Question

In a large room, a person receives direct sound waves from a source 120 m away from him. He also receives waves from the 25 m high ceiling at a point half way between them. The two waves interfere constructively for wavelengths of

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

Answer 1

The correct option is D

$10cm,5m,\frac{10}{3}m,etc.$

Step 1: Given data

In a large room, a person receives direct sound waves from a source 120 m away from him.

He also receives waves from the 25 m high ceiling at a point halfway between them

Step 2: Formula used

Using the formula that demonstrates the relationship between the path difference and the phase difference, the constructive interference of the wavelength of sound waves may be calculated.

$\delta =\frac{2\pi }{\lambda }\times ∆x[where\delta =phasedifferenceofsoundwaves,\lambda =wavelengthofsoundwaves,∆x=pathdifference]$

Step 3: Calculating path difference

When a man receives sound from two sides, the route difference of the sound wave is calculated by taking the square root of the sum of the individual squares of the distance and subtracting it from the overall distance.

$$\begin{gathered} ∆x=2(\sqrt{{25}^2+{60}^2})-120 \\ ∆x=2(\sqrt{625+3600})-120 \\ ∆x=2(\sqrt{4225})-120 \\ ∆x=2\left(65\right)-120 \\ ∆x=130-120 \\ ∆x=10 \end{gathered}$$

Step 4: Calculating constructive interfaces

$$\begin{gathered} \delta =\frac{2\pi }{\lambda }\times ∆x \\ 2n\pi =\frac{2\pi }{\lambda }\times ∆x[As,\delta =2n\pi ] \\ n=\frac{∆x}{\lambda } \\ \lambda =\frac{∆x}{n} \end{gathered}$$

For, the constructive interference, the value of $nis1,2,3\dots \dots \dots$ and also substituting the path difference in the above equation, then

$\lambda =10forn=1$

For the next interference,
$\lambda =5forn=2$

For the next interference,
$\lambda =\frac{10}{3}forn=3$

Hence, option D is the correct answer.