Is it possible for two waves travelling at different speeds to be in phase? Why?
You can imagine two waves that are in phase at the point where they originate. But everywhere else the phases will have a non-constant difference.
Does this means at the detector, the interference pattern will keep changing?
Yes. You can write down the amplitude at the detector for each of the contributing waves (##\ \ A(x,t) = A_0 \; cos(\omega t - kx)\ \ ##) and see they have a difference that depends on time.
Ohh, ryt, thanks. Its starting to make sense now, my last question is that the constant bright and dark fringes are for waves at the same speed only?
My telepathic capabilities are rather limited . What fringes ?
You only see bright and dark fringes when your waves are light waves (because brightness and darkness are about light intensity) and light waves all travel at the same speed, so the question as asked doesn't make sense.
However, if you are asking about a stable pattern of high and low amplitudes at various points in space... No, such patterns do not require that the waves all travel at the same speed. A superposition of standing waves of different frequencies in a dispersive medium ("dispersive" just means that the speed is different for different frequencies, and a standing wave is a superposition of left-moving and right-moving travelling waves) will do the trick.
This question is not detailed enough to be sure what you want to know. You need to tighten up your specification of the problem. Are we dealing with two waves on two, one dimensional paths (e.g. two wires) or two waves, travelling in space, with three dimensional wave fronts?
The waves can only be in phase if they are the same frequency. (I presume that is what you assume.) If they are travelling at different speeds (two paths in different media, I presume) their phase relationship will remain constant at any particular point and any interference pattern (in phase regions and anti phase regions etc.) will remain stationary.
If the two waves are not of the same frequency then the relative phases will be changing at a rate governed by the frequency difference and the locations of the maxes and mins will march along rather than staying stationary.
Can you draw a diagram to support your particular query?
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