“Consider this image of a wave. Which wave would cause destructive interference at every point of the wavelength and completely cancel out the wave?”
The Correct Answer and Explanation is :
To achieve destructive interference at every point of the wavelength and completely cancel out a given wave, the second wave must be an exact replica of the original wave but inverted in phase. This means that the two waves should have the same amplitude, frequency, and wavelength, but their crests and troughs must be aligned in such a way that they are perfectly out of phase by 180 degrees.
Explanation:
Destructive interference occurs when two waves meet and combine to form a new wave with a lower amplitude than either of the original waves. When waves are perfectly out of phase, the crest of one wave coincides with the trough of the other. For example, if the original wave has a peak (crest) of +A and a trough of -A, then the second wave, which we will call Wave 2, must have a peak of -A and a trough of +A.
Mathematically, if we denote the original wave as ( y_1(x, t) = A \sin(kx – \omega t) ), where ( A ) is the amplitude, ( k ) is the wave number, ( \omega ) is the angular frequency, and ( x ) and ( t ) are position and time respectively, the inverted wave can be expressed as ( y_2(x, t) = -A \sin(kx – \omega t) ).
When these two waves are combined, the resulting wave ( y_{total}(x, t) = y_1(x, t) + y_2(x, t) ) becomes:
[
y_{total}(x, t) = A \sin(kx – \omega t) – A \sin(kx – \omega t) = 0
]
This shows that at every point in the wavelength, the two waves cancel each other out completely, resulting in a total amplitude of zero. Therefore, to ensure complete cancellation of the wave at all points, the secondary wave must be an exact inverted replica of the original wave.