Two waves of intensity ratio \( 1 : 9 \) cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is \( I_1 \) (b) Waves are coherent is \( I_2 \) and differ in phase by \( 60^\circ \) If \( \frac{I_1}{I_2} = \frac{10}{x} \) then \( x = \) _____.
Correct Answer: 13
Solution and Explanation
Incoherent Wave:
The intensity, denoted as \( I_1 \), is the aggregate of individual intensities \( I_A \) and \( I_B \):
\[ I_1 = I_A + I_B \quad \Rightarrow \quad I_1 = I_0 + 9I_0 = 10I_0 \]
Coherent Wave:
The intensity, denoted as \( I_2 \), is calculated using the formula:
\[ I_2 = I_A + I_B + 2 \sqrt{I_A I_B} \cos(60^\circ) \]
Upon substitution of values and simplification:
\[ I_2 = I_0 + 9I_0 + 2 \sqrt{I_0 I_0} \cdot \cos(60^\circ) = 13I_0 \]
Intensity Ratio:
The ratio between \( I_1 \) and \( I_2 \) is determined as:
\[ \frac{I_1}{I_2} = \frac{10I_0}{13I_0} = \frac{10}{13} \]
Top Questions on Waves
- Two coherent loudspeakers \(L_1\) and \(L_2\) are placed at a separation of \(10\,\text{m}\) parallel to a wall at a distance of \(40\,\text{m}\) as shown in the figure. On a width \(AB\) on the wall, 10 maxima and minima are found. If the velocity of sound is \(324\,\text{m s}^{-1}\), find the frequency of sound. (Given: \( \sqrt{5}=2.23 \)).
- The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is \( \frac{5}{x} \). The value of \( x \) is ________.
- In an open organ pipe \( \nu_3 \) and \( \nu_6 \) are 3rd and 6th harmonic frequencies, respectively. If \( \nu_6 - \nu_3 = 2200\ \text{Hz} \), then the length of the pipe is _________ mm. (Take velocity of sound in air as \(330\ \text{m s}^{-1}\).)
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 \( \sqrt{5} = 2.23 \))
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