Air particles in a lattice: Considerations for sound wave simulations in physics education

Many existing sound simulations show air particles as semi-stationary, as if suspended in a lattice. While this portrayal is based mainly on practical considerations, it can inadvertently reinforce non-normative conceptions. We share our observations of middle school science students interacting with lattice-style sound simulations, attending to the effect such representations had on their conceptualizations of sound. We discuss the implications of these results for the implementation of digital simulations in lessons on sound waves and provide suggestions for instructors to frame these tools and guide students’ attention toward relevant aspects of the visualization.


I. INTRODUCTION
The advent of the Digital Age granted physics educators access to a useful and powerful new tool: the digital simulation [1][2][3].Physics phenomena are often complex and even unobservable, but simulations can strip away irrelevant or distracting information and make hidden events visible [4].These affordances are especially useful for novices who have not yet developed the discernment to know where to look when observing phenomena, or constructed reliable mental models for visualizing abstract mechanisms.Moreover, simulations can be designed to connect with students' prior knowledge and lived experience in order to address common misunderstandings [3].When simulations are engaging, interesting, and designed with learning objectives in mind, they can add a rich new dimension to traditional instruction [1].
For all their utility, digital simulations are not a silver bullet and present several key challenges.Novices tend to interpret visual representations as literal analogs of the physical system, like seeing crossed lines on a graph as a physical intersection [5].Naive learners can also trust simulations to a fault, assuming the experts who designed them could not be mistaken.This can even lead to students interpreting erroneous simulation data (caused by bugs or malfunctioning programs) as legitimate and meaningful [1].On top of these risks, instruction that seeks to simplify or reduce the complexity of natural phenomena (as most simulations do) can inadvertently misrepresent the physics at play [6].Given these hazards, the design and implementation of digital simulations in physics education must proceed with care.
Understanding sound requires connecting a microscopic model of particle interactions to a macroscopic model of everyday experience.Previous work in the domain has identified a number of widespread non-normative conceptions [6][7][8][9][10].Students routinely imagine air molecules to be more-orless stationary, swaying back and forth as sound waves pass by [7].Similarly, they may characterize sound as a dominolike series of collisions [11].Students may also misinterpret sinusoidal wave diagrams that portray abstract measurements of sound (e.g.pressure) as literal pictures of the sound waves [6,10].The variety and nuance of these beliefs means that students who provide technically-correct answers may not have necessarily arrived at a normative understanding of the wider topic [10,12].
Digital simulations of sound waves have been successful at improving students' conceptualizations of the phenomenon [14].An archetypal example can be found among the PhET Interactive Simulations [2,13,15] developed by the University of Colorado Boulder, shown in Figure 1.As with most simulations of this design, the air molecules oscillate in place without travelling around the room.Simulations such as these can provide an informative (and even captivating) visual for enhancing classroom instruction.However, our team observed a concerning trend when utilizing simulations for sound waves in a middle school science class; the semistationary, coupled nature of the particles encouraged and re- inforced a lattice-style mental model for sound waves travelling through air.In this paper, we share our observations and discuss accommodations or adaptations for integrating sound wave simulations into classroom instruction.

II. CONCEPTUAL FRAMEWORK
The analysis discussed here was informed by the constructivist theory of knowledge in pieces (KiP) [16], which describes naive knowledge as a loose network of individual resources accumulated through experience and interaction with the physical world.When learners reason through novel problems, they are guided primarily by whatever resources are readily available.Importantly, KiP foregrounds the salient features of the learning context; resources are activated by contextual cues, and thus the resources which scaffold sensemaking will depend on the aspects of the problem-solving context which attract the learner's attention.Instruction is then most likely to succeed when it actively directs students' perceptions to the most relevant contextual cues.
Individual resources can have a surprising range of applicability and may be seen in use across multiple contexts.According to KiP, many student misconceptions are the result of students activating resources (which have been productive elsewhere) in an incompatible context.This situates KiP as an anti-deficit perspective, which focuses primarily on the utility of student ideas [17].KiP research which identifies student difficulties looks for alternate contexts in which seemingly detrimental student ideas could be beneficial, rather than discounting and discarding them entirely.

III. SETTING
The data for this study come from one classroom of 31 eighth-grade students who were participating in an instructional module pertaining to sound waves.The students participated in physical demonstrations and activities using musical instruments and slinkies to learn the basic characteristics of frequency, wavelength, and amplitude.Next, the students were shown two digital simulations [18,19] of a speaker instigating sound waves through particles of air.Finally, the students designed their own interactive simulation in NetTango [20] (a block-based interface to NetLogo [21], an agent-based computational modeling environment) to explore the way changes to pitch and volume would affect the behavior of the particles of the medium.
Our observations focus on three questions from the students' worksheet, which contained their predictions and reflections from the various activities.The first (Q1) was posed after the students observed the digital simulations and asked them to predict what would happen to the air particles if the speaker was louder.The remaining questions were given after the students had created their own NetTango simulation for sound waves.These questions asked the students to describe the simulation's behavior when the volume (Q2) or the pitch (Q3) was altered.
To identify patterns in reasoning, the students' answers were categorized by two independent coders based on the common terms or phrases which appeared in their explanations.The coders then engaged in social moderation [22] until complete agreement was reached.

