Question

In a hall, a person receives direct sound waves from a source $120m$ away. He also receives waves from the same source which reach him after being reflected from the $25m$ 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. $30,20,\mathrm{10.....}$
• D. $35,25,\mathrm{15.....}$

Answer 1

Answer: (B)

Path difference $\Delta x(SA+AP)-SP$
$\Rightarrow \Delta x=(65+65)-120$
$\Rightarrow \Delta x=10m$
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
$=(2n)\frac{\lambda }{2},n=0,1,2,3\dots$
$10-\frac{\lambda }{2}=(2n)\frac{\lambda }{2};n=0,1,2,\dots$
$\Rightarrow 10=(2n+1)\frac{\lambda }{2};n=0,1,2,\dots .$
$\Rightarrow \lambda =\frac{20}{2n+1};n=0,1,2,\dots$
$\Rightarrow \lambda =20,\frac{20}{3},\frac{20}{5},\frac{20}{7}\dots$