Standing waves and resonances are at the heart of musical instruments, electronic technology, and atomic systems. In this posting we examine standing waves in another dimension, adding the complex dimension. In the last posting, we introduced standing waves. We examined the phasors for standing waves at isolated points along the wave. We can extend this to a continuous representation of the phasors, similar to what we did for traveling waves in the posting on Waves using complex phasors.
The animation below as it initially starts, shows a single traveling wave in a rope in front of a gridded wall. We are assuming that there is no reflection of waves at the far end, in all aspects of this animation. Mouse over the animation to start it, and mouse off to suspend it, and back on again to allow it to continue. This is useful if you want to examine it stopped. You can click on the animation to cause the waves to be restarted from zero. If you click on "right", a traveling wave will be launched from the right. The "both" button will give you two traveling waves, traveling in opposite directions. Once they meet and interfere, you will see a standing wave as discussed in the last posting Superposition and standing waves.
The man and woman are holding on to either ends of the rope. Waves on the rope are the only "real" waves in this animation. The rest of the "waves", that you are about to examine, are waves in the phasors, which in turn are just a way of keeping track of waves, but are not real, physical waves themselves...just like a clock face is not the time, but it shows a representation of the time. On the other hand, even though waves in phasors are not physical waves, understanding them is central to the understanding of waves.
If you turn on (click on) "both" and "components", you will see the two traveling waves that make up a standing wave. As before, you can observe the two traveling waves sliding in opposite directions to cause (or add up to) the standing wave. As before, the appearance of the two traveling waves is quite different than the standing wave. The traveling waves maintain their shapes as they slide along, while the standing wave does not move, but oscillates in place. I should state again, that only the wave on the gray rope is real or physical. The pink and bluish "waves" show what you would observe if you had only one of the traveling waves. The two waves represent the mathematical dissection of the standing wave.
Click "components" again to turn off that feature for a minute. Now click on "phasors". This will show you a continuous helical representation of the complex phasor of the wave, similar to that seen in Waves using complex phasors. I show phasor diagrams every so often on the x axis. In principle there should be a phasor diagram for every possible point on the x axis, but that would obscure the phasor waves. Alternately, I could have not shown the phasor diagrams and said the phasor wave is plotted in a special "space", where one axis is imaginary displacement of the wave, one is the real displacement of the wave, and one is the propagation direction. To see such a plot of the phasor wave in this special space, click on "phasor dials" to hide the phasor diagrams.
When the animation starts afresh, (click on it to restart it if needed) and if the "left" or the "right" button is activated, a helical phasor wave will be launched from the appropriate end. This phasor wave represents the loci of all the tips of the phasor vectors in the phasor diagrams for varying x and time. When "both" is active, helical phasor waves are launched from both ends and come together in the center to interfere. What results is quite unintuitive. It is not at all a helix. Instead, the resulting phasor wave lies entirely in a plane, a plane that rotates about the axis of propagation. It looks like the standing wave at its maximum (see the standing wave to the left of the phasor) but now instead of oscillating in place, the complex phasor wave spins around the x axis. The phasor looks a bit like a potato masher twisting along its axis. This illustrates on of the important features of a standing wave, that the phase is the same (or 180° different, equivalent to a minus amplitude) at all point along it. This phase changes in time, but it is not a function of position, as it is with a traveling wave.
If you read the labeling on the phasor diagram axes carefully, you will see that we have twisted the axes of the complex plane by 90° from the normal orientation; so that the real axis is vertical. In this orientation, the actual standing wave is the projection of the phasor wave, projected to the left. That is to say, the actual wave is like a shadow of the phasor if the light were coming in horizontally and casting the shadow of the phasor onto the vertical tiled surface.
You can now turn on the "components". The result is somewhat complicated .... quite a tangle so don't be too disappointed if you have trouble following my explanation. If you skipped some of the discussion above, you might want to revisit it. You need to understand that before working on the full complexity of phasors with the components showing. It also might be useful to mouse off the animation from time to time to stop it and examine all the details better. The animation shows the red and blue helical phasor waves for the two traveling waves, in addition to their sum, the standing wave. Note that there are red and blue vectors on the phasor diagrams (the "phasor dial" need to be on to see this). If you study the animation carefully in this setting, you will see that the phasor wave of one component is a helix traveling in one direction, and that of the other traveling wave is a helix traveling in the other direction. At any particular plane, for example, at one of the planes of the phasor diagrams, the three waves are represented by vectors, all of which are rotating in the same direction (counterclockwise) and at the same rotational speed. At different phasor diagrams, the angle between the red and blue vectors differ and consequently their sum varies.
You may have noticed that the wave motion slows down when the "phasor" button is activated and that the "components" slow it down even more. These additional features increase the amount of computations that your computer is doing and can result in a slower animation. A faster computer will slow down less.
The math of complex helical phasors for standing waves
In the previous posting Superposition and standing waves, we derived equations for the cosine representation of a standing wave:
The complex representation of the same wave was derived to be:
We see that in the cosine representation, the x function, cos κx, is multiplied by the time oscillating function, cos ωt, producing an oscillating version of the original x function. This is what we see in the animation above for the actual wave, the one on the left against the gridded wall. On the other hand, in the complex representation, the space function cos κx is multiplied by the complex rotor function eiωt. The complex rotor function has a constant magnitude of one and rotates in time in the complex plane as seen in a previous animation on Complex phasors. It is this complex rotor that is responsible for spinning the phasor in the complex plane. Thus in the complex representation, we see a constant space function rotating in the complex plane as shown in the animation above.
As is always the case, the actual wave is the real part of the complex phasor. So it makes sense that the actual wave is the horizontal projection of the complex phasor wave, remembering that our real axis is vertical.
Physical helical waves on a rope
One additional point, that I might clear up is that you can easily make an actual or physical helical wave on a rope, and also that a superposition of two such waves propagating in opposite directions looks just like the phasor wave in the animation above. You might get a rope and a friend and try this. At the same time, such a helical wave is not exactly the same thing as what we are trying to portray in the animation above. In the animation, the rightmost wave is the complex phasor wave representing the actual flat standing wave to the left. One of its axes is imaginary. The helical wave in the animation is solely a mathematical representation of a real planar wave on a rope. The complete complex phasor representation of an actual helical wave on a rope would require five dimensions and thus is very hard to illustrate. Most scientists and engineers simply write the complex formula for the phasor of this wave and stop there. If they really want a picture or animation, they would draw a complex phasor drawing for each component of the wave, that is, two complex phasor wave drawings. We'll stop here, since this is probably a little too much complication for this level. Even though these complex phasors can be difficult to draw, animate, or visualize, wave related calculations are made much easier by their use.
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