Wolfgang Bock

Martin Bracke
email: bracke@mathematik.uni-kl.de

Patrick Capraro

Jean-Marie Lantau

On the positive influence of product-orientation in mathematical modeling: A research proposal

Keywords: Product-orientation, modeling, best-practice examples

Motivation

The term mathematical modeling as we have observed in 25 years of modeling projects [START_REF] Bock | Applied school mathematics -Made in Kaiserslautern[END_REF], bears for many students and teachers the notion of something visible and touchable. In our opinion mathematical modeling can and has to be considered in a broader fashion, also allowing what [START_REF] Blum | How do students and teachers deal with mathematical modelling problems[END_REF] call a mathematical model, namely the mathematical description of a real process. Experts in applied mathematics, i.e. scientists working with mathematical modeling, however describe the necessity of an end product to be presented as solving a client's problem in a suitable way (see Neunzert & Prätzel-Wolters, 2015), like a real prototype made from metal or plastic or developing a software or a process simulation, all based on the mathematical model. In the last CERME, [START_REF] Bock | Product orientation in modelling tasks[END_REF] defined the term product, closely related to the notion given in economics or marketing:

Definition

In a mathematical modeling situation (with an authentic and real-world problem) a product is a deliverable in the language of the client which satisfies the needs incorporated in the task given by the client to the provider in such a way that the client can use it directly for his purposes.

Based on former modeling week reports since 1993, [START_REF] Bock | Product orientation in modelling tasks[END_REF] proposed a second cycle within the modeling cycle based on client interaction. This poster is an outline for our future research, i.e. the analysis of the positive influence of product-orientation in mathematical modeling processes.

Best-practice examples for product-orientation in modeling

During different modeling activities [START_REF] Bock | Applied school mathematics -Made in Kaiserslautern[END_REF] a positive influence of productorientation for the motivation and the quality of the mathematical modeling process was observed in contrast to projects without product orientation. Recently several product-oriented modeling activities have been carried out [START_REF] Lantau | Effective factors of a teacher training concerning the implementation of interdisciplinary STEM-modelling projects[END_REF][START_REF] Lantau | The design of a successful teacher training to promote interdisciplinary STEM modeling projects[END_REF]. During a modeling week (3.5 full days + 1 day for presentations), 5 students of Grades 11 and 12 were asked to build a 3d scanner. They were provided with electronical devices and components, such as Raspberry Pis, microcontrollers, sensors, etc. The students handled electric circuits and learned to control the stepper motor, the sensors and data processing. As the construction is mainly a problem of electrical engineering and programming, the data processing contains many mathematical questions. In fact it is impossible to build a working device without having a good mathematical model and awareness of the mathematical toolbox! Due to the project-orientation however, mathematical difficulties had no need to be pointed out explicitly, since the product itself is going to raise the right questions.

A positive influence of product-orientation in mathematical modeling has been observed. The next step is to analyze this fact both in an empirical as well as in a theoretical way. The aims are: 1. To model and understand the underlying processes; 2. To analyze the influence of product-orientation on students' learning processes; 3. To analyze the influence of product-orientation on the quality of results; 4. To develop new methods for interventions and guidance in mathematical modeling. For this we plan to analyze several product-oriented, long-term modeling projects. We are going to use videotaping, questionnaires, interviews. Moreover we plan to analyze of the modeling processes using a new model of dynamical network graphs. Finally, it will be important to create and study adequate forms of teacher training suitable to train teachers in creating and supervising productoriented modeling projects. First steps have already been undertaken (cf. [START_REF] Lantau | Effective factors of a teacher training concerning the implementation of interdisciplinary STEM-modelling projects[END_REF][START_REF] Bock | Mathematical modeling of musical fountains and light organswhere is the M in interdisciplinary STEM projects[END_REF][START_REF] Lantau | The design of a successful teacher training to promote interdisciplinary STEM modeling projects[END_REF].

Acknowledgement

This work is partially supported by the European Social Fund (ESF) project SchuMaMoMINT.


Jérémy Sautel

Andréane Bourges

Aude Caussarieu

Nicolas Plihon

Nicolas Taberlet
email: nicolas.taberlet@ens-lyon.fr

The physics of a popsicle stick bomb

Popsicle sticks can be interlocked in the so-called cobra weave to form a chain under tension. When one end of the chain is released the sticks rapidly disentangle, forming a traveling wave that propagates down the chain. In this paper the properties of the traveling front are studied experimentally and classical results from the theory of elasticity allow for a seemingly impossible dimensional analysis. The study presented here can help undergraduate students familiarize with experimental techniques of image processing. Most importantly it demonstrates the predicting power of dimensional analysis and scaling laws.

I. INTRODUCTION

A. What is a popsicle stick bomb?

Wooden popsicle sticks can be bent and interlocked into a pattern known as the "cobra weave" (due to the resemblance with the patterns formed by the scales of an actual cobra), see figure 1. In the displayed setup, each individual stick is bent to rest alternatively on top of and under four perpendicular sticks (as shown at the bottom of figure 2) but other geometrical configurations could also be woven. When weaving the chain one realizes that a large potential energy is stored into the sticks and the ends of the chain have to be held still. When the weight holding the sticks down on one end of the chain is removed, the so-called stick bomb detonates: the sticks pop-up and a traveling wave is formed.

This system is reminiscent of similar mechanical chain reactions in a line of dominoes [1][2][3] , or in an assembly of mousetraps 4,5 , both systems displaying propagation fronts and conversion of the stored potential energy (whether gravitational or elastic) into kinetic energy. This stick bomb also involves a competition between elastic and gravitational energy. This interesting concept is present in many domains of physics, from the well-known problem of the bounce of a ball, involving mechanical physics 6,7 , to the realization of a precise measurement of the gravity constant by making ultra-cold atoms bounce on an atomic mirror, in quantum mechanics 8 . In this article the speed of the traveling wave and the height reached by individual sticks are studied through simple hands-on experiments.

B. International Physicists Tournament

The work presented here was done in preparation for the International Physicists Tournament (http:// iptnet.info/), a world-wide competition for undergraduate students. Each national team is composed of six students who work throughout the academic year on a list of seventeen open questions and present their findings during the tournament.

Unlike the typical physics exam, the problems must not only be presented, but also challenged and reviewed by the other participants, allowing students to respectively assume the roles of researchers, referees and editors. In addition to the challenge that the tournament represents, it provides students with an exciting and eye-opening experience in which they learn how to design experiments with the aim of solving physics problems, and to constructively criticize scientific solutions.

The authors would highly recommend participation in the IPT as a rare learning opportunity for undergraduate students.

C. What undergraduate students can learn from this problem

This problem can provide an amusing introduction to solid mechanics, giving a simple example of the conversion of stored elastic energy into kinetic and gravitational potential energy 9 in which students can measure an energetic yield and consider sources of dissipation. The problem is also an interesting illustration of the theory of elasticity since the study of the statics and dynamics of a single bent stick fits in the Euler-Bernoulli theory for beam deflection [START_REF] Landau | Theory of Elasticity[END_REF][11][12] . Moreover, students may familiarize with experimental techniques such as particle tracking or with the interpretation of a space-time diagram 13 . Finally, this paper may help undergraduate students grasp the importance of dimensional analysis [14][15][16] as well as the predicting power of scaling laws 17,18 .

D. Outline

The aim of this study is to understand the dependence of the velocity and of the height of the cobra wave on the parameters of the popsicles and of the weaving. In a nutshell, section II examines the elasticity of one unique stick, section III focuses on the traveling wave obtained with one given set of identical sticks, while the properties of the sticks (dimensions and material) are varied in section IV.

To model the popsicle stick bomb and understand why a wave propagates, sticks are modeled by beams in the theory of elasticity framework whose important results are recalled in section II. We will first focus on the static aspects by considering the mechanical equations, boundary conditions and the energy stored in a beam. The dynamics of a stick will also be discussed for a clamped beam, by studying the frequency (in the linear approximation) of the first transverse vibration mode. We then explain how the parameters of the sticks are measured and discuss the efficiency of the conversion of stored elastic energy into kinetic and gravitational energy.

Experimental data obtained for the traveling waves using a given set of wooden sticks will be exposed in part III where the experimental methods used to measure the speed and height of the traveling wave are explained.

