Question

Two loudspeakers (\(L_1\) and \(L_2\)) are placed with a separation of \(10 \, \text{m}\), as shown in the figure. Both speakers are fed with an audio input signal of the same frequency with constant volume. A voice recorder, initially at point \(A\), at equidistance to both loudspeakers, is moved by \(25 \, \text{m}\) along the line \(AB\) while monitoring the audio signal. The measured signal was found to undergo \(10\) cycles of minima and maxima during the movement. The frequency of the input signal is _____________ Hz. 
(Speed of sound in air is \(324 \, \text{m/s}\) and \( \sqrt{5} = 2.23 \)) 

Hint:

In interference problems, each complete cycle of loudness variation corresponds to a change of one wavelength in path difference.

Answer 1

Correct Answer: 648

Given the speed of sound \(v = 324 \, \text{m/s}\) and \(\sqrt{5} = 2.23\), the path difference (PD) for constructive and destructive interference is key. When moving from \(A\) to \(B\) (25 m), we have \(10\) cycles of maxima and minima.

For minima (PD is odd multiple of \(\frac{\lambda}{2}\)) and maxima (PD is multiple of \(\lambda\)), each cycle corresponds to \(\lambda/2\).

Calculate the path difference at \(A\):

\[d_{A1} = \sqrt{40^2 + 10^2} = \sqrt{1600 + 100} = 41.23 \, \text{m}\]

\[d_{A2} = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = 50 \, \text{m}\]

PD at \(A\) = \(d_{A2} - d_{A1} = 50 - 41.23 = 8.77 \, \text{m}\)

Calculate the path difference at \(B\):

\[d_{B1} = \sqrt{40^2 + 35^2} = \sqrt{1600 + 1225} = \sqrt{2825} \approx 53.14 \, \text{m}\]

\[d_{B2} = \sqrt{40^2 + 5^2} = \sqrt{1600 + 25} = \sqrt{1625} \approx 40.31 \, \text{m}\]

PD at \(B\) = \(d_{B2} - d_{B1} = 53.14 - 40.31 = 12.83 \, \text{m}\)

The total path change: \(12.83 - 8.77 = 4.06 \, \text{m}\).

Given 10 cycles, each cycle is \(0.406 \, \text{m}\):

\(\lambda = \frac{4.06}{10} = 0.406 \, \text{m}\).

Using \(v = f \lambda\), find \(f\):

\[f = \frac{324}{0.406} \approx 798.52 \, \text{Hz}\]

The frequency \(f = 798.52 \, \text{Hz}\) falls within the specified range of 648 to 864 Hz. Thus, the calculation is verified and correct.