Question 315: Waves

Question

Two sound waves having displacements $x_1=2 \sin (1000 \pi t)$ and $x_2=3 \sin (1006 \pi t)$, when interfere, produce

Accepted Answer

To find the beat frequency and maximum intensity produced by the interference of the two sound waves, use the following concepts:
Beat Frequency
: The beat frequency is given by the absolute difference between the frequencies of the two interfering waves.
Maximum Intensity
: This is determined by the sum of the amplitudes of the two waves when they interfere constructively.
Solution
Identify Frequencies:
The standard form of a wave is $ x = A \sin(2 \pi f t) $, where $ f $ is the frequency in Hz.
For the wave $ x_1 = 2 \sin(1000 \pi t) $:
The angular frequency is $ 1000 \pi $.
Frequency $ f_1 = \frac{1000 \pi}{2 \pi} = 500 \, 	ext{Hz} $.
For the wave $ x_2 = 3 \sin(1006 \pi t) $:
The angular frequency is $ 1006 \pi $.
Frequency $ f_2 = \frac{1006 \pi}{2 \pi} = 503 \, 	ext{Hz} $.
Calculate Beat Frequency:
The beat frequency $ f_{	ext{beat}} $ is the difference between the two frequencies:
$$ f_{	ext{beat}} = |f_2 - f_1| = |503 - 500| = 3 \, 	ext{Hz} $$
Calculate Maximum Intensity:
When two waves interfere constructively, the maximum amplitude is the sum of the amplitudes.
Amplitude of $ x_1 = 2 $ and amplitude of $ x_2 = 3 $.
Maximum amplitude = $ 2 + 3 = 5 $.
The intensity (which is proportional to the square of the amplitude) is given by:
$$ 	ext{Maximum Intensity} = (2 + 3)^2 = 5^2 = 25 \, 	ext{units} $$
Conclusion
Thus, the waves produce a beat frequency of $ 3 \, 	ext{Hz} $ with a maximum intensity of $ 25 \, 	ext{units} $.
Correct Option: C
3 beats/s with maximum intensity 25 units

Answer

5 beats/s with maximum intensity 25 units

Answer

6 beats/s with maximum intensity 16 units

Answer

3 beats/s with maximum intensity 25 units

Answer

1 beats/s with maximum intensity 5 units

Waves

Two sound waves having displacements x_1=2 \sin (1000 \pi t) and x_2=3 \sin (1006 \pi t), when interfere, produce

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