Finally, we will derive in part IV scaling laws allowing us to predict the speed and height of a cobra wave, by knowing only some intensive (Young's modulus, density) and extensive (geometrical dimensions) parameters of one stick. The predicting power of the scaling law and the influence of dimensionless parameters are discussed.

II. ELASTICITY OF A SINGLE STICK

A. Euler-Bernoulli beam theory

This section analyses the elastic properties of one single stick. Unless otherwise mentioned the classical results presented in this paragraph are taken from "The Theory of Elasticity" by Landau and coworkers [START_REF] Landau | Theory of Elasticity[END_REF] . a. General hypotheses The Euler-Bernoulli beam theory is a simplification of the theory of elasticity limited to small deformations and in which the shear stress in neglected. In this framework there exists a neutral axis along the beam whose length remains constant and to which the cross section remains perpendicular. In the case where the beam is only constrained by point supports (i.e. with no distributed load or torque) the static beam equation z(x) follows:

d 4 z dx 4 = 0 (1)
meaning that between two consecutive contact points, the equation of the beam is a third degree polynomial whose coefficients are given by the boundary conditions. Equation 1 is derived analytically but a hand-waving argument can help understand its fundamental physical meaning. The local curvature c = d 2 z dx 2 of the stick is the first relevant derivative of the profile z(x) that plays a crucial role. Indeed, a stick can be translated (affecting the absolute value of z) or rotated (changing dz dx

) without any changes to the physics of the bending (except for the unrealistic case of extremely flexible sticks whose shape is affected by gravity). In terms of curvature, equation 1 therefore simply reads:

d 2 c dx 2 = 0.
b. Elastic potential energy A point contact at a given position x 0 simply imposes the position z(x 0 ) and continuity of the first and second derivatives of z. Note that the force locally imposed by the point contact can be computed from the third derivative of z. More importantly for the present article, the elastic potential energy stored in the bent beam is given by:

U el = L 0 1 2 EI d 2 z dx 2 2 dx
where E is the Young's modulus of the material and I the second moment of area, given by I = lh 3 /12 for a rectangular beam of constant section l × h and length L.

Again, this equation can be understood in term of the curvature c of the sticks:

U el ∝ 1/2 c 2 dx.
The elastic potential energy is given in equation 2 for various configurations: deflection of δ at the end of a clamped beam (or cantilever), at the center of a 2-point bent beam (x = 0 and x = L), for a 4-point bent beam δ l FIG. 2. Sketches of a single stick in four configurations. For clarity the thickness, h, and the deformation, δ, are largely exaggerated.

(x = 0, L/3, 2L/3 and L), and for a beam woven between four perpendicular sticks. Note that the last case requires a careful analysis since one has to determine on which edges of the central sticks the beam rests (the contact points are indicated by dots in figure 2).

                       U c el = 3 2 EI L 3 δ 2 clamped, U 2p el = 24 EI L 3 δ 2 2 points, U 4p el = 432 EI L 3 δ 2 4 points, U 4s el 872.4 EI L 3 δ 2 4 sticks. (2)
The numerical prefactor in the last equation is obtained using the dimensions of the sticks shown on figure 1; l = 10 mm and L=110 mm. Indeed, when taking the width of the sticks, l, into account, one realizes that the first obvious effect is that their length is reduced by 2l since the outermost contact points are at a distance l from the ends. Moreover the width, l, also induces a larger deformation and hence a greater elastic energy in the central part of the bent stick. The energy was computed using Maple for various values of l/L and we empirically found that accounting for l is equivalent to reducing its length to L -2.3l, meaning that the elastic potential energy is given by :

U 4s el (l/L) = U 4p el 1 (1 -2.3 l/L) 3 ≡ U 4p el f (l/L). ( 3
)
The prefactor in the last line of equation 2 is then given by: 872.4 432f (10/110).

c. Vibrations The equation of motion describing the dynamics of a uniform beam of mass m undergoing transverse deformations (forbidding any twisting) can be easily derived from the lagrangian of a stick and is given by 10 :

EI ∂ 4 z ∂x 4 + m L ∂ 2 z ∂t 2 = 0 (4)
Solving equation 4 allows one to compute the eigen modes of a vibrating beam and the corresponding frequencies. For instance, the angular frequencies of the eigen modes of a beam clamped at one end (cantilever) are given by ω n = α n EI mL 3 with α 1 3.52, α 2

22.0, α 3 61.7, when for a free beam (under no constrants): α 1 22.4, α 2 61.7, α 3 121.0. Note again that all the eigen frequencies are proportional to a characteristic frequency ω * ≡ EI mL 3 with a geometrical prefactor that depends on the boundary conditions. It is worth mentioning that ω * can be easily deduced from a dimensional analysis of equation 4.

d. Equivalent spring-mass system Obviously the energy stored varies widely from one configuration to another but the reader must note that the elastic potential energy is always proportional to a characteristic energy 1 2 kδ 2 , where k ∝ EI/L 3 , with a geometrical prefactor. The characteristic energy can be regarded as the potential energy stored in a linear spring. A dimensional analysis seems in principle impossible because of the large number of lengths in the problem (length, width and thickness of the beam, pitch of the pattern). However the theory of elasticity provides one with an effective spring constant which then allows for a dimensional analysis. A characteristic elastic energy stored in a stick in cobra weave is therefore given by U * el ≡ Elh 5 L 3 since in this case, the deflection is given by: δ = h.

Even more interestingly for the scaling laws discussed in section IV, one can notice that ω * ∝ k/m, the angular frequency of a linear spring-mass system. Although seemingly impossible, a dimensional analysis of the problem is rendered possible by the theory of elasticity. Having understood this subtle point, one finds itself equipped with a characteristic time for the deformation of the beam woven in the cobra pattern: T * = m/k. Note moreover that this characteristic time T * can be written as a function of the material properties (Young's modulus, E, and density, ρ):

T * = L h L ρ E = L h L c 0 .
where the ratio c 0 = E/ρ has the dimension of a velocity and is the speed of sound of a material whose Poisson's ratio, ν, equals zero. For the more realistic value of ν = 0.3 for hard wood, the speed of sound (compression waves) is given by

E ρ 1-ν (1+ν)(1-2ν) 1.16 c 0 .
Interestingly, the characteristic time can therefore be seen as the product of the length-to-thickness aspect ratio of the sticks, L/h, and the time it takes a sound wave to propagate along the stick. Mass and dimensions of the sticks can be easily measured using a scale and a caliper. The Young's modulus, E, or equivalently, the effective spring constant, k, were measured using two separate methods on a beam clamped at one end. The first method that was used consists in simultaneously measuring the force, F , applied at the free end of the stick (using a dynamometer) and the resulting deflection δ. The relation between the two is given by the Euler-Bernoulli theory: F = 3EIδ/L 3 . We have checked that the relation between the force and the deflection remains linear for small deflections (δ < L/10) and the slope gives a direct measurement of the effective spring constant, which leads to the value of the Young's modulus.

The second method consists in studying the first mode of vibration of a clamped stick. The frequency of this mode is given in the previous paragraph and allows one to compute the Young's modulus (or equivalently the effective spring constant). This second method is far more accurate than the first method. Figure 3 shows the position of the free end of a wooden stick of length L = 85 mm after it was given an initial deflection of δ = 5mm. The data was obtained by tracking the position of the end of the stick using the "Analyze Particle" tool in Im-ageJ, a free software developed by NIH, from a film taken at 1000 FPS with a resolution of 10 × 1500 pixels.

It is worth noting that the vibrations of the stick are rapidly damped. In principle the damping can originate both from air resistance (drag) and from internal dissipation. A rapid estimation of the drag force (based on the Reynolds number) shows that the effective quality factor of the vibrations should be of the order of 10 6 . It is very clear from figure 3 that the damping is much faster (the amplitude is halved in less than 10 oscillations), which indicates that the main source of damping is the internal dissipation within the stick, whose influence will be discussed in section IV. Although this dissipation mechanism is not expected to be linear, the damping is remarkably close to an exponential decay (see figure 3).

C. Energy release and efficiency

In this section we will discuss the efficiency of the conversion of elastic potential energy into gravitational energy.

