Colorado State University

What happens when you hit a water balloon with a baseball bat? Why does tightening and loosening a guitar string change the sound it produces? Can you actually catch a wave? Read on for an investigation into the answers to these questions and more!

Your students probably know oscillations and waves when they see (or hear!) them, and we can rather easily build a formal physical model on this intuitive understanding. The resources on this page will help your students understand oscillations and waves in terms of energy and develop facility with the basic concepts we use to describe these phenomena.

Theory

An oscillation is quite simply a motion that predictably plays out over and over (think swinging pendulum); a wave is just an oscillation that’s going somewhere (pendulum that always pays attention in class, asks thoughtful questions, and gets straight A’s). Here we’ll discuss oscillations in which the position graph of the oscillating object is a sine or cosine wave; this is simple harmonic motion (SHM). 

Also characteristic of SHM is repeated transformation of potential energy to kinetic, then of kinetic to potential. Returning to our pendulum: As you pull the pendulum back, its gravitational potential energy increases. Once you let go, the gravitational potential energy is transformed into kinetic energy. The kinetic energy is maximal at the very bottom of the pendulum’s motion; it is transformed back to gravitational potential energy as the pendulum begins the upward portion of its motion.

Let’s continue thinking about energy as we shift our attention to waves. Let’s say you add energy to a pond by throwing a pebble into it. The pebble will disturb the surface of the pond, and energy will be carried away from the pebble — the source of the disturbance — by waves in the water. Consider a single water molecule a short distance from where the pebble landed. As the wave passes by this water molecule, the molecule will be pushed upward (gaining potential energy), then downward; it will pass a point of maximal kinetic energy and gain potential energy again as it is pushed down below the level of the surface at rest. Hopefully it’s clear that this molecule is undergoing SHM as the wave passes! However, it’s important to note that the water does not move away from the point at which the pebble struck, as the wave does — the medium oscillates in place as the wave passes through it. 

These are generalizable characteristics of traveling waves: they carry energy, and as they pass through a medium they cause every point in it to undergo SHM. (Electromagnetic waves are traveling waves that carry energy, but do not require a medium to propagate. We have much more to say about these in Everything You Need to Know About Electromagnetic Waves!) Another important characteristic of traveling waves: They can interact with other traveling waves. Let’s go back to the pebble in the pond. Now your friend throws another pebble in, a short distance away from yours. As the waves spread out from the sites where each pebble hit, they’ll eventually be in the same place at the same time. If both waves push the water down at this place, the trough will be lower than if just one wave was passing (the same principle applies if both waves push the water up). We call this constructive interference. If, at the place where the waves meet, one is pushing the water up and the other is pushing the water down, the surface of the water will remain at its normal position. This is destructive interference. 

Traveling waves can also interact with themselves. When you pluck a guitar string, you send a wave down the string. When the front edge of the wave reaches the tuner, where the string is fixed to the neck of the guitar and can’t vibrate, the wave reflects off this fixed point and starts heading back toward your fingers. However, on the way it meets the portion of the wave still making its way up the neck of the guitar. In some places this wave and its reflection will constructively interfere and in some places they will destructively interfere; the result is what we call a standing wave. The standing wave will be maintained — i.e., waves will continue to reflect off boundaries at each end of the string and interfere — until all the initial energy input has been dissipated as thermal or other forms of energy. (It’s important to note here that standing waves don’t have to be on a string — it’s also possible to set up, e.g., a standing sound wave in the air between two boundaries.) We refer to the points of complete destructive interference — where the string doesn’t move at all — as nodes, and to those of maximum constructive interference — where the string is maximally displaced from equilibrium during each cycle — as antinodes. For a given wave speed and boundary separation distance, only certain standing wave sizes (properly, wavelengths — we’ll discuss this a bit more below) are allowed. We refer to these allowed states as modes. The mode number m of a standing wave is the same as the number of antinodes in that wave.

We can characterize any oscillation or wave by its period (T) and frequency ( f ). One period is the amount of time to complete one full cycle — how long it takes the pendulum to get back to where you let it go from, or how long it is between successive instances when a point in the water is at its maximum height. Frequency describes the rate of cycling — how many oscillations occur in a given time, how many waves pass a point per unit time. We can additionally describe oscillations and waves by their amplitude (A): the maximum displacement from equilibrium either an oscillating object or the medium in which a wave is traveling achieves. We have one more descriptor for waves: wavelength (λ, the Greek letter lambda). This is the distance which the wave travels in one full cycle — i.e., in one period. Thus, if we know the wavelength and the period, it’s quite easy to find the velocity of a wave: v = λ÷T. Since frequency is simply the inverse of period, this also means that v = λ･f. 

Application

What follows is a set of videos focused on oscillations and waves that will help you create an interactive and effective learning experience for your students on this topic. We’ve organized the videos in terms of “the 5 E’s”: Engage, Explore, Explain, Extend, and Evaluate. This structure has worked well for us in teacher workshops, and we hope it works well for you too! However, we always welcome your feedback. Please contact us anytime to let us know what’s working, what’s not, and how we can make this resource more useful for you. Enjoy!

Engage

The purpose of the Engage segment is to pique students’ interest. We don’t need to directly transmit concepts at this stage, merely give students a chance to think and get excited about the topic we’ll be discussing.

Video: Energy and Oscillations

Hitting a giant water balloon with a baseball bat: Kind of like dropping a pebble in a pond, but better. Striking the balloon with the bat adds energy to the water and the balloon. As a wave travels outward from the point of impact, the balloon applies a restoring force that pushes the water back toward its equilibrium position. This gives us some really interesting patterns on the surface of the balloon — as long as the restoring force is great enough to contain the oscillating water!

