Answer
Two waves of significantly different frequencies cannot be coherent because the phase relationship between the two waves will change rapidly over time, causing the waves to appear out of phase with each other. In order for two waves to be coherent, their phase difference must remain constant over time. This can only happen if the two waves have the same frequency and a fixed phase relationship. Therefore, if two waves have significantly different frequencies, they cannot be coherent.
Work Step by Step
When we talk about waves being coherent, we mean that they have a consistent phase relationship with each other. This means that the peaks and troughs of the waves match up at the same point in time. However, if two waves have significantly different frequencies, their peaks and troughs will not match up at the same point in time, as the faster wave will have completed more cycles than the slower wave. This means that the phase relationship between the two waves will constantly be changing, making it impossible for them to be coherent. Think of it like trying to clap in sync with someone who is clapping at a different tempo than you, it will never match up.
