP and being reflected from the ceiling at point A following the path SAP. M is mid-point of `SP` (i.e., `SM=MP`) and `angleSMA=90^@`
Path difference between waves `Delta x=SAP-SMP`
We have `SAP=SA+AP=2(SA)`
`=2sqrt((SM)^2+(MA)^2)=2sqrt(60^2+25^2)=130m`
Path difference `=SAP-SMP=130-120=10m`
`:.` Path difference due to reflection for ceiling `=(lamda)/(2)`
`:.` Effective path difference `Delta x=10+(lamda)/(2)`
For constructive interference
`Delta x=10+(lamda)/(2)=nlamdaimplies(2n-1)(lamda)/(2)=10(n=1,2,3`..)
Wavelength `lamda=(2xx10)/((2n-1))=(20)/(2n-1)`
wavelengths are `lamda=20`, `(20)/(3), (20)/(5), (20)/(7)`,`(20)/(9)`,..
`=20m,6.67m,4m,2.85m,2.22m`,.