When two sound waves travel in the same direction in a...

When two sound waves travel in the same direction in a medium, the displacement of a particle located at $\mathrm{x}$ at time $\mathrm{t}$ is given by $\mathrm{y}_{1}=0.05 \cos (0.50 \mathrm{p} \mathrm{x}-100 \mathrm{pt}) \&$ $y_{2}=0.05 \cos (0.46 p x-92 p t)$, where $y_{1}, y_{2}$ and $x$ are in meter and $t$ is in seconds. How many times per second does an observer hear the sound of maximum intensity ? (A) 4 (B) 8 (C) 12 (D) 16

When two sound waves travel in the same direction in a medium, the displacement of a particle located at $\mathrm{x}$ at time $\mathrm{t}$ is given by $\mathrm{y}_{1}=0.05 \cos (0.50 \mathrm{p} \mathrm{x}-100 \mathrm{pt}) \&$
$y_{2}=0.05 \cos (0.46 p x-92 p t)$, where $y_{1}, y_{2}$ and $x$ are in meter and $t$ is in seconds.
How many times per second does an observer hear the sound of maximum intensity ?
(A) 4
(B) 8
(C) 12
(D) 16

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Key Concepts

Beat Frequency

Beat frequency is the phenomenon that occurs when two waves of similar, but not identical, frequencies interfere. The result is a modulation of the sound intensity, where the amplitude fluctuates at a rate equal to the difference between the two frequencies. This concept is essential for understanding why an observer hears periodic increases in sound intensity when multiple wave sources are present.

Angular Frequency to Frequency Conversion

Angular frequency (measured in radians per second) and ordinary frequency (measured in hertz) are related by the equation f = ?/(2?). This conversion is crucial when working with wave equations expressed in terms of angular frequencies, as it allows one to determine the actual oscillation rate of a wave, which is necessary for calculating phenomena like beat frequency.

Superposition Principle and Wave Interference

The superposition principle states that when two or more waves overlap in a medium, the resultant displacement at any point is the algebraic sum of the displacements due to the individual waves. This principle underlies the interference of waves, including the formation of beats when waves of slightly different frequencies constructively and destructively interfere over time.

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