Wave represented by the equation \( y_1 = A \cos(kx - \omega t) \) is superimposed on another wave to form a stationary wave such that the point \( x = 0 \) is a node. The equation representing the wave is given by
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- \( A \cos(kx + \omega t) \)
- \( -A \cos(kx + \omega t) \)
- \( A \sin(kx + \omega t) \)
- \( -A \sin(kx - \omega t) \)
The Correct Option is B
Solution and Explanation
When two waves interfere to form a stationary wave with a node at \( x = 0 \), the equation of the resulting wave can be obtained by adding the two individual waves. The general equation for a stationary wave is: \[ y = 2A \cos(kx) \cos(\omega t) \] Given the form of the wave \( y_1 = A \cos(kx - \omega t) \), for \( x = 0 \) to be a node, the second wave must have the opposite sign of the first, resulting in the equation \( -A \cos(kx + \omega t) \).
Hence, the correct answer is (b).
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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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