Question:
Two sources of sound are emitting progressive waves \(y_1 = 4sin708\,πt\) and \(y_2 = 3sin700\,πt\). The sources are placed close to each other. The number of beats heard per second and intensity ratio between waxing and wanning are respectively
Two sources of sound are emitting progressive waves \(y_1 = 4sin708\,πt\) and \(y_2 = 3sin700\,πt\). The sources are placed close to each other. The number of beats heard per second and intensity ratio between waxing and wanning are respectively
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Frequencies come from the angular frequencies, and intensity goes as amplitude squared.
Updated On: Oct 1, 2026
- \(4\) , \(16:9\)
- \(8\) , \(16:9\)
- \(8\) , \(49:1\)
- \(4\) , \(49:1\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
For \(y = A\sin\omega t\), the frequency is \(f = \frac{\omega}{2\pi}\). Beats per second equal the difference in frequency. Intensity is proportional to the square of the amplitude.
Step 2: Key Formula or Approach:
Maximum amplitude \(= A_1 + A_2\) and minimum amplitude \(= A_1 - A_2\).
Step 3: Detailed Explanation:
\(f_1 = \frac{708\pi}{2\pi} = 354\) Hz and \(f_2 = \frac{700\pi}{2\pi} = 350\) Hz.
Beats per second \(= 354 - 350 = 4\).
\[ \frac{I_{max}}{I_{min}} = \frac{(4 + 3)^2}{(4 - 3)^2} = \frac{49}{1} \]
The ratio \(16:9\) is the ratio of the intensities of the two sources, not of the maximum and minimum intensity.
Final Answer:
There are \(4\) beats per second and the intensity ratio is \(49:1\), option (D). \[ \boxed{4,\ 49:1} \]
For \(y = A\sin\omega t\), the frequency is \(f = \frac{\omega}{2\pi}\). Beats per second equal the difference in frequency. Intensity is proportional to the square of the amplitude.
Step 2: Key Formula or Approach:
Maximum amplitude \(= A_1 + A_2\) and minimum amplitude \(= A_1 - A_2\).
Step 3: Detailed Explanation:
\(f_1 = \frac{708\pi}{2\pi} = 354\) Hz and \(f_2 = \frac{700\pi}{2\pi} = 350\) Hz.
Beats per second \(= 354 - 350 = 4\).
\[ \frac{I_{max}}{I_{min}} = \frac{(4 + 3)^2}{(4 - 3)^2} = \frac{49}{1} \]
The ratio \(16:9\) is the ratio of the intensities of the two sources, not of the maximum and minimum intensity.
Final Answer:
There are \(4\) beats per second and the intensity ratio is \(49:1\), option (D). \[ \boxed{4,\ 49:1} \]
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