Question

Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone. For what frequencies does their sound at the speakers produce (a) constructive interference and (b) destructive interference?

Short answer

Any frequency which is an integer number of 819 will produce constructive interference.

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

Step-by-step solution

Condition of destructive interference

Condition of destructive interference is$\mathbf{\Delta l}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathbf{}\mathbf{\lambda }$$\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 $\mathbf{\Delta l}\mathbf{=}\mathbf{n\lambda }\mathbf{;}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$, where$\mathbf{∆}\mathit{l}$ is the path difference and$\mathit{\lambda }$ is the wavelength of the sound emitted from the speakers.

Calculate the constructive interference

From the graph, we can calculate d.

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

Notice that$\Delta l=d-4.5$

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

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

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

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

Calculate the destructive interference

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

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

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

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