91Ó°ÊÓ

Condition of destructive interference is$\mathbf{Δ\pm ô}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathbf{}\mathbf{λ}$$\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$ and for constructive interference is $\mathbf{Δ\pm ô}\mathbf{=}\mathbf{²Ôλ}\mathbf{;}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$, where$\mathbf{∆}\mathit{l}$ is the path difference and$\mathit{λ}$ is the wavelength of the sound emitted from the speakers.

From the graph, we can calculate d.

$$\begin{gathered} d=\sqrt{2^2+4.5^2} \\ d=4.92m \end{gathered}$$

Notice that$\mathrm{Δ}l=d-4.5$

$$\begin{gathered} 4.92-4.5=n\mathrm{λ} \\ n\mathrm{λ}=0.42 \\ \mathrm{λ}=\frac{0.42}{n} \end{gathered}$$

But we need to calculate the frequency which can be calculated by dividing the speed of sound in air by the wavelength.

$$\begin{gathered} f=\frac{v}{\mathrm{λ}} \\ f=\frac{n×344}{0.42} \\ f=n819Hz;(n=1,2,3,4....) \end{gathered}$$

So, any frequency which is an integer number of 819 will produce constructive interference.

For destructive interference,$\mathrm{Δ}l=\frac{n}{2}\mathrm{λ};(n=1,3,5,...)$

Now, we know that $\mathrm{λ}=\frac{v}{f}=\frac{344}{f}$.

$$\begin{gathered} f=\frac{n×344}{2×0.42} \\ f=n409.5Hz;(n=1,3,5,...) \end{gathered}$$

So, the frequency that will produce the destructive interference is n409.5Hz.