Question

In a hall, a person receives direct sound waves from a source $120 \,m$ away. He also receives waves from the same source which reach him after being reflected from the $25\, m$ high ceiling at a point half-way between them. The two waves interfere constructively for wavelengths (in metre) of

• A. $10,5,\frac{5}{2}$
• B. $20,\frac{20}{3},\frac{20}{5}.....$
• C. $30,20,10.....$
• D. $35,25,15.....$

Answer 1

Answer: (B)

$20,\frac{20}{3},\frac{20}{5}.....$

Answer 2

Path difference $\Delta x(S A+A P)-S P$
$\Rightarrow \Delta x=(65+65)-120$
$\Rightarrow \Delta x=10 \,m$
But at A the wave suffers reflection at the surface of rigid/fixed end or denser medium hence the wave must suffer an additional path change of $\frac{\lambda}{2}$ on a phase change of $\pi$.

$\Rightarrow$ Net path difference
$=\left(10-\frac{\lambda}{2}\right)$
For maxima (constrictive interference) Net path difference
$=(2 n) \frac{\lambda}{2}, n=0,1,2,3 \ldots$
$10-\frac{\lambda}{2}=(2 n) \frac{\lambda}{2} ; n=0,1,2, \ldots$
$\Rightarrow 10=(2 n+1) \frac{\lambda}{2} ; n=0,1,2, \ldots .$
$\Rightarrow \lambda=\frac{20}{2 n+1} ; n=0,1,2, \ldots$
$\Rightarrow \lambda=20, \frac{20}{3}, \frac{20}{5}, \frac{20}{7} \ldots$