Two waves of the same wavelength, traveling in opposite directions, will interfere so as to produce a standing wave. No problem; I can get a bit of string and secure one end and wiggle the other until I get nodes and antinodes in the string. I imagine that I can bump things up to 2D and do the same with a sheet of metal, and get nodes that are lines across the metal surface instead of points on a string. Sand scattered across the sheet would settle into lines along these linear nodes.
Now I want to make a standing 2-D wave from three intersecting waves instead of two - so I take a hexagonal sheet of metal, and wiggle the edges that are 120º apart from each other. I think that the nodes form a set of lines at 60º to one another, like triangle grid paper, and the antinodes are points at the center of each triangle. But that’s just by intuition. Am I right, or are the nodes the centers of the triangles and the antinodes are along the lines? What do I see if I sprinkle sand across this sheet - lines or points?
What you describe is one possible mode of the system, yes: nodal lines running across the center of the hexagon, aligned with the points along the edge of the shape.
I’m guessing you’ve seen Chladni figures. For your case, each triangle would be vibrating 180 degrees out of phase with its neighbors, and this mode could be excited on a hexagonal Chladni plate by bowing at the center of the base of one of the triangles (with the added help of the enforced node at the center-mount.
I’m not sure, though, that this mode can be viewed as the superposition of three plane waves in the way described in the OP. I’ll have to think about it more.
That visualization is awesome! I slowed it way down, and it really does look like there are no nodes in the sense of points that stay at zero amplitude. It does seem like you should be able to find points on the plane where the three waves were 120º out of phase with each other so as to cancel out, though.
So, to be clear, you have a hexagonal plate ‘floating’ unconstrained in free space where you are introducing transverse (perpendicular to the surface) waves at three non-opposing edges and asking about the interference patterns that would be seen on the plate? Assuming that the surface is stiff or taut (such that you can neglect out-of-plane rotation) and the waves are in the same phase, you’ll see an interference pattern where each wave interacts, so a set of triangular points at different nodes and antinodes plus some ripple patterns where the interacting waves produce fringe patterns. You’ll also have waves reflecting off of the opposite edges of the plate (assuming they are unbounded) that will likely produce a lot of ‘noise’ near the points of the hexagon and different characteristics depending on the modal response of the free edges.
This isn’t quite the same setup as your description but Chladni Plates will give you an idea of what different patterns can look like:
Well, I was thinking more about an infinite plane with three waves of the same wavelength traveling at 120º to each other, but the hexagon made it easier for me to visualize the mechanics, moving up from the string→square sheet→hexagon. So no Chladni nodes from fixed points and no reflections off the edges of the plate.
It seems to me that on an infinite plane the phase of the waves wouldn’t matter. Wouldn’t the pattern just shift around a bit if the phases changed?
I take that back (the no nodes). It that visualization it looks like the nodes are points in a triangular lattice rather than lines, and the antinodes are points on a triangular lattice as well.
The animation above doesn’t have the reflected waves that are needed for standing wave patterns. You’d want to add in the same terms again with a flipped sign on the time.
Damnit, ninja’d by @Pasta with the Chladni figures!
As @MikeS says, the patterns that you get are a result of superposition, which is just a fancy way of just saying “addition of wave amplitudes”. Because the waves are dynamic the amplitudes will change but as long as they are in phase you will see consistent patterns of sand on the surface of the plate where the superposition of the waves results in minimum root mean square displacement. Note that the scenario you’ve concocted is essentially aphysical; although it can be represented mathematically (as done upthread) there isn’t really any way to apply a forced displacement along three edges without any other constraints that isn’t also going to result in some rigid body motion, so the plate would either have to have a lot of inertia or much of the input is going to go into the entire plate going up and down.
If we’re assuming an infinite domain and no boundary conditions then you don’t have to worry about edge effects or reflections, but yes, the phase synchronization definitely matters. The symmetry of the pattern only occurs because of all three waves being in phase. If one or two are out of phase then it is no longer radially symmetric about the centroid and you’ll get some very different pattern.
I take it back: there are fixed points for the superposition of three plane waves (even ignoring the important issue about reflections from the other side raised by the other posters upthread.) I don’t know of a good way to prove it, but it appears that the fixed points are discrete (i.e., points, not lines) and they form a hexagonal lattice. I’ve edited my visualization to include those fixed locations.
See my note about the reflected waves. If you make that change, you’ll see nodes that are loops with shapes that vary depending on the relative phases of things.
I don’t .. think.. that we need to have reflected waves on an infinite plane to get nodes. Can’t I find spots where all three waves are 120º out of phase and cancel out? And other spots where they are always in phase, and interfere constructively?
On the infinite plane I would intuit that the waves being out of phase (but coming in at the same angles) would just shift the lattice a bit in one direction or another. So not symmetric around the origin, but still the same pattern.
In general you do, although with three waves I imagine you can effect the same sort of cancellation with care in choice of phases and amplitudes.
Consider the simplest plane wave cos(x-t). That wave travels. But cos(x-t) + cos(x+t) is standing. (You can edit @MikeS’s simulation to be these waves to visualize this.)
