In a large room, a person receives direct sound waves from a source $120$ metres away from him. He also…
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 reaches him after being reflected from the $25$ metres high ceiling at a point midway between them. The two waves interfere constructively for wavelength (in m) of
20, 20/3, 20/5, etc
10, 5, 2.5, etc
10, 20, 30, etc
15, 25, 35, etc
Solution
Let $S$ be the 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, $\Delta x = SAP - SMP$
(Diagram shows source $S$ and listener $P$ separated by $120\text{ m}$ with ceiling reflection point $A$ at height $h = 25\text{ m}$ above midpoint $M$.)
Now, $SAP = SA + AP = 2 (SA)$
$= 2\sqrt{(SM)^2 + (MA)^2} = 2\sqrt{(60^2 + 25^2)} = 130\text{ m}$
$\therefore$ Path difference, $\Delta x = SAP - SMP = 130 - 120 = 10\text{ m}$
Path difference due to reflection from ceiling $= \frac{\lambda}{2}$
$\therefore$ If $\Delta x = \frac{\lambda}{2}, \frac{3\lambda}{2}, \frac{5\lambda}{2}$, etc., interference will be constructive
$\therefore \lambda = 20\text{ m}, \frac{20}{3}\text{ m}, \frac{20}{5}\text{ m}$, etc.