To measure this ratio we performed a very simple experiment, considering the case of a stick which is stuck between two nails, as shown on figure 4 and bent when one applies a force F , with the help of a dynamometer, at the free end of the stick. The energy stored into the stick is not computed from the theory (which relies on a number of assumptions) but instead directly measured as the work provided by the dynamometer:

U el = δ 0 F dδ = 1 2 F δ.
When the dynamometer is removed the stick rises, reaches a maximum point, and then falls down. We measure the maximum height H reached by the center of mass of the stick, in order to compute the maximum gravitational energy E g = mgH. Repeating this experiment for several values of F , one is able to plot E g as a function of U el in order to determine the efficiency of the conversion of elastic energy into gravitational energy. The plot is represented on figure 4, and gives a constant efficiency of roughly 0.25. Only a fraction of the available energy is therefore used to reach the maximum height. In this particular case 75% of the energy is converted into vibrations (mostly transverse but possibly in twisting modes as well) and rotation (in all directions) or dissipated (because of air friction or internal dissipation).

This empirical efficiency of 25% is specific to the setup used here and one should therefore expect a different efficiency for a stick in the cobra weave. The simple geometry presented in this section optimizes the height reached by a stick whereas a more symmetrical shape, such as the cobra weave (see figure 2) would favor vibrations at the expenses of the height. Moreover, in the cobra weave solid friction between the sticks can be very high, and is another source of energy dissipation. Hence, the efficiency of the conversion of elastic energy into gravitational energy for an actual popsicle stick bomb is expected to be even lower than 25%. For the sticks used in figure 7 the efficiency was found to be close to 5%.

III. SPEED AND HEIGHT OF THE WAVE FOR ONE SET OF STICKS

The height and velocity of the wave may depend on numerous parameters. The material properties and the dimensions of the sticks are discussed in section IV. Here instead we focus on results obtained with a given set of wooden sticks, varying only the pitch of the cobra pattern.

A. Experimental methods

A 3-m long chain of sticks is woven keeping the extremities clamped using large weights. When the weight is removed at one end, the stick bomb detonates and the propagation of the wave is filmed using a fast camera at rates up to 2000 FPS with a resolution of 1500×10 pixels.

The films obtained were processed using the Orthogonal View function in ImageJ, in order to perform a spatiotemporal analysis. This function consists in choosing one fixed horizontal line for all the pictures located approximately at half-height of the wave. A threshold is applied to make the picture black and white. The profile along this line is then plotted for all the frames which therefore creates space-time diagram (see figure 5).

FIG. 5. Space-time diagram of the wave using wet wooden sticks. The two lines can be interpreted as the horizontal position of the ascending (front) and descending (rear) of the "hump" seen on figure 1, measured at half-height. A steady state is reached after the front has traveled 1.5 m (dashed line), corresponding to roughly 100 sticks.

When the chain reaction is triggered at one end of the chain a traveling wave is triggered. One can easily see the evolution of the front and rear position of the wave as a function of time. A steady state, in which the half width of the wave remains constant and the trajectory of the front is a straight line, is reached after the front has traveled for no less than 1.5m (dashed line). It is therefore important that the length of the chain is longer than this transient length (which may depend on the type of sticks used).

The velocity is measured as the slope of the trajectory in the steady regime. The height of the wave is directly measured on the frames (see figure 1), considering for each of them the altitude reached by the center of mass of a highest stick. The results are averaged over the steady state.

The authors need to mention that for convenience the sticks used in figures 1 and 5 were made of remarkably flexible wood and were soaked in water for several days. This results in low speed (1.7 m/s) which helps taking sharp images and produces clear space-time plots. However in the following more rigid sticks are used.

B. Results and discussion

The results presented in this section were obtained using one set of rigid sticks (Wood 2 in table I), soaked in water for three weeks (thickness h = 2.3 mm, width l = 10 mm, length L = 114 mm, density ρ = 970kg.m -3 and Young's modulus E = 5.0 GPa , measured using the method presented in section II B). pitch L FIG. 6. Sketch of the cobra weave: the black lines correspond to the sticks of length L. The pitch is defined as the distance between the extremities of two consecutive sticks:

pitch = 2L 3 sin α 2 .
This section is devoted to the study of the influence of the pitch of the pattern (see figure 6) on the speed and height of the wave. The pitch appears to be a crucial parameter and can be easily changed by varying the angles between the sticks when building the chain. One can easily show that this pitch is given by pitch = 2L 3 sin(α/2) where α is the angle between the sticks (see figure 6). The same set of sticks was used to build several chains with different values of α, and the results of this study are shown on figure 7.

The first conclusion that can be drawn is that the height is unaffected when the pitch is varied whereas the speed of the wave clearly increases with increasing pitch. As discussed in section II for the simpler case of a single stick, the height reached by the wave finds its origin in the elastic energy stored when the chain is built. In a steady state each individual stick can on average only hope to recover its own elastic energy. When neglecting both the width l of the bent stick (i.e. considering 2D cross sections as in figure 2) and the twisting of the sticks, the elastic energy stored in the bending is independent of the pitch. This leads to a constant height when the pitch is varied. The data presented on figure 7 also enables one to estimate the efficiency of the energy conversion. On average, the efficiency is roughly 5%, considerably less that the 25% obtained with the more favorable setup of figure 4.

As discussed in section II the motion of a stick occurs over a typical time T * , specific to each set of sticks, with numerical prefactors that depend on the exact configuration. For a given set of sticks, when the pitch is varied, a dimensional analysis therefore indicates that the speed of the wave should scale as the ratio: pitch/T * . This predicted linear dependence of the velocity as a function of the pitch is indeed experimentally confirmed (blue line on figure 7). For the sake of clarity, the width l of the sticks is not shown on figure 6 but clearly, the pitch has to exceed this value. Therefore, the linear fit is obtained using an offset of l = 10 mm.

In conclusion, a dimensional argument allows one to predict both the linear dependence of the speed and the constant height as a function of the pitch. of the wave on the pitch of the pattern. The speed V depends linearly on the pitch, however with an offset corresponding to width of the sticks l. The height H seems to be independent of the pitch. The blue line indicates an efficiency of 5%.

IV. SCALING LAWS A. Various materials and dimensions

The previous section presented data obtained with the same wooden sticks as the pitch of the cobra weave was varied. In addition to these experiments the dimensions and material of the sticks were varied (table I gives a summary of the sticks used) while the angle α of the cobra weave is kept constant (to 90 • ). In order to vary their Young's modulus, the wooden sticks were soaked in water, for durations ranging from a few days to a few weeks. This process did make the wooden sticks less rigid (their Young's modulus was computed in a simple clamped configuration using the method presented in section II B) but it also affected their density (up to a factor 2) and size (with an increase up to 10%). For all sets of sticks the Young's modulus E and density ρ were independently measured prior to any experiment and the corresponding so-called speed of sound c 0 = E/ρ is reported in table I. Plastic sticks of various sizes, made of PVC (polyvinyl chloride), were also used. Figure 8 shows the experimental results obtained using these various sticks plotting both the height, H, and velocity, V , of the wave. The relation between H and V is not expected to be simple and the data are not expected to fall onto a master curve. The figure is only intended as a synthetic way to present the experimental results. However a general trend can be seen as the stiffer and lighter sticks tend to both travel faster and reach a greater height, but the next paragraph reveals that thickness-to-length aspect ratio h/L plays a crucial role.

B. Characteristic velocity and height

As shown in section III, the velocity of the wave is proportional to the pitch of the pattern divided by the characteristic time of vibration of a stick. Therefore a relevant characteristic velocity can be defined as pitch/T * ∝ L k/m ∝ E/ρ(h/L) ≡ V * = c 0 h/L. While the linear dependence on the pitch was confirmed in section III, the more complex dependence on the material parameters through the speed of sound c 0 and on the aspect ratio h/L is examined in next paragraph. It is important to emphasize that the characteric speed, V * , is not a theoretical prediction of the actual speed since unknown numerical prefactors have been left out.

Obviously the gravitational potential energy, E g = mgH, gained by a single stick finds its origin in the elastic energy stored during the weaving. Comparing these two energies is equivalent to comparing the height H with a characteristic height defined as H * ≡ U el /mg = c 2 0 g (h/L) 4 . The characteristic height, H * , can then be seen as the maximum average height reach by the sticks if all the potential energy were converted into gravitational energy. Again, the predicted dependence of the height on the material parameters and dimensions of the sticks requires an experimental validation.

