Interference: Calculating the direction of the resultant wave

[franco1991]

Two coherent waveforms (could be pulses or wavetrains), that intersect at an angle, as in the picture below.

Assume that we are able to make the waveforms in phase and superposed at the point of intersection, such that they constructively interfere.

In what direction does the resultant wave propagate? Both waves, though superposed, are traveling in different directions. Does the resultant wave travel in one of the two component (original) waveforms' directions, or does it propagate forward at a 90 degree angle from the point of intersection, such that the resultant wave is traveling in a slightly different direction that either of the original waveforms (forward from the point at which they were superposed)?

 

[sophiecentaur]

Are A and B, slits or just points in space?

 

[franco1991]

Just points in space, not slits.

 

[sophiecentaur]

OK then. There is no 'resultant wave'. There are just the same two waves which add vectorially at every point in space to produce resultant fields. The interference pattern will be a 'waffle iron' shape, observed from side on, with many localised peaks and troughs. If you put the proverbial 'projection screen' in some position, you'll get the well known stripes pattern.

You are getting a combination of a standing wave and the two progressive waves. In the limit, when the two waves are traveling in opposite directions (θ=180°) and with equal amplitude, you get just a standing wave and no net energy flow.

 

[franco1991]

I appreciate your help. I have a questions and points of clarification though.

1.) You say "at all points in space". But they only are superposed at the point of intersection, so they can only sum vectorially at that point of intersection. Say, for example, that they are either coherent laser beams or that they each travel thru waveguides, which intersect at the point shown in the diagram, rather than spreading out thru space in circles (such that they can only superpose at the point of intersection; they don't overlap at any point but the point of intersection).

2.) In the scenario I'm asking about, the waves never travel in opposite directions, θ never equals 180°, so they never form standing waves. They are each propagating towards the point of intersection (thus in the same general direction), just at different angles.

3.) By resultant fields, you mean that there are multiple resultant waves in diff. points of space, not just one?

So, at the point of intersection, are the waves summing their respective amplitudes? If they aren't, why, since they are satisfying the conditions of interference (in phase and superposed)? And if they DO, then what direction does the new wave travel (the original question)?

 

[sophiecentaur]

What "new wave"? Neither of the incident waves can affect the other. Don't confuse fields adding up with a new progressive wave. Where could it come from?
If you have two narrow beams, the resulting power distribution will not lie outside the two. The pattern is multiplicative - beam pattern times fringe pattern.
Your diagram implied infinite extent for the waves which is why I used the phrase "over all space". But the pattern would be limited to within the beams.
What I wrote applies to all values of angle. 180 is just the extreme case.

 

[franco1991]

I meant resultant wave rather than new wave - a single wave that is the sum of each incident wave's amplitude, rather than a new wave created out of nothing (thus violating energy conservation laws).

Say that rather than wavetrains they are pulses. Would the two pulses sum their amplitudes at the point of intersection, but then continue on in their original directions (with their original amplitudes) after they've intersected, such that there is no "permanent" resultant field with an amplitude that is the sum of the incident waves' amplitudes? (permanent in that it continues to propagate away from the point of intersection with the summed amplitude).

I find it difficult to understand how the resultant fields (the parts of the wave that have summed each incident wave's respective amplitude) are limited to just the point of intersection, because parts of the beam do constructively interfere and sum the amplitudes of the original waves, the summed fields have to propagate forward, and if they do propagate forward they need a determinate direction, don't they?
A wave (according to my understanding) with a summed amplitude must propagate forward, it can't revert to one of the original wave's original amplitudes without interfering destructively with another wave, and it has to propagate in some direction, so it must be in the direction of one of the original waves, or a new direction. So I think I'm confused by your saying that the resultant field (interference) is limited to within the beams. I can see that it's creation is limited to within the beams, but doesn't it propagate forward in some direction with a summed amplitude?

I suppose I'm confused because I'm picturing it (your explanation) as the pulses (or beams) interfering (creating a resultant field of "pulse"), but then having that resultant field "split apart" again into the original two waves.

Is it that waves have to be perfectly parallel to actually produce a resultant wave that is the sum of their respective amplitudes? That a wave or field cannot intersect another at an angle and produce a single resultant wave that has a summed amplitude?