IV. OBSERVATIONS
The instructional module for the sound wave unit included links to two publicly available digital simulations of longitudinal sound waves travelling through air [18,19].Both were structurally analogous to Figure 1: the air particles were represented by a cloud of dots with a few particles highlighted in red.A speaker pushed the nearest air particles, and each particle oscillated around an equilibrium point at their original position while transferring its motion to the neighboring particles.This caused a wave with dense crests and sparse troughs to propagate through the medium.The highlighted red particles showed how individual particles remained near their initial positions while the wave crest moved continuously across the room.The simulations were looped and could not be altered.The students were asked a series of questions relating to these simulations, including prompts about what each piece of the simulation represented, how the highlighted dots moved, and how energy was translated through the system.As a final prompt, Q1 gave the students a hypothetical consideration: "what do you think would happen to the air particles (dots) if the speaker was louder?"Afterwards, the students used the block code in the Net-Tango modeling microworld to construct their own model of a speaker, medium, and listener as seen in Figure 2. The students added each agent to the model and specified their parameters.When the agents were deployed correctly, the simulation represented air particles as green dots which oscillated horizontally when perturbed.The students were challenged to alter the volume and pitch of the speaker and document the results.Q2 and Q3 asked for a description of the behavior of the particles when the volume or pitch was increased, respectively.
The most common responses, their frequencies, and a prototypical example of each can be found in Table I.It should be noted that not all students answered each question, and responses could belong to multiple categories.The faster responses explicitly referenced the increased speed of the particles.The movement responses mentioned a variety of changes in the motion of the particles, including that they would "go farther," "vibrate more," or move "more freely."The energy category included responses that explicitly mentioned "energy," as well as those which expressed similar notions like moving "harder" or "with more force."The frequency and amplitude categories counted responses which commented on the changing dimensions of the waves, such as being "closer together" or "bigger," respectively.The density responses described how the particles would be "more condensed" or "closer together."Finally, the more air responses stated that the number of air particles would increase.
Across all three questions, the faster category dominated the responses, showing up more than twice as often overall as the next most common category, movement.The movement, energy, frequency, and amplitude categories were seen in roughly similar amounts.The prevalence of frequency responses was much higher in Q3 than Q1 or Q2 due to the nature of the prompts; likewise, amplitude was more common in Q1 and Q2 than in Q3.Lastly, the density and more air responses were significantly less common than any other category.The unmatched prevalence of the faster responses across all three questions necessitates an immediate recognition: the notion that air particles move faster in a louder or higherpitched sound wave is not true!Sound waves traveling through air are alternating waves of high and low pressure.As a speaker moves forward, some number of air particles that would not have contacted the speaker had it been stationary are instead reflected back into the same region as other unaffected particles, causing an increase in density.Then, as the speaker retreats, some number of air particles that would have reflected back if the speaker been stationary are allowed to travel slightly further "into" the space previously occupied by the speaker, causing a decrease in density.When sound is louder, the speaker moves with a larger amplitude, increasing the number of particles affected and causing the magnitude of the density fluctuations to increase.When sound is higher pitched, the rate at which the speaker oscillates increases.This creates more alternating density zones in a given period of time, which also causes each zone to be thinner.
Meanwhile, the speed of the air particles is dictated by their kinetic energy, which is measured via temperature; if the temperature of the air remains constant, the average speed of the particles will also be constant.Air particles at standard temperatures and pressures move at roughly the speed of sound, between 300 and 400 meters per second.Determining the speed of a typical speaker is challenging as it will vary based on the size of the diaphragm and the pitch and volume being produced; However, conservative estimates put most speakers' speeds at roughly an order of magnitude or more slower than the speed of sound [23].As such, the speed of the air molecules which happen to ricochet off of a speaker's diaphragm will be relatively unchanged.Furthermore, the speaker would contact air particles just as often on its forward and backward strokes, resulting in a near-zero net impact on the average speed of the particles on the whole [24].
Digital simulations which attempt to model sound waves in air are faced with several pragmatic impossibilities: not only would a simulation which depicts air particles at a somewhataccurate speed and randomness be computationally taxing, the volatility of the display would likely render it incomprehensible.As is the case with many physics concepts, sim-plifications are necessary when designing usable sound wave simulations.As seen in the examples provided [13, 18-20], the most common means for addressing these concerns is to program air particles which are semi-stationary and oscillate in tandem with their neighbors, creating the illusion that they are connected in a loosely-coupled lattice.This accommodation successfully avoids the previously mentioned pragmatic concerns, but inadvertently makes the simulation more susceptible to misinterpretation.
