The loudspeakers $L_1$ and $L_2$ driven by a common oscillator and amplifier are set up as shown in the…

The loudspeakers $L_1$ and $L_2$ driven by a common oscillator and amplifier are set up as shown in the figure. As the frequency of the oscillator increases from zero, the detector at $D$ recorded a series of maximum and minimum signals. A diagram shows two speakers L1 and L2 separated vertically by 9 m. L1 is connected horizontally by a line of length 40 m to detector D, and L2 is connected diagonally to D.
  1. (a) $165\text{ Hz}$
  2. (b) $330\text{ Hz}$
  3. (c) $495\text{ Hz}$
  4. (d) $660\text{ Hz}$

Solution

We have, $L_2D = \sqrt{(40)^2 + (9)^2} = 41\text{ m}$ Path difference, $\Delta x = L_2D - L_1D = 1\text{ m}$ For maximum, $\Delta x = 2n \frac{\lambda}{2}$ For $n = 1 \Rightarrow 2(1)\frac{\lambda}{2} = 1 \Rightarrow \lambda = 1 \Rightarrow f = \frac{v}{\lambda} = 330\text{ Hz}$

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