What phase difference between the waves would result in the combined wave having an amplitude of 1.37 times that of the two individual waves?

Question

10-08-2023
Physics

High School

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What phase difference between the waves would result in the combined wave having an amplitude of 1.37 times that of the two individual waves?

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atlas533 atlas533 · Answer 1 · 2024-07-06T21:23:22-04:00

The required phase difference for a combined wave amplitude of 1.37 times the individual waves can be found using the equation AR = 2A cos(φ/2) and solving for φ, where AR is the amplitude of the resultant wave, A is the amplitude of individual waves, and φ is the phase difference.

The phase difference between waves that results in a combined wave amplitude of 1.37 times that of the individual waves can be found using the superposition principle for waves. When two waves interfere constructively, the amplitude of the resultant wave is the sum of the individual amplitudes. Conversely, with destructive interference, the amplitude is the difference between the individual amplitudes.

Using the formula AR = 2A cos(φ/2), where AR is the amplitude of the resultant wave, A is the amplitude of the individual waves, and φ is the phase difference, we can find the required phase difference. Plugging in AR = 1.37A, we get 1.37A = 2A cos(φ/2). This equation can be solved for φ, yielding the phase difference required for the combined wave's amplitude to be 1.37 times that of the individual waves.

To solve for the phase difference φ, first divide both sides by 2A: cos(φ/2) = 1.37/2. Then, take the arccosine (inverse cosine) of both sides to find φ/2, and finally multiply by 2 to find the full phase difference φ.