To summarize we have identified both a characteristic horizontal velocity and a characteristic height of the traveling wave:

     V * = c 0 h L H * = 36 f l L c 0 2 g h L 4 (5)
The two scales are specific to each set of sticks and depend solely on their speed of sound and aspect ratio. For dry wood the speed of sound is of the order of 4000 m/s and the aspect ratio of the wooden popsicle sticks is h/L = 2/110, leading to a characteristic velocity for the wave of V * 70 m/s, which gives the right order of magnitude (see figure 8). The characteristic height is H * 5.7 m, which again gives the right order of magnitude given the low efficiency of the energy conversion (see figure 7).

C. Collapse

As one can see on figure 8, the experimentally gathered data displays an important spreading of the values of V and H between the different types of sticks used. Namely, the speed ranges from 7 m/s to 36 m/s and the height from 15 cm to 100 cm. When the same data is rescaled using the characteristic quantities defined in the previous paragraph (specific to each type of sticks) a good collapse is obtained. In particular, the reduced velocity V /V * only varies by a factor 3. The overall collapse seems to be less efficient regarding the reduced height but a close inspection reveals that the data splits in two separate groups, accordingly to the material used (wood or PVC). Within each of these groups, the collapse is far better (at most a factor 2 for the reduced velocity, and a factor 3.5 for the reduced height). This shows that the characteristic height and speed provide one with two accurate scales for the popsicle stick bomb.

The existence of these two sub-sets, based on the material used, can be explained and expected. The dimensional analysis used to derive the characteristic scales inherently excludes some phenomena among which two might play a crucial role: on the one hand, the solid friction between the sticks (either static or dynamic) and on the other hand, the internal dissipation. Both phenomena can be characterized by dimensionless parameters: respectively the friction coefficients (µ s and µ d ) and the quality factor Q (or damping) of the oscillations (see figure 3). It is clear that increasing either the friction or the damping can only lead to lower values of the height and the velocity since both solid and internal frictions are a source of energy dissipation. Performing accurate and reproducible measurements of friction coefficients can be a difficult endeavor but in our experiments it is obvious that the rough wooden popsicle sticks used have greater friction coefficients than the smooth PVC sticks. Moreover, measurements of the decay rate of the free vibrations of a clamped beam clearly indicate that the internal dissipation of wood is greater than that of PVC. Altogether these observations justify the two subsets of rescaled data, and clarify why both the rescaled height and speed of wood should be less than those of PVC.

The results and conclusions presented in this paragraph can illustrate the predicting power of scaling laws. Indeed, one could have reasonably computed (within error bars) the speed and height of the stiffer dry sticks, knowing only their dimensions and the corresponding speed of sound, from the results obtained with more flexible wooden sticks. Alternatively, the height and speed of the wave of for instance thicker sticks can be extrapo-lated.

V. CONCLUSION

In summary, the linear theory of elasticity allows us to derive scaling laws for the height and speed of the traveling wave. These predictions are validated through a series of experiments varying the pitch of the weave as well as the dimensions and materials of the sticks. This problem is a good example of how dimensional analysis can shed new light on a complex phenomenon.

Several aspects of the popsicle stick bomb are not addressed in this paper and might deserve further attention. It can be seen on figure 1 that the pattern is not symmetrical. Indeed, the wave differs depending on which end of the chain is released, i.e. on the direction of propagation. Equivalently, if the chain is flipped over before the release, the wave properties are affected. Note that the scaling laws should still apply although the numerical prefactors might differ. In particular, it would be interesting to study whether the rescaled velocity and height of this so-called "reverse" wave fall in the same region as those of the "regular" wave. The influence of the support has not been investigated. Indeed, the speed and height might vary whether the chain is set on a hard surface (such as a laboratory table) or on a softer surface (such as a rug or carpet). Numerous home videos found online also show that the wave can propagate without a direct support. It would therefore be interesting to study the case of a chain hanging vertically. Finally, the pattern of the weave can be varied. A cobra weave can be created by interlocking one stick to three or five (instead of four) perpendicular sticks and a square pattern can also be built. The elastic energy and characteristic time would clearly vary but once more it would be interesting to check if the scaling laws hold.

FIG. 1 .

 1 FIG. 1. Top: top-view of the popsicle sticks interlocked in the cobra weave. Bottom: side-view showing the wave propagating.

  B. Measurement of the mechanical parameters position y (mm)

FIG. 4 .

 4 FIG.4. Gravitational energy of a stick as a function of the elastic energy stored, in the case of the simple setup shown on the upper part of the figure. A stick is placed between two nails and bent at its free end. When the force is released, the sticks takes off vertically. The data shows a remarkably linear behavior which defines a constant conversion efficiency of roughly 25%.

FIG. 7 .

 7 FIG.7. Dependence of the speed (top) and height (bottom) of the wave on the pitch of the pattern. The speed V depends linearly on the pitch, however with an offset corresponding to width of the sticks l. The height H seems to be independent of the pitch. The blue line indicates an efficiency of 5%.

TABLE I

 I

			. Parameters of sticks
	Name	h (mm)	l (mm)	L (mm)	c0 (m/s)
		± 0.1	± 0.1	± 1	± 100
	Wood 1	2.2	9.5	113	4600
	Wood 2	2.4	10.0	114	2000
	Wood 3	2.3	10.0	114	2300
	Wood 4	1.1	5.3	136	6500
	PVC 1	2.0	9.7	100	1400
	PVC 2	2.0	9.7	120	1400
	PVC 3	2.0	9.7	140	1400
	PVC 4	2.0	9.7	133	1400

  FIG.8. Top: speed V and the height H of the wave for each type of sticks tested. The squares represent the four types of PVC sticks, and the circles the four types of wooden sticks. The points are widespread and no simple relationship between the two quantities is expected. Bottom: Plot of the rescaled data, V /V * and H/H * . One can notice a good collapse of the data which gathers according to the material used.

		100
	Height H (cm)	50
		0
		0	10	20		30	40
				Speed V (m/s)
		0.5
		0.4		PVC		Wood 1 Wood 2
	H/H*	0.2 0.3	Wood			Wood 3 Wood 4 PVC 1
						PVC 2
		0.1				PVC 3
						PVC 4
		0
		0	0.2	0.4	0.6	0.8	1
				V/V*

ACKNOWLEDGMENTS

The authors acknowledge support from the PALSE program of the University of Lyon Saint-Etienne, the University Lyon Claude Bernard, the Société Française de Physique and from the École Normale Supérieure de Lyon and its Physics Department. The authors are grateful to both local and international organizing committees of the International Physicists Tournament for having put together an exciting event.


Jennifer Dröse
email: jennifer.droese@math.tu-dortmund.de

Comprehending mathematical problem texts -Fostering subjectspecific reading strategies for creating mental text representations

Keywords: Situation model, mental text representations, subject-specific reading strategies, strategy training, comprehending mathematical problem texts

The comprehension of mathematical problem texts has been identified as one of the essential steps towards solving modelling problems successfully. Understanding in this context means constructing an adequate situation model by using reading and understanding strategies. A separate strategy training might foster those strategies as studies indicate. Therefore, the presented study reports on a design research project with the aim of fostering subject-specific reading and comprehension strategies. Presented are further specifications of the processes of constructing mental text representations as well as insights into the use of strategies. The case study indicates that the fostered strategies contribute to different steps in the process of constructing mental text representations. Those strategies seem to be constantly used by the students to deal with word problems within the teaching-learning arrangement and might therefore be relevant for modelling processes in general.

