Of Particular Significance

How To Make A Standing Wave Without Edges

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 03/25/2024

I recently pointed out that there are unfamiliar types of standing waves that violate the rules of the standing waves that we most often encounter in life (typically through musical instruments, or when playing with ropes and Slinkys) and in school (typically in a first-year physics class.) I’ve given you some animations here and here, along with some verbal explanation, that show how the two types of standing waves behave.

Today I’ll show you what lies “under the hood” — and how you yourself could make these unfamiliar standing waves with a perfectly ordinary physical system. (Another example, along with the relevance of this whole subject to the Higgs field, is discussed in chapter 20 of the book.)

Strings, Balls and Springs

It’s a famous fact that an ordinary string bears a close resemblance to a set of balls connected by springs — their waves are the same, as long as the wave’s shape varies slowly compared to the distance between the balls.

Figure 1: (Top) A length of string. (Bottom) A set of balls connected by springs. The vertical waves of the ball-spring system are similar to the vertical waves on the string; see Figure 2.

The string remains continuous, rather than fragmenting into pieces, because of its internal atomic forces. Similarly, in the ball-spring system, continuity is assured by the springs, which prevent neighboring balls from moving too far apart.

Both systems have familiar standing waves like those on a guitar string, but only if their ends are attached and fixed to something. The most familiar standing wave, shown for each of the two systems, is displayed below.

Figure 2: The classic standing wave for the string and for the the ball-spring system

A Different Set of Balls and Springs

Figure 3 shows a different system of balls and springs, unlike a guitar string. Here, the two sets of springs have distinct roles to play.

Figure 3: A set of balls connected to each other by horizontal springs and to the ground by vertical springs. The waves of this system are less familiar.
  • The horizontal springs again assure continuity — they prevent neighboring balls from moving too far apart, and keep the set of balls behaving like a string.
  • The vertical springs provide a restoring effect — they pull or push each ball back toward the position it holds in the figure.

Compare The Waves

These systems can exhibit many types of waves, depending on whether their ends are fixed or allowed to float (“boundary conditions”). We can have some fun with all the different options at another time. But today I just want to convince you of the most important thing: that the first system of balls and springs requires walls for its standing waves, while the second one does not.

I’ll make waves analogous to the ones I made in last week’s post on this subject. In the animations below, horizontal springs are drawn as orange lines, while vertical ones are drawn as black lines.

First, let’s take the system with only horizontal springs, distort it upward only in the middle, and let go. No simple standing wave results; we get two traveling waves moving in opposite directions and reflecting off the walls (shown as red, fixed dots.)

Now let’s take the system that has vertical springs as well. In particular, let’s make the vertical springs strong, so that the restoring effect is powerful. Again, let’s distort the system upward at the center, and let go. Now the restoring force of the vertical springs creates a standing wave. That wave is nowhere near the walls, and doesn’t care that there are walls at all. It gradually spreads out, but maintains its coherence for many vibration cycles.

The stronger the vertical springs compared to the horizontal springs, the faster the vibration will be, and the slower the spreading of the wave — and thus the longer the standing wave will maintain its integrity.

The Profound Importance of the Restoring Effect

The key difference, then, between the two systems is the existence of the restoring effect of the vertical springs. More specifically, the two types of springs battle it out, the restoring effect fighting the continuity effect. Whether the former wins or the latter wins is what determines whether the system has long-lasting unfamiliar standing waves that require no walls.

In school and in music, we only encounter systems where the restoring effect is absent, and the continuity effect is dominant. But our very lives depend on the existence of a restoring effect for many of nature’s fields. That effect provides the key difference between photons and electrons (see chapters 17 and 20) — the electromagnetic field, whose ripples are photons, experiences no restoring effect, while the electron field, whose ripples are electrons, is subject to a significant restoring effect.

As described in chapter 20 of the book (which gives other examples of a systems with unfamiliar standing waves), this restoring effect is intimately tied to the workings of the Higgs field.

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20 Responses

  1. ear Prof, about another subject (about waves, but other kind of waves), I don’t know if is that your study area, but I ask you about a possible relativity basic error: we know that the particles masses are an universal constants, such as the mass of the electron, neutrinos, quarks, bosons, and just like protons and Neutron, besides other particles, so that is, they are all constant masses. In this case there will be a large basic error of relativity when assigning mass increase as its speed increases and even when it is close to speed of light. So, either it is constant, or is not. And the situation gets worse with gravity, as Einstein will have serious errors in his gravity equation, as well as Newton with his basic equation, when both made fickle equations as the mass particle has a lot of speed, then their gravity no works. Then I ask you prof, whom is wrong with all that? The physics basic fundamentals and the standard model, the relativity, or all that are wrong (or do I’m confused in all this)? Thanks. Have a good lectures.

    1. These issues are all explained in the book, mainly Chapters 5 and 8. Due to the carelessness of many scientists and journalists, you are suffering from a very common confusion, regarding the different kinds of mass. In short, you are conflating different types (rest mass, relativistic mass and gravitational mass) in your question. They behave differently; rest mass is constant, the others are not.

        1. And this system would produce hundreds of relativistic questions, such as who would attract who, having a “resting” mass system and another in extreme movement, if the rocket reached a mass close to the earth mass, and I ask where it would move in this relativistic gravity equation, and how would it be the equation in the constant mass increase on the rocket, and if this object near the speed of light, does it would be able to attract another mass or it violates this equation, then asking, how much the structure mass of each proton’s rocket goes self increase, or if they will always be constant masses, this all among hundreds of questions that would be generated in this system. An Curiosity: Perhaps from SciFi (or useless), some speculators says that in this hypothetical system, this orbital rocket at near the speed of light, would generate a huge Boreal Aurora around all the Earth, like Sun does it, illuminating it a lot and until night, of course, unfounded exaggerations …. Regards and Thanks.

