Two sound waves of equal displacement amplitude A and nearby frequencies f₁ and f₂ are initially in phase. Which expression correctly represents their resultant displacement and displays the beat modulation explicitly?
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Question

Two sound waves of equal displacement amplitude A and nearby frequencies f₁ and f₂ are initially in phase. Which expression correctly represents their resultant displacement and displays the beat modulation explicitly?

Options

G1 Physics · Accepted Answer

Let y₁ = A cos(2πf₁t), y₂ = A cos(2πf₂t). Using cosα+cosβ = 2cos[(α−β)/2]cos[(α+β)/2], we obtain y = 2A cos[π(f₁−f₂)t] cos[π(f₁+f₂)t]. The rapidly oscillating factor has frequency (f₁+f₂)/2, while the loudness maxima repeat at the beat frequency fᵦ = |f₁−f₂|. Correct option: A.