Introduction

"Understanding the content of a task is fundamental for solving it (…)" (Leiß, Schukajlow, Blum, Messner, & Pekrun, 2010, p. 123). There is a wide range of mathematical tasks used to link or apply mathematical models, concepts and procedures to reality, ranging from simple word problems to complex modelling problems. Greefrath and Vorhölter (2016) categorize different accesses to mathematical modelling referring to Griesel (2005) when describing the term modelling in general as "the process of developing a model based on an application problem and using it to solve the problem" (Greefrath & Vorhölter, 2016, p. 8). Many kinds of those application problems are presented to the students in text format. This requires reading and understanding the problem text as the basic step in the modelling and solution process [START_REF] Plath | The impact of linguistic complexity on the solution of mathematical modelling tasks[END_REF]. Understanding a text in general and a (modelling or word) problem text specifically goes along with understanding the described situation, i.e. constructing an adequate situation model [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF][START_REF] Tapiero | Situation models and levels of coherence: Towards a definition of comprehension[END_REF][START_REF] Weimer | The role of understanding in solving word problems[END_REF]. As [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF] point out precisely the step of constructing a situation model is one of the most challenging ones in mathematical modelling. Reasons for difficulties in modelling and word problem solution processes have been hypothesized to lie in "a lack of strategic understanding both of the language involved and of the situation denoted by its verbal description" (Reusser, 1987, p. 474). Furthermore, Leiß et al. (2010, p. 119) point out: "strategies for constructing an adequate situation model have a significant influence on the modelling competence." Meanwhile, there are indications that the use of these strategies can be trained separately [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF]. Therefore, specific teaching-learning arrangements for fostering students' reading strategies are required as a first step towards fostering modelling competencies in general. The presented study reports on a design research project in which a teaching-learning arrangement was iteratively developed and investigated, that aims at fostering subject-specific reading strategies for fifth graders by using strategic scaffolding [START_REF] Prediger | Low achieving eighth graders learn to crack word problems[END_REF]. So, following the distinction of Blomhøj and Bergman Ärlebäck (2018) on research in the field of modelling and application, this project can be located within the development, implementation and evaluation of a specific aspect of modelling activities (namely fostering understanding strategies for constructing a situation model).

After further specifying the theoretical background of generating a situation model in the following section, the methodological framework and empirical insights are presented.

Theoretical Background

Mental text representations as units of understanding

There exists a wide consensus within reading theories that during reading processes (of mathematical problems) four text representations are to be mentally constructed [START_REF] Tapiero | Situation models and levels of coherence: Towards a definition of comprehension[END_REF][START_REF] Kintsch | Understanding and solving word arithmetic problems[END_REF][START_REF] Reusser | From text to situation to equation[END_REF]: the representation of ( 1) the text-surface, (2) the propositional structure, (3) the situation described in the text and (4) the problem. The term mental text representations resumes those four kinds of representations. Steps towards constructing those representations have also been included in different conceptualizations and models describing modelling processes (cf. modelling cycle by Blum and Leiß, further illustrated in [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF] or solution processes of mathematical word problems (see [START_REF] Reusser | From text to situation to equation[END_REF]. As different terms have been allocated to those four representations, their characteristics are briefly outlined with reference to Reusser's model for treating word problems (1987), which is chosen due to its focus on reading and comprehension processes.

(1) The representation of the text-surface (not directly displayed in the model) is mainly the representation of the surface structure, text order, syntax and wording of the text [START_REF] Tapiero | Situation models and levels of coherence: Towards a definition of comprehension[END_REF]. Only directly expressed propositions are represented here [START_REF] Kintsch | Towards a model of text comprehension and production[END_REF].

(2) The representation of the propositional structure (named text base in Reusser's model) is a coherent text representation constructed by forming and organizing propositional units into a global structure [START_REF] Kintsch | Understanding and solving word arithmetic problems[END_REF].

(3) In the representation of the situation (named situation model), the information given by the text are integrated with the readers' prior knowledge or goals [START_REF] Kintsch | Understanding and solving word arithmetic problems[END_REF] to make inferences of different kinds (drawn from text or prior knowledge) [START_REF] Weimer | The role of understanding in solving word problems[END_REF]. Reusser (1989) therefore defines the situation model as "the personal cognitive structure to which the process of understanding is directed. The situation model is the cognitive correlate to the situation structure either supposed from the author's point of view or understood from the reader's point of view" (Reusser 1989, p. 136ff., translated by Plath & Leiß, 2018, p. 161).

(4) The representation of the problem (named problem model in Reusser's model) is an extension of the situation model focussing especially on the aspect of the goal (generated or given problem question) guiding the construction process [START_REF] Kintsch | Understanding and solving word arithmetic problems[END_REF][START_REF] Reusser | From text to situation to equation[END_REF].

As stated above, for mathematical modelling tasks, the process of constructing a situation model is the most challenging one [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF]. Meanwhile, most of the mathematical problem texts examined by researchers are restricted to one mathematical relation mathematized in one step (c.f. [START_REF] Reusser | From text to situation to equation[END_REF]. Therefore, they do not further specify the process of building situational or problem representations [START_REF] Kintsch | Towards a model of text comprehension and production[END_REF]. As many applied mathematical problem texts display more than one relation and multiple steps, the model needs to be differentiated to take the processes of constructing mental text representations into consideration [START_REF] Prediger | Low achieving eighth graders learn to crack word problems[END_REF].

Processes of constructing mental text representations

Considering the situational or problem representation of multi-step problems, different mental elements must be related to each other. In line with [START_REF] Schnotz | Aufbau von Wissensstrukturen. Untersuchungen zur Kohärenzbildung beim Wissenserwerb mit Texten[END_REF], these mental elements form the basic units of the writer's and the reader's communication process: (A) the writer identifies the relevant units of the mental representation, constructs an adequate language representation for each of them and integrates them into a text, (B) after that, the reader needs to identify the relevant textual or language units, interpret their meaning and integrate them into a mental representation [START_REF] Schnotz | Aufbau von Wissensstrukturen. Untersuchungen zur Kohärenzbildung beim Wissenserwerb mit Texten[END_REF]. In order to include these considerations into Reusser's model, the model is extended as shown in Figure 1 in order to capture the reader's processes of achieving local or global coherence.

The detailed specification of the reader's activities and processes are integrated. Following from the investigation of [START_REF] Weimer | The role of understanding in solving word problems[END_REF]: (1) relevant elements have to be identified, (2) elements have to be structured and related and (3) elements have to be interpreted and new elements have to be inferred (see Figure 1). Apart from these processes, the kinds of mental elements are further specified. Those relevant mental elements can be of These kinds of processes of text comprehension are initiated and monitored by reading strategies [START_REF] Tapiero | Situation models and levels of coherence: Towards a definition of comprehension[END_REF][START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF]. Whereas the steps in the model themselves cannot be taught directly, reading strategies can be fostered in separate strategy trainings [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF]. Most relevant for the texts in view are the comprehension strategies identified by Prediger and Krägeloh (2015, p. 951):

(S1) Focus on the question to find relevant information (S2) Focus on the information together with their meaning (S3) Focus on relations connecting the information [START_REF] Reusser | From text to situation to equation[END_REF] indicates that strategies might support special processes in his model. One might consider that the named strategies can be linked to the processes as follows: S1 to the process of question generation and (1) identifying elements, S2 to the process of (1) identifying elements and S3 to the process of (2) relating elements (see Figure 1). However, further investigations are required on how these strategies can support the displayed processes to construct a situation or problem model and how the use of strategies might differ when dealing with different problem texts. For considering those research aspects and getting first insights, the focus is laid on word problems where the construction of the mental text representations is the main aim rather than on rich modelling tasks, which require additional processes after understanding the problem text.

Research demands and questions

The considerations described above called for designing a teaching-learning arrangement to foster students' comprehension strategies. The corresponding qualitative analysis of the initiated comprehension processes pursued two research questions:

(RQ1) How can the three comprehension Strategies S1, S2, S3 contribute to students' mental construction processes described in the extended model for treating word problems?

(RQ2) How do students' use of strategies differ for different word problems?

Methodology of the design research study

Design research as methodological framework. Due to the dual aims of designing a teachinglearning arrangement to foster students' use of subject-specific reading strategies as well as describing and analyzing the use of comprehension strategies, which is in line with the considerations by Blomhøj and Bergman Ärlebäck (2018) concerning theory-practice relations, the framework of topic specific design research has been chosen. It iteratively connects (1) the specification and structuring of the learning content, (2) the creation of the teaching-learning arrangement, (3) the use of the teaching-learning arrangement in design experiments, which are afterwards analyzed to (4) generate contributions to local theories on the learning content and learning processes (Prediger & Zwetschler, 2013).

Design experiments as a method for data collection. In the presented study, six design research cycles have been conducted, five in laboratory setting with 2-3 students each and the last one in a classroom setting. Each cycle consisted of approx. 4-7 sessions of 90 minutes each. This paper focuses on Cycle 5 in which six pairs of students have worked on the teaching-learning arrangement (in total 4320 minutes of video data).

Methods of qualitative data analysis.

