91Ó°ÊÓ

Constructive interference occurs when the difference of the distances of each source from point P is an integer number of wavelengths. The interference is destructive when this difference of path lengths is a half integer number of wavelengths.

The wavelength is,

$\mathrm{v}=\mathrm{λ´Ú}$

where v is the wave velocity, f is the frequency and $\mathrm{λ}$is the wavelength of the wave.

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The value of $x$ ranges from 0 to L, where $L=2.00m$is the distance between the speakers.

The difference in path length is

$\begin{array}{c}\mathrm{Δ}l=(L−x)−x\\ =L−2x\\ x=(L−\mathrm{Δ}l)/2\end{array}$

destructive interference occurs for path difference $\mathrm{Δ\pm ô}=(\mathrm{n}+(1/2\left)\right)\mathrm{λ}$and constructive interference for $\mathrm{Δ\pm ô}=\mathrm{²Ôλ}$.

(a) Destructive interference: $\mathrm{n}=0\mathrm{gives}\mathrm{Δ\pm ô}=0.835\text{″¾}\mathrm{andx}=0.58\text{″¾}.$

And at $\mathrm{n}=-1\text{ g¾±±¹±ð²õ}\mathrm{Δ\pm ô}=-0.835\text{″¾}\mathrm{andx}=1.42\text{″¾.}$

For any other value of $n$, P will not lie between the speakers.

(b)Constructive interference: $\mathrm{n}=0\text{ g¾±±¹±ð²õ}\mathrm{Δ\pm ô}=0\text{″¾and}\mathrm{x}=1.00\text{″¾}.$

And at$\mathrm{n}=1\text{gives}\mathrm{Δ\pm ô}=1.67\text{″¾and}\mathrm{x}=0.17\text{″¾}.$

At $\mathrm{n}=-1\text{ g¾±±¹±ð²õ}\mathrm{Δ\pm ô}=-1.67\text{″¾and}\mathrm{x}=1.83\text{″¾.}$

For any other value of $n$, P will not lie between the speakers

(c)Treating the speakers as point sources is a poor approximation for these dimensions, and sound reaches the given points after multiple reflections from the walls, ceiling or floor.