Q39E - University Physics with Modern Physics

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?

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.