For the qualitative data analysis, a category-led deductive procedure has been chosen. The use of strategies has been coded (according to a categorical scheme presented in Dröse and Prediger (in press)). Afterwards, the moments in which the use of strategy was identified, have been categorized with respect to the processes of identifying text, situation or propositional elements; inferring situational elements or relating mental elements (see Figure 1).

Here, the case study of Metin and Yaren (both boys are 11 years old and attend a secondary school in a German urban area) is presented, based on 120 minutes of video data (partly transcribed).

Empirical insights: Yaren's and Metin's construction of mental representations

The case of Yaren and Metin provides an insight into their process of constructing mental text representations and their use of Strategies S1-S3 concerning RQ1. The two boys work on Task 3 "Bear child" (Fig. 2) in Sequence 1 and on Task 5 "Parrot feeding" (Fig. 3) in Sequence 2.

Bear child

The brown bear child in the zoo weighted 700g at his birth. After his birth it lost 200g because it was ill. Over the last weeks, his weight has increased by 4000g and it is healthy again.

How much does the brown bear child weight now?

Parrot Feeding

In the zoo, the children are feeding the parrots. Felix fills parrot Tobi's bowl with 10g sunflower kernels and 15g berries. In the bowl, there have already been 3g feed before the feeding. Parrot Piet eats 5g of parrot Tobi's feed.

How much feed does parrot Tobi have in the end? 4] 104 Teacher: So, he weighted 700g at his birth, then he loses 200g, then this was 4000g altogether. Does that fit? 105 Metin: Yes, that fits (…) 107 Metin: or if, wait look, if he loses 200g of 700g, then that would be 500g, so then this have to be 4500g?

When the teacher verbalizes the relation that Metin has visualized (l. 104), Metin implicitly infers the two situational elements, "state after the bear was ill" and "state now that the bear is healthy again" (l. 107) that are not explicitly mentioned in the problem text. After finishing his visualization with cards and arrows, Yaren explicitly describes the relation between the two states implicitly mentioned above (l. 206).

206 Yaren: (…) It has weighted 500g and then it got healthy and got 4000g, because it was healthy afterwards (…). Regarding RQ1 the sequence gives an example for a pattern we have often found in the data: When students activate the Strategies S2 (focus on meaning of information) and S3 (focus on relation of information) as triggered here by the cards and arrows, they can identify not only text elements, but infer also further situational elements and create an appropriate situation and problem representation.

Sequence 2: Expressing relationships between identified and inferred elements in Task 5

After solving Task 3 and 4, Metin and Yaren work on Task 5 "Parrot feeding" (Figure 3). At first, they write down the relevant text elements, and then the teacher asks them to express the given relations.

1 Teacher: Tell me, what happens to the 10g and the 15g in the task? What about them? (…) 3

Metin: So at first, there were 10g sunflower kernels, then 15g berries and then he has eaten 5g of Parrot Piet's feed. 4

Yaren: (…) Ah yes, then there were 20g left. 5

Metin: (…) Ah so (…) now I understand (…) I found the result. He had 25g in total [points at the cards and arrows] and then (…)

When being asked to articulate the actions taking place in the task, Metin uses the Strategies S2 (focus on meaning) and S3 (focus on relations). He puts the given information into the described order to symbolize the relation between the text elements (l. 3) and the inferred situational elements (l. 5). Yaren explains that there are only 20g left (l. 4). The teacher points to one information that has already been written down but not used, so far: Metin's explanation reveals that he has focused on the meaning of the 3g first (S2) and then tries to relate the text element to the information already given (S3). In this sequence, the teacher is not expressing the relation among elements herself as she has done in Task 3; instead, first she prompts the students to do so (l. 1). Later on, she prompts the students to take the meaning of the text element into consideration (l. 101). Being familiar with the scaffold, the boys use Strategies S2 and S3 to relate mental elements and the teacher's prompt for activating S2 suffices to activate S3 as well (l. 108). Concerning RQ1, this shows how both strategies are intertwined in the process of relating elements. This observation reoccurred in further data.

Overview of used strategies to construct and process mental elements

Concerning RQ2, Table 1 gives a short overview on the strategies, which Metin and Yaren used to construct mental elements. For each task, the relevant text and situational elements as well as the relations between them have been specified. For the Tasks 3 -5 those text elements can be either the starting point or a change of the situation (Change 1, Change 2), as described in the problem text. The situational elements, that have to be inferred, can be the state of the situation either after the first or after the second described change (State 2, State 3). After specifying the relevant mental elements, the coded strategies (S1 -S3) were assigned to the prevailing mental elements, when the

Discussion and outlook

The case study of Metin and Yaren provides first insights into the use of the fostered strategies to identify, infer and relate mental elements. The analysis reveals the contribution of the identified strategies to the inference of implicit situational elements (Sequence 1) and the processing of relations between these elements (Sequence 2). Furthermore, the importance of the interplay between Strategies S2 (focus on meaning) and S3 (focus on relations) can be lined out (RQ1). The use of strategies to identify, infer or relate the prevailing mental elements seems to become more stable during the process of task comprehension (RQ2).

These results are of particular interest for research on modelling against the backdrop of difficulties in constructing a situation model [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF]. Therefore, two indications might follow from the empirical insights: (1) as the strategies named above might contribute to generating a situation model it is possible that those strategies belong to the strategies having an impact on modeling competencies (see [START_REF] Leiß | The role of the situation model in mathematical modelling[END_REF], (2) the use of those strategies might be fostered by strategic scaffolding and their use seems to become more stable independent of the tasks' characteristics. So, the use of those strategies might be extended to rich modelling taskspresented in written format and with more than one relation describedto facilitate the understanding of the situation.

The presented study is limited at first because of the small number of students that have been analyzed as well as the laboratory setting in which the research was conducted. Further research needs to extend the investigation of the use of strategy to more students and different problem texts. It should especially take the interplay between the named strategies as well as between the teacher and students into consideration. At least the investigation of reading strategies might be extended from fostering reading strategies for word problems to reading strategies for modelling problems.

  (a) textual, (b) propositional or (c) situational type. At first, the (a) text elements and (b) propositional elements have to be (1) identified from the problem text or the text base, according to the problem question. Afterwards, (a) text and (b) propositional elements can be (2) related to (3) interpret or (3) infer new (c) situational elements that have not been identified before. Then those (c) situational elements can be (3) related and (3) integrated to form the problem model (see Figure 1).

Figure 1 :

 1 Figure 1: Extended model for treating word problems (adapted from Reusser, 1997; Prediger & Krägeloh, 2015)

Figure

  Figure 2: Task 3 Figure 3: Task 5

Figure

  Figure 4: Metin's concept map of the situation

101

  Teacher: Okay and now, you have one card left written 3g on it. Do you need this? 102 Metin: No, not really… but [is reading the text]. Yes, we can use it, because (…) 108 Metin: Ah, Felix has given him something and he has had 3g before.

Table 1 : Strategies used to assimilate elements of different kinds

 1 (S3 represents unsuccessful strategy use)For answering RQ2 the following differences and similarities in the use of strategies can be observed: The analysis of Table1reveals that both boys use Strategy S2 to identify relevant text elements, having also focused on the question (S1). Furthermore, in Task 3 all strategies are used to infer and relate mental elements, while in Task 4 or Task 5 mainly Strategies S2 and S3 are used. This underlines the close entanglement of those strategies. With regard to RQ2 the boys' use of the identified strategies seems to become more stable with frequent use of the scaffold.

Acknowledgment

The study is conducted within the MuM-research group (mathematics learning in language diversity) in cooperation with the project Mastering Math funded by the Deutsche Telekom Foundation (2011-2020, grant holder Susanne Prediger and Christoph Selter).

strategies are used before a successful identification, inference or relation of this mental element. The use of those strategies is displayed in Table 1 without focusing on its frequency or its order.


Nicolas Plihon
email: nicolas.plihon@ens-lyon.fr

Jérémy Ferrand

Tristan Guyomar

Flavien Museur

Nicolas Taberlet

Why do aged fluorescent tubes flicker?