          1. Last question: is there possible a proton close to light speed transform in a Black Hole on its extreme mass? Or relativity equations are wrong? (sorry a lot of questions but this is a very interesting matter)

            1. The answer to both equations is “NO”; you are confusing different types of mass.

              If you keep not trusting me on this point, I’m going to start wondering why you are asking me questions.

          2. You are confused about mass; there are different types and you are assuming they are the same, but they are not. I suggest you read the book, then come back and ask your questions, which right now are too confused to answer. Alternatively, I have a less clear and complete discussion here: https://profmattstrassler.com/articles-and-posts/particle-physics-basics/mass-energy-matter-etc/more-on-mass/the-two-definitions-of-mass-and-why-i-use-only-one/

  2. Imagine a two dimensional ball-spring system arranged on a closed spherical surface like a soap bubble. If I displaced a ball a small distance perpendicular to the surface, I’d expect the displacement to propagate outwards as a circle, converging on the diametrically opposite end, and getting ‘reflected’ back.

    If instead the initial displacement was parallel to the surface, I’d expect some sort of standing wave to result, possibly more complicated than the one above. Now let’s add a more rigid ball-spring system coupled to the first. At the very least it should drastically slow down the spreading of the waves of the outer system from what I’m visualizing about it so far, if it’s initially displaced parallel to the surface.

    1. Your first paragraph is right, but your second is not. You should think about how slinkys work. If you stretch a slinky horizontally and displace it vertically at one location, it will make waves that propagate outward from that point. Those are “transverse” waves, like ocean waves. If you displace a slinky horizontally at one location, it will again make waves that propagate outward from that point. Those are “logitudinal” waves, like sound waves. You will not get standing waves in general.

      1. I’m thinking of something on the lines of Daniel A. Russell’s animation for ‘Standing Sound Waves (Longitudinal Standing Waves)’, from Pennsylvania State University: https://www.acs.psu.edu/drussell/demos/standingwaves/standingwaves.html

        Or from Classroom Physics Demos: ‘Standing Waves on a Spring Nodes and anti-nodes can be seen in a Slinky’ https://demos.smu.ca/demos/waves/111-standing-waves-on-a-spring

        I see a slinky spring as a lumped one dimensional object able to vibrate in three dimensions; with vibrations along and parallel to its length being longitudinal. Hence for a bubble’s surface, waves propagating along and parallel to its surface are two-dimensional longitudinal waves IMO, those perpendicular to its surface being transverse waves. I’d expect to see resonant nodes and anti-nodes on its surface like the demo above, but granted I might be wrong on this.

        1. This is all true, but you’re forgetting the slinky has ends and you are looking at waves with long wavelengths, which are sensitive to the walls. If you make the slinky very long, and make a small disturbance, you will not make standing waves; you will simply make a pulse that travels outward in both directions from the original disturbance. This is true for both transverse and longitudinal waves; there is no fundamantal difference.

          All of this is independent of the point that I am making: namely, there are substances and systems that can have standing waves even if the system is infinitely long. This is not true of sound waves, slinky waves, or light.

          I do suggest you buy a slinky and play with it. There’s no substitute for direct physical intuition. For standing waves without walls, the easiest system I know that one can build is the one that appears in Figure 48 of the book, but it would have to be quite long and carefully calibrated to really be intuitive. I tried a couple of times to make something simpler, but never really made it work in an intuitively useful way.

  3. The restoring force makes sense, a ‘wall’ in a sense, an anchor point in a slightly abstract way. I can understand that as the restoring force drops to zero the minimum possible frequency drops to zero as well. So a massless particle has no lower bound on its energy, no upper bound on its wavelength.

    But you say that the EM field has no restoring force, but don’t photons behave the same way as electrons, in that it’s possible to have a wave of a certain frequency (dependent on the photon’s energy) without walls? As far as I can understand a photon wave, which should then have none of this restoring-force nature and behave like the ‘regular string’. I must be missing something here but am not sure of what.

    1. No, photons cannot have a *standing wave* of a certain frequency without walls, whereas electrons can. Photons can have any energy, but they are always traveling waves. That’s equivalent to the statement that photons can never stop, while electrons can; the former (in empty space) must always be traveling waves, while the latter need not be. And this is the key difference between particles with zero rest mass and particles with non-zero rest mass.

    1. A restoring force is always necessary. It need not be normal to the direction of wave travel; that’s a specific feature of this example. In fact, in a true standing wave, it will not be traveling at all.

  4. Dr.Strassler:
    Can you highlight some of the major differences between Quantum Mechanics and Quantum Field theory? I know they are different, but I think many people believe they are the same theory. I think many people also believe that string theory is the same as quantum field theory, But isn’t string theory, as it currently stands, a “deeper” theory that attempts to explain the origin of the fundamental fields in QFT?

    1. These are article-level questions; I can’t answer them in comments. I may write posts about them in the future, as they are good topics. For now, I’ll just say that string theory has many applications and settings, and in your last two sentences you are mixing two of them… not surprising, given that many writers aren’t clear on the matter. Note that I did mention this distinction in my recent post, https://profmattstrassler.com/2024/03/15/searching-for-suep-at-the-lhc/ .

    1. Well, the problem with visualizing things that way is that the walls would be determined by the width of the standing wave, so each standing wave (of which there might be many — perhaps a whole set of electrons floating in a room) would have its own walls. Normally, when we speak of walls, we mean boundaries to the system that exist independently of which objects happen to be within the system at a particular moment. Moreover, real walls have energy and mass, whereas the ones you’re referring to don’t; they are imaginary, in that sense. So I wouldn’t recommend that sort of intuition. The standing wave just exists without walls, that’s all.

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