Explore

In the explore segment, students get to experience firsthand the principles we hope to teach them. We still don’t need to explicitly describe any concepts yet — the students will start working these out as they explore.

Video 1: Mimicking Mallards

A very easily assembled simple harmonic oscillator. We didn’t talk about this in the Theory section, but it’s a straightforward extension of the basic principles we did discuss, and it particularly encourages students to explore energy in oscillations. We coupled two pendulums to one another by hanging two weighted rubber ducks, affixed to strings of precisely the same length, from a third string stretched across the top of a wire mesh basket. When you set one of the pendulums in motion, things at first seem more or less as expected. However, after a few moments, the amplitude of the first pendulum’s oscillation begins to decrease and the second pendulum starts to swing. At some point the second pendulum’s oscillation will have approximately the same amplitude as the first pendulum’s originally did, and the first pendulum will cease swinging. Just after this instant, the amplitude of the second pendulum’s oscillation begins to decrease, and the first pendulum starts swinging again. This is all about energy. As the first pendulum swings, it tugs on the string it’s hung from. This top string in turn tugs on the second pendulum, thus transferring energy from the first pendulum to the second. Because the pendulums are the same length, the frequency at which the second pendulum is tugged is one at which it very much “likes” to oscillate (i.e., it is a resonant frequency of the pendulum), so it quickly begins to do so. Also because the pendulums are the same length, the period and frequency of the oscillation should be about the same as it is for the first pendulum (an interesting question you could tweak the design to allow students to explore: do differences in the masses at the end of the pendulums matter?). Of course, this won’t be perfect, and the oscillation will eventually cease, due to “losses” of energy to forms such as thermal and spring potential (in stretching of the strings) during the transfers.

Video 2: Dark Vibrations

An open-ended exploration of standing waves constructed from readily available materials. We attached elastic strings to the heads of a vibrating back massager. Because the vibrating string is fixed at each end, it’s possible to set up a standing wave it it. Pulling on the string changes its tension, which changes the speed of the wave traveling on it, which changes the wavelength. Thus, this setup allows students to experiment with different standing wave modes. (This is also why you tune a guitar string by changing its tension.) It’s also noteworthy that not every tension produces a standing wave — there are “sweet spots” that yield one of the allowed wavelengths for a mode, and only at these tensions will standing waves form.

Explain

Now it’s time to codify and formalize the students’ observations. The following videos can help you elucidate and demonstrate the basic concepts underlying the discoveries students made as they explored.

Video 1: Catching a Wave

A rolled-up giant sheet of spandex is a lovely tool for experimenting with both traveling and standing waves (if you can find a few large sheets of spandex — online or at a fabric store — this would fit quite well in the Explore portion of a lesson as well). In this segment we demonstrate that adding energy to a medium can produce a traveling wave in the medium, that a traveling wave causes points in the medium to oscillate in place as it passes, that a traveling wave can reflect off a boundary, and that increasing the tension in a string increases the speed of a wave traveling on it.

Video 2: Constructive and Destructive Interference

A very clear demonstration of constructive and destructive interference of two waves on a string. If each wave is pushing the string in the opposite direction when they meet, the string is not displaced from its equilibrium position at the meeting point (destructive interference). However, if both waves are pushing the string in the same direction, the string’s displacement from equilibrium is larger than it would be if only one wave was present (constructive interference). In either case, both waves emerge unaltered and traveling in the same direction they were before the interaction.

Video 3: Superposition

Back to the giant water balloon for a rather more exciting demonstration of constructive interference. Multiple people along the perimeter of the balloon simultaneously create waves that travel inward; these add at the center to launch a small stuffed animal placed there!

Extend

Once the basic principles are clear, we want to encourage students to expand their thinking and ask questions that go beyond the scope of what we’ve already discussed.

Video: Bubble Trumpet

Standing waves on strings are pretty cool, and pretty important in the context of music, but so are standing sound waves in air! It’s possible to create a standing sound wave in a tube that’s open at one or both ends because sound, unlike the waves we’ve considered thus far, is a longitudinal wave, not a transverse one — it causes the medium to oscillate in the plane of the direction of travel instead of in a plane perpendicular to the direction of travel (however, there’s still no net movement of the medium in the direction the wave is going). Thus, as a sound wave travels through air it creates a series of high and low pressure regions. (If you haven’t already, check out Everything You Need to Know About Sound for much more information on this topic!) Generally, the open end of a tube will be at atmospheric pressure, so it can act as a node of a standing sound wave. Let’s consider a vuvuzela (stadium horn): one end is clearly open, and the other — the one the player blows into — is clearly closed. The closed end, perhaps counterintuitively, will be an antinode of the standing sound wave, since it is a hard stop for the air molecules pushed around by the sound wave (when the particles collide with the closed end, the pressure will be at its maximum; the particles are subsequently pulled back into the region from whence they came, leaving minimal pressure at the closed end). All this is to say that it’s quite possible — and indeed functionally essential — to set up a standing sound wave in a vuvuzela. It’s also possible for said wave to pass through a bubble on the bell of the vuvuzela and produce a clear standing wave pattern on its surface — without popping it!

Evaluate

It’s important to figure out what students understand after the first four E’s. Tests and quizzes are one option, but there are many others.

This one is up to you — what works best for you and your students? We, for example, have had good results with turning the tables and letting the students make a video at this stage. We’d love to hear what works in your classroom!