Common experience shows that aged and defective fluorescent tubes or bulbs may flicker and emit a clicking sound while struggling to light up. In this article, the physical mechanisms controlling the initial illumination of a functioning fluorescent tube are investigated using a simple and affordable experimental setup. Thermionic emission from the electrodes of the tube controls the startup of fluorescent tubes. The origin of the faulty startup of aged fluorescent tubes is discussed and flickering regimes using functional tubes are artificially produced when decreasing, in a controlled manner, electron emission by the thermionic effect. The physical parameters controlling the occurrence of flickering light are discussed, and their temporal statistics are reported.

Introduction

Physical phenomenon and article outline

Fascinating phenomena may emerge from the interplay between electromagnetism, fluid dynamics and quantum physics in plasma physics. Several experimental devices have recently been designed as educational tools at the undergraduate level [START_REF] Gekelman | Ion acoustic wave experiments in a high school plasma physics laboratory[END_REF][START_REF] Lisovskiy | Validating the collision-dominated Child-Langmuir law for a DC discharge cathode sheath in an undergraduate laboratory[END_REF][START_REF] Gekelman | Using plasma experiments to illustrate a complex index of refraction[END_REF][START_REF] Lisovskiy | Validating the Goldstein-Wehner law for the stratified positive column of DC discharge in an undergraduate laboratory[END_REF][START_REF] Wissel | The use of DC glow discharges as undergraduate educational tools[END_REF][START_REF] Lisovskiy | In-depth treatment of discharge ignition data during undergraduate laboratory work[END_REF][START_REF] Gekelman | Drift waves and chaos in a LAPTAG plasma physics experiment[END_REF]. Widely spread, fluorescent tubes or other plasma-based lighting devices remain an affordable an simple way to introduce several fundamental concepts of plasma physics [START_REF] Auerbach | Fluorescent Lamp. A Teaching Device[END_REF][START_REF] Field | Microwave measurements of a fluorescent lamp plasma[END_REF][START_REF] Orr | Cyclotron resonance effects in a fluorescent lamp plasma[END_REF][START_REF] Nassar | Introductory physics of harmonic distortion in fluorescent lamps[END_REF][START_REF] Eldakli | Optogalvanic effect and laser-induced current oscillations in hollow-cathode lamps[END_REF]. In this article, we analyse the startup of fluorescent tubes with an emphasis on the physical understanding of the flickering operation encountered during the startup of aged or faulty fluorescent tubes.

The basic description of how fluorescent tubes operate can be found in Ref. [START_REF] Bloomfield | How things work: the physics of everyday life[END_REF][START_REF] Lister | The physics of discharge lamps[END_REF]. Fluorescent tubes are long glass tubes filled with noble gas (usually Argon) at low pressure (around 100 Pa). When a plasma is sustained between the electrodes at each end of the tube, UV radiation from the plasma is absorbed by fluorescent coating within the glass tube, which re-emits white light.

Igniting and sustaining the plasma in the tube, while the electrodes are fed by electric power, requires external components -in particular a ballast used to limit the plasma current and an ignition-helping system known as the starter. This article focuses on the simple configuration involving a magnetic ballast and a starter, as shown in Fig. 1. The electrodes (enclosed in the tube) are small foiled tungsten filaments coated with a thin layer of highly emissive oxides in order to increase electron emission from the thermionic effect. The starter consists of a separate element in which a bimetallic strip is enclosed in a low pressure noble gas capsule. When a voltage of the order of a few hundred of volts is applied across the starter, the gas it contains is ionised. The bimetallic strip heats up due to ionic bombardment on each side of the strip which act as electrodes of the starter plasma, and then deforms and closes the circuit. As a consequence a large current (around 1 A) flows through the electrodes of the tube, which are heated-up by the Joule effect. As no plasma is sustained in the (now closed) starter, the bimetallic strip cools down and eventually opens up (Joule heating in the starter is negligible as compared to the heating due to plasma bombardment). This sudden current interruption induces a large over-voltage across the large inductance of the magnetic ballast, which allows the system to reach the breakdown voltage of the noble gas enclosed in the fluorescent tube and to sustain a plasma within the tube. In this article, we provide hints that the startup of the functional tubes is controlled by the intensity of thermionic emission from the filaments. The steady-state operation of fluorescent tubes fed by the mains (240 V., 50 Hz in Europe, Africa or Asia or 120 V, 60 Hz in the USA) involves shimmering at twice the AC line frequency [START_REF] Vollmer | Flickering lamps[END_REF] (100 Hz in the work reported in this article). Fluctuations of light emission of light source at the mains frequency (or its harmonics) has been studied in several contexts [START_REF] Bacalla | On the intensity profile of electric lamps and light bulbs[END_REF][START_REF] Andersson | Amplitude modulation of light from various sources Lighting[END_REF]. Fluorescent tubes are based on a plasma glow discharge which is known (back to the early years of plasma physics) to be strongly intermittent [START_REF] Kock | The Effect of Inductance on the Intermittent Glow Discharge Physics[END_REF][START_REF] Boyle | SelfPropagating Intermittent Discharge[END_REF], inducing intermittent light emission. On the other hand, fluctuations of light emission of fluorescent tubes may also originate from moving non-linear potential structures in the plasma [START_REF] Liu | Experimental investigations on moving striations in a 50 Hz ac fluorescent lamp[END_REF]. However, to the best of our knowledge, the defective startup of aged fluorescent tubes has not been reported elsewhere.

This defective startup, resulting in flickering light emission and pulses of sound emission is a widely observed phenomenon when the tube struggles to light-up. The flickering light emission appears to be random in time, with bursts of emitted light typically separated by a duration ranging from a few tenths of a second to a few seconds.

As no steady state regime is reached, the starter continuously switches between the open and closed states, leading to the typical sound clicks (synchronised with the flickering of the tube). This peculiar regime is studied in details in this article, introducing an experimental method which reduces the efficiency of the thermionic emission. This is achieved by lowering the temperature of the filaments of functional tubes when adding an additional resistance in the electrical circuit. The article is organised as follows: plasma breakdown, plasma I-V characteristics of the glow discharge and a detailed analysis of the normal startup are described in Section 2. Section 3 then introduces an experimental setup which simulates the defective startup of fluorescent tubes resulting in random flickering light emission. The physical origin of this regime is then analysed and the statistics of the flickers are detailed. Concluding remarks are then provided in Section 4.

What students can learn from this problem

The work presented in this article was done in preparation for the International Physicists Tournament http://iptnet.info, a world-wide competition for undergraduate students. Unlike the typical physics exam, the problems must not only be presented, but also challenged and reviewed by the other participants allowing students to respectively assume the roles of researchers, referees and editors. In addition to the challenge that the tournament represents, it provides students with an exciting and eye-opening experience in which they learn how to design experiments with the aim of solving physics problems, and to constructively criticise scientific solutions. This article reports some of the results obtained from a series of six 4-h long labs, where students solved the question of the origin of the flickers as a problem-based learning.

The investigation of the dynamics of fluorescent tubes is a simple hands-on experiment for the introduction of plasma physics at the advanced undergraduate level. A basic understanding of electrokinetic and electromagnetism concepts and experimental techniques are required to carry this study out. Basic concepts of plasma physics, such as gas discharge ignition and sustainment, can be experimentally introduced. Other advanced such as the thermionic effect are also at grasp using this simple experiment, including the investigation of the effect of temperature on the resistance and electron emissivity of a filament. This article also provides an introduction to the statistical analysis of long time series and rare-events.

Normal operation of the fluorescent tube

Characterisation of the fluorescent tube plasma

Experiments have been carried out using commercial T8 types fluorescent tubes (of diameter 26 mm and length 590 mm) Philips TL-D 18W powered from the mains (U mains,rms = 240 V and f mains = 50 Hz) through a magnetic ballast Mazda 36N20-01, as shown in Fig. 1. A Sylvania FS11 starter primes the tubes at startup. Current and voltages have been measured using current sensors and differential voltage sensors respectively, and the intensity of light emitted by the tube has been measured using a Thorlabs PDA10A-EC photodiode. In the remaining of Sec. 2, we investigate the normal startup operation described in the introduction. Note that the additional resistance displayed in Fig. 1 is introduced to simulate flickers (see Sec. 3) and was set to R = 0Ω for the results presented in Sec. 2.

