Question

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

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

Answer 1

The correct option is A

20, 20/3, 20/5 etc

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 $\angle SMA={90}^{\circ }$

Path difference between waves $△x=SAP-SMP$

We have $SAP=SA+AP=2\left(SA\right)$

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

$\therefore$ Path difference = SAP - SMP = 130 - 120 = 10 m

Path difference due to reflection from ceiling = $\frac{\lambda }{2}$

$\therefore$ Effective path difference $△x=10+\frac{\lambda }{2}$

For constructive interference

\(\triangle x=10+\dfrac{\lambda}{2}=n\lambda\Rightarrow (2n-1)\dfrac{\lambda}{2}=10(n=1,2,3....)

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

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