Destructive interference occurs when two waves have opposite displacements. Which statement is true about the resulting displacement?

Destructive interference occurs when two waves have opposite displacements. Which statement is true about the resulting displacement?

• A. The resulting displacement is zero at all points.
• B. The resulting displacement has a smaller magnitude than either wave.
• C. The resultant displacement is the sum of the displacements.
• D. The amplitude is greater than either wave.

Answer: (B)

When two waves meet, their displacements add at each point in space and time. This is the superposition principle. If the waves have opposite displacements, the total displacement is the algebraic sum with signs: y = y1 + y2. Destructive interference means they tend to cancel, so this sum behaves like subtraction. If the two amplitudes are equal, they cancel completely and the resultant displacement is zero; if they’re not equal, the cancellation is partial and the magnitude equals the difference between the two amplitudes. So the most general and correct statement is that the resulting displacement is the sum of the displacements (with the appropriate signs). Note that complete cancellation everywhere would require equal amplitudes and perfect opposite phase at every point, which isn’t generally the case. The resultant magnitude isn’t necessarily smaller than both original wave magnitudes in all situations, and it cannot exceed the larger of the two in the opposite-phase case.

When two waves meet, their displacements add at each point in space and time. This is the superposition principle. If the waves have opposite displacements, the total displacement is the algebraic sum with signs: y = y1 + y2. Destructive interference means they tend to cancel, so this sum behaves like subtraction. If the two amplitudes are equal, they cancel completely and the resultant displacement is zero; if they’re not equal, the cancellation is partial and the magnitude equals the difference between the two amplitudes. So the most general and correct statement is that the resulting displacement is the sum of the displacements (with the appropriate signs).

Note that complete cancellation everywhere would require equal amplitudes and perfect opposite phase at every point, which isn’t generally the case. The resultant magnitude isn’t necessarily smaller than both original wave magnitudes in all situations, and it cannot exceed the larger of the two in the opposite-phase case.