2.1.1. I-V characteristic of the fluorescent tube plasma. Typical time series of the voltage U tube and current I tube across the tube during its steady-state normal operation are displayed in Fig. 2(a) over 20 ms. The I --V characteristics shown in Fig. 2(b) was extracted from these time series and is typical of plasma glow discharges [START_REF] Raizer | Gas discharge physics[END_REF], with a local negative resistance ∂U tube /∂I tube < 0. This typical I --V characteristic with a large voltage required for plasma breakdown and a negative local resistance implies that sustaining the plasma in the tube is an easy task once it has been created. The negative resistance ∂U tube /∂I tube < 0 also requires current limitation, achieved by the magnetic ballast, whose resistance R b and inductance L b have been measured to be 30 Ω and 1 H respectively. This results in a ballast inductance Z b :

Z b = R 2 b + (2πf mains L b ) 2 ∼ 315Ω
larger than the tube impedance (of the order of 50 Ω), which limits the rms intensity of the current to typically 240/Z b ∼ 0.8 A. 2.1.2. Influence of filament temperature on plasma breakdown. The ignition, or breakdown, of a plasma within a tube of length d at pressure p submitted to an electric field E = U tube /d is a well-known physical phenomenon [START_REF] Wissel | The use of DC glow discharges as undergraduate educational tools[END_REF][START_REF] Lisovskiy | In-depth treatment of discharge ignition data during undergraduate laboratory work[END_REF][START_REF] Raizer | Gas discharge physics[END_REF][START_REF] Yip | Comment on "The use of dc glow discharges as undergraduate education tools[END_REF]. It involves (i) exponential growth of electrons accelerated by the electric field from ionising collisions with the neutral gas, and (ii) electron emission at the cathode. At low cathode temperature, the latter is dominated by secondary electron emission from ion bombardment, usually characterised by a coefficient γ, which specifies the number of secondary electrons per incident particle. The voltage breakdown between two electrodes spaced apart by distance d is then expressed from Paschen's law [START_REF] Yip | Comment on "The use of dc glow discharges as undergraduate education tools[END_REF]:

U b = Bpd ln(Apd) -ln [ln(1 + 1/γ)] , (1)
where A is the saturation ionization of the gas under consideration and B is related to the ionisation energy of the gas. For noble gases, both of these coefficients increase with the atomic mass [START_REF] Lieberman | Principles of plasma discharges and materials processing[END_REF].

When electrodes are heated up, a second process, thermionic emission, adds up to secondary electron emission. The ability of a material to emit electrons from the thermionic effect increases with its temperature T following Richardson's law as T 2 e -W/k B T with k B the Boltzmann constant and W the work function [START_REF] Azooz | An experiment on thermionic emission: back to the good old triode[END_REF][START_REF] Kittel | Introduction to Solid State Physics[END_REF]. In our experimental setup, the increase of the filament temperature T f ilament is expected to increase the value of the effective γ coefficient and to reduce the breakdown voltage. This effect is directly measured using the dedicated setup displayed in Fig. 3(a). Both electrodes are Joule-heated by the same DC current, driven from two independent DC power supply. The measurement of the voltage U f ilament across the filament allows for the computation of the resistance of the filament R f ilament and its temperature is inferred assuming a local power balance between Joule heating and losses by radiation (an assumption valid at high enough filament temperature and low enough pressure in the tube) [START_REF] Carlà | Stefan-Boltzmann law for the tungsten filament of a light bulb: Revisiting the experiment[END_REF][START_REF] De Izarra | Calibration and temperature profile of a tungsten filament lamp[END_REF]. Assuming that the core filament is pure tungsten, the temperature is computed from the following equation [START_REF] Zanetti | Temperature of incandescent lamps[END_REF][START_REF] Prasad | A laboratory experiment on the application of Stefan's law to tungsten filament electric lamps[END_REF]:

T f ilament = T 0 R f ilament R 0 1/1.2 (2)
where R 0 is the resistance at room temperature T 0 . Eq. 2 is valid above typically 150 K (see for instance the log-log plot in page 1092 of Ref. [START_REF] Desai | Electrical Resistivity of Selected Elements[END_REF]). A high-voltage DC power supply (up to 3 kV) is used to bias the fluorescent tube between the hot electrodes.

The DC voltage at which plasma breakdown occurs is reported as a function of T f ilament in Fig. 3(b). The filament temperature is thus shown to have a strong impact on the breakdown voltage. An abrupt decrease of breakdown voltage is evidenced above 1400 K. This filament temperature is associated to a visual modification of the light emitted by the tube, even in the absence of a external bias between the filaments. Below 1400 K, the light emitted by the filaments is due to thermal radiation and filaments appear reddish (see Fig. 3, bottom left photograph). Above 1400 K, the temperature is high enough to trigger electron emission and the voltage across the filament locally sustains a plasma; UV radiated by the plasma excite the phosphor coating, which in turn emits white light (see Fig. 3, bottom right photograph).

T f ilament (K) U breakdown (V)

Time analysis of normal startup

Let us now investigate the temporal dynamics of normal startup as the electric power of the tube is fed by the mains as displayed in Fig. 4. At time t = 0, the mains are switched on: the line voltage (rms voltage of 240 V) is applied to the tube and the starter, but no current flows. At t ∼ 0.05 s (identified by vertical lines in Fig. 4), the gas enclosed in the starter capsule becomes ionised: a weak current of the order of 50 mA crosses the starter during the positive voltage half-wave and the bimetallic strip enclosed in the starter heats-up. At t ∼ 0.31 s (identified by vertical lines in Fig. 4) the deformation of the bimetallic strip is large enough to close the electrical circuit (referred to as closed starter in the remainder of this paper) and a current of typically 1 A flows in the starter and the filament electrodes. This large current leads to Joule heating of the filaments, while the starter cools down (as no plasma is present in the starter). Finally at t ∼ 0.88 s (identified by vertical lines in Fig. 4) the starter opens, leading to a large over-voltage across the tube, exceeding the breakdown voltage and igniting the plasma in the tube. Note that light emission is modulated at 100 Hz, with maxima for each half-wave of the mains voltage [START_REF] Vollmer | Flickering lamps[END_REF]. The startup process thus strongly relies on the pre-heating of the electrodes when the starter is closed. The temporal evolution of the current flowing in the starter, the filament resistance and the filament temperature during the pre-heating phase is displayed in Fig. 5. The filament resistance was measured as the ratio of the envelope of the filament voltage over the envelope of the starter current (this is valid as long as no plasma is present in tube, i.e. when I tube = 0), and reaches a few tens of Ohms (error bars are displayed as the grey area). The resistance of both electrodes (typically less than 50Ω) remains at least five times lower than the ballast impedance (Z b ∼ 315Ω); as a consequence the amplitude of the current flowing in the filament remains roughly constant during pre-heating, while the voltage across the filament increases (not shown).

The time evolution of the filament temperature T f ilament , computed from Eq. 2, is displayed in Fig. 5(c). After a few tenths of seconds of preheating, T f ilament reaches values around 1500 K , i.e. at the onset of emission by the thermionic effect, which decreases the breakdown voltage to ignite the plasma, as described in subsection 2.1.2.

Flickering regimes

Flickering regimes observed at the startup of aged fluorescent tubes involve faulty startups, i.e. plasma breakdown is reached transiently but the plasma within the tube is not steadily sustained. From the previous observations, this is likely to be related to inefficient pre-heating of the filaments. In aged fluorescent tubes, thermionic emission might be decreased from sputtering of the filament emissive layer, oxidation of the emissive layers, or deposition of other materials on these emissive layers. Note that sputtering or evaporation of the filaments would lead to a gradual decrease of the filaments' diameter and to a gradual increase of R f ilament (T 0 ). However, this effect is likely to be counter-balanced by the sputtering of the strongly emissive layers: the thermionic effect from the bulk filament material is likely to be lower than that from the emissive layer. On the other hand, modification of the gas composition could also lead to a change of breakdown voltage. Ageing of the starter could also lead to the decrease the duration of closed time, which would then lead to less efficient pre-heating. However, we postulate that the leading process controlling the tube ignition is thermionic emission from the electrodes. In this framework, faulty startups using functional fluorescent tubes can be obtained when decreasing the temperature of the filament (since thermionic emission depends exponentially on the material temperature according to Richardson's law). In order to experimentally simulate the inefficient pre-heating of the filaments, an additional high-power resistance R was inserted in series with the ballast, as shown in Fig. 1. This additional resistance reduces the amplitude of the current flowing in the filaments, which lowers the efficiency of the preheