Conditions for waves to be coherent?

[MBBphys]

Homework Statement

Hello,
For two waves to be coherent, they must have the same frequency right? Does this on its own implies a constant phase different between any point on one wave and any on the other.

So, for example, if we had two waves with different wavelengths and velocities but equal frequencies traveling in opposite directions towards each other, and both had the same amplitude, would a stationary/standing wave be produced? A friend of mine said no, but I thought as the frequencies are the same, there is a constant phase difference between any two points as aforementioned, so a stationary wave would be produced. They didn't really explain why hence I turn to you :)

Thanks!

Homework Equations

Not really any relevant equations for this I know of (or at least, at A Level in the UK, there aren't any equations we learned).

The Attempt at a Solution

I thought that, instead of looking at waves, if we look at two points, one on each of the two waves concerned, it shouldn't matter if they have differing wavelengths and velocities, they will still have a constant phase difference if they have equal frequencies, because this means they take the same time to go through one complete cycle of the wave, hence difference between the fractions of the wave cycle passed will always be the same?

 

[olivermsun]

I guess there are a few things that you may want to consider for this question.
First of all, can you write an equation that represents a wave with a given frequency? (For example, take a simple sinusoidal wave moving along a string.)
How do you show "coherence," i.e., constant phase difference between any two points (either in time or space) along the wave?
Now consider: does your equation change for a wave moving in the opposite direction? For example, does the "frequency" change? How about the coherence?