Question

In a large room, a person receives direct sound waves from a source $120\text{m}$ away from him. He also receives the sound waves from the same source, which reaches him after being reflected from the $25\text{m}$ high ceiling at a point midway between them. The two waves interfere constructively for wavelengths (in $\text{m}$) of -

• A. $20,\frac{20}{3},\frac{20}{5}...$
• B. $10,5,2.5...$
• C. $10,20,30...$
• D. $15,25,35...$

Answer 1

The correct option is A $20,\frac{20}{3},\frac{20}{5}...$

Let $S$ be the source of sound and $P$ be the listener. The waves from $S$ reach point $P$ directly following the path $SMP$ and also after reflecting from the ceiling at point $A$ following the path $SAP$. $M$ is the mid-point of $SPi.e.SM=MP$ and $\angle SMA={90}^{\circ }$


Path difference between waves,

$\Delta x=SAP-SMP$

We have,

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

$=2\sqrt{(SM{)}^2+(MA{)}^2}$

$=2\sqrt{{60}^2+{25}^2}=130\text{m}$

$\therefore$ Path difference,

$\Delta x=SAP-SMP=130-120=10\text{m}$

We know that if, $\Delta x=\lambda ,2\lambda ,3\lambda ,...$, the interference will be constructive.

$\therefore \lambda =10\text{m},5\text{m},\frac{10}{3}\text{m},...$

Hence, option $\left(A\right)$ is the correct answer.