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

Let S be source of sound and P the person or listner.
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 $\angle \mathrm{SMA}={90}^{\mathrm{o}}$
 Path difference between waves $\mathrm{\Delta x}=\mathrm{SAP}-\mathrm{SMP}$
 We have  $\mathrm{SAP}=\mathrm{SA}+\mathrm{AP}=2\left(\mathrm{SA}\right)$
 

$=2\sqrt{[{\left(\mathrm{SM}\right)}^2+{\left(\mathrm{MA}\right)}^2]}=2\sqrt{({60}^2+{25}^2)}=130\mathrm{m}$

$\therefore$Path difference = SAP – SMP = 130 -120 = 10 m
Path difference due to reflection from ceiling = $\frac{\mathrm{\lambda }}{2}$
$\therefore$Effective path difference $\mathrm{\Delta x}=10+\frac{\mathrm{\lambda }}{2}$

For constructive interference 

$\mathrm{\Delta x}=10+\frac{\mathrm{\lambda }}{2}=\mathrm{n\lambda }\Rightarrow (2\mathrm{n}-1)\frac{\mathrm{\lambda }}{2}=10(\mathrm{n}=1,2,3....)$

$\therefore$Wavelength $\mathrm{\lambda }=\frac{2\times 10}{(2\mathrm{n}-1)}=\frac{20}{2\mathrm{n}-1}$. The possible wavelength are $\mathrm{\lambda }=20,\frac{20}{3},\frac{20}{5},\frac{20}{7},\frac{20}{9},$  …..

= 20 m, 6.67 m, 4 m, 2.85 m, 2.22 m, …..