We believe that, when confronted with the sensemaking challenge in the sound wave module, the students were generating mental models for sound waves which took the simulated air particles to be literally analogous to real-world air particles.As the students observed the digital simulations, the motion of the air particles would stand out as an easily perceptible feature and would cue familiar resources regarding classical motion.According to KiP, these resources would then be the primary driving force behind the students' reasoning as they navigated this novel context.When Q1 asked the students to predict the effect of increasing the speaker's volume, it would not be surprising to see the speed of the particles play such a prominent role in the students' answers, given how attention-grabbing the motion was in comparison to other visible features in the simulation.
When answering Q1, the students were theorizing about increasing the speaker's volume but could not confirm their predictions in the moment as the digital simulations were not interactive.However, the NetTango coding environment let students alter the parameters of the environment and observe the consequences immediately.Thus, for Q2 and Q3, the students could observe the simulation's actual response to the varying volume and pitch.As the interactive simulation modeled air particles as semi-stationary and loosely coupled, increasing the volume and pitch did, in fact, cause the programmed particles to move at a higher speed.This confirmation would reinforce the previously activated resources as reliable and trustworthy, solidifying their connection within the students' knowledge network.
We refer to the normative model for sound waves in air as the "Gas" model and the proposed students' model as the "Lattice" model, visualized in Figure 2. The Gas model describes air particles as moving independently and randomly, with sound waves being formed of alternating zones of high FIG.3. Comparison of air particle behavior and sound waves of varying amplitude between "Gas" and "Lattice" conceptual models and low density.In this model, a louder sound wave would have a larger disparity between the high and low density regions.In comparison, the Lattice model envisions air particles as being connected to their neighbors and subject to restorative forces that allow them to move about or oscillate while always returning to an equilibrium position.In the Lattice model, a louder sound wave would displace the particles further from their equilibrium position.
In alignment with our KiP framework, we wish to emphasize the utility of the students' Lattice model even if it may be non-normative within the context of air particles.While semi-stationary particles may be inaccurate for gases and other fluids, the Lattice model is remarkably accurate for solids.Going further, simulations which depict coupled particles can still help students visualize critical aspects of the microscopic mechanisms for sound in air particles, namely the lateral propagation of high-density regions moving independently of individual particles.As such, we believe that the utilities and relevancies of lattice-style simulations more than justify their place in physics instruction; their inclusion just needs to be accompanied with proper framing to help students navigate the challenging nuances.Here, we offer a few suggestions for instructors using digital simulations for sound wave instruction in physics classrooms.
First, instructors should introduce the simulation with proper context.Depending on the placement of the sound instruction within the larger curriculum, students may have already been introduced to a particle-based microscopic view of the various states of matter.If this is the case, the instructor should have students recall the chaotic, random motion of particles seen in gases and compare it to the motion of the particles in the simulation.Instructors can directly address the simplifications made when modeling gas particles and ask students to predict how these simplifications may affect the validity of conclusions drawn based on the simulation.
Second, instructors should help students attend to the most relevant aspect of these simulations when discussing gases in particular.According to KiP, students' sensemaking is widely shaped by the characteristics and properties of a phenomenon which receive the greatest attention or trigger recollection of familiar patterns [16].This means that instructors can drastically influence the direction of students' thought process by priming relevant connections or placing focus on important facets of the phenomenon.Instructors should foreground the density fluctuations over the lateral motion of the individual particles.Students can explore how the oscillating motion of a speaker would create such regions, and then be challenged to consider how their proposed mechanism compares with both the lattice-style simulation and the random motion of real-world gas particles.This conceptualization focusing on propagating density waves can then be connected to and supplemented by other classic demos or thought experiments, such as the suspended particle or candle flame vibrating at the mouth of a speaker [7,10]; instead of imagining the vibrations as being in sync with oscillating, semi-stationary air particles, the motion can be explained by passing density waves creating pressure gradients in alternating directions.
Third, instructors should promote the Lattice model for sound waves within solids, where particles are connected to their neighbors via intermolecular forces; such a model provides valuable insights when discussing topics such as the longitudinal waves traveling across the string in the "lover's telephone" [9].The Lattice model can also be given practical importance by connecting it to the study of seismic waves: primary waves (P-waves) from earthquakes are longitudinal and are accurately represented by the model, while secondary waves (S-waves) are transverse.Interestingly, this modeling may help explain why P-waves can travel through both the solid and liquid regions in earth's core while S-waves cannot: students would need to consider how the coupled motion within the solid regions would be converted into density fluctuations in the liquid regions (essentially, considering the solid material at the boundary to act like an oscillating speaker).
With these adjustments, instructors can make students explicitly aware of the differences between the simulations and reality, indicate which characteristics of the simulation are relevant to fluid-based contexts, and validate the Lattice model seen in the simulation for solid-based contexts.We believe these accommodations can allow instructors to make the most of digital simulations in their classrooms while making use of productive student ideas and validating students' conceptualizations.

FIG. 1 .
FIG. 1. Sound wave simulation developed for PhET Interactive Simulations at University of Colorado Boulder [13]

TABLE I .
Common response categories for selected questions