I love the demonstration of two-source interference of sound waves - two speakers and a sine wave generator is all it takes to get one of those moments that kids remember for a long while after the course is over (now, if we can just get them to remember why it happens!).
The echoes in my lab aren't too bad, and we can actually map an interference pattern reasonably well, in large areas. At some point, I'd like to try this in a big space and have them mark nodes (quiet points) and antinodes (loud points) with cones or something, and see if we can recreate the classic illustration.
In the meantime, we mapped out a few points that were relatively easily found:
At this point, we already knew all about order lines and how these loud and soft points came to be; we had analyzed a diagram of the interfering waves, determined where constructive and destructive interference were happening, and noticed the pattern in the 'lines' of nodes and antinodes (and even brought those shapes back to the base definition of a hyperbola!).
This was their first chance to apply that to a live problem: what's the frequency of that annoying hum that I'm playing through the speakers, anyway? Are these measurements all that we need to determine that? After all, there's nothing about time here at all, so determining the frequency seems daunting at first.
"What order line is that first soft point on?" is the real catalyzing question that I ask the groups a few minutes into their whiteboarding, if they haven't figured out how to apply the two-source model yet.
From there, there are a few steps of reasoning, some triangle manipulation, and some unit conversions needed. The triangles and unit conversions shouldn't be an issue at this point, though they still are for some that haven't reached an unconscious competence level. These assumed skills can really derail you. Especially if a problem is already difficult enough to tax your faculties to the max to begin with, adding a protracted wrestle with a unit conversion or diagramming effectively is likely to push your brain into a useless fried state (like okra).
Once we've connected that the order number is important, that connects to something about the distances from the speakers to the point, so we need to find those distances, using the Pythagorean theorem:
OK, great, but what does the order number tell us? This is a place where the sound is quiet because there's destructive interference, which happens because the waves travel different distances to get here. The waves began synchronized, but since they've traveled different distances, they're at different points in their oscillation between high and low pressure (they're out of phase). At this place, the wave from the left speaker has traveled half of a wavelength further than the wave from the right speaker (that's the .5 in the order number!). Now we know two different ways to write the difference in travel distances:
That's quite a set of inferences and observations, each necessary to solve the problem. This number of inferences and observations is necessary for many problems that students try to solve, but we don't often think about the chain of logic so literally. A great deal goes on behind the scenes... or doesn't. Note too that there's knowledge from previous courses, previous terms and units of this course, and the current situation, but that it all has to fit together; this doesn't happen if students view understanding as disposable. That sort of mindset is like tying your leg to a tree and trying to go for a run - no matter what direction you go, you can only go so far.
I think that, if students can get comfortable spelling out their reasoning like this, then they can get better at making a linear argument, which is what all problem-solving is - each step must be supported by knowledge or information and must lead to the next step. Scatter-shot thinking is rampant, and it does not help problem-solving. Perhaps conscious effort on constructing a "chain of reasoning" can help a student be able to do this type of thinking unconsciously, too!