A wave is represented by the equation y(x, t) = 3.2 cm sin(1000 rad/s t − 50 m⁻¹ x). Another wave, with the same wavelength and frequency, has an amplitude of 4.2 cm. If the two waves interfere constructively, then which equation could represent the superposition of the two waves? A) y(x, t) = 7.4 cm sin(1000 rad/s t + 50.0 m⁻¹ x) B) y(x, t) = 1.0 cm sin(1000 rad/s t − 50.0 m⁻¹ x) C) y(x, t) = 1.0 cm sin(1000 rad/s t + 50.0 m⁻¹ x) D) y(x, t) = 7.4 cm sin(1000 rad/s t − 50.0 m⁻¹ x)

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Quizplus · Accepted Answer

Constructive interference occurs when the waves are perfectly in phase with each other, meaning the crest of one wave aligns with the crest of the other wave and the trough of one wave aligns with the trough of the other wave. In this case, the two waves have the same wavelength and frequency, so the phase difference between them is solely determined by the difference in amplitude. Since the second wave has a larger amplitude than the first wave, the resulting superposition should have an amplitude equal to the sum of the amplitudes of the two waves, which is 3.2 cm + 4.2 cm = 7.4 cm. The phase difference of 50 m⁻¹ between the two waves corresponds to a displacement of one-quarter wavelength, so to produce constructive interference, the two waves must be added with the same sign. This means that the phase in the resulting wave must be -50 m⁻¹, which corresponds to a positive sign in front of the x term. Therefore, the correct equation for the superposition is y(x, t) = 7.4 cm sin(1000 rad/s t − 50.0 m⁻¹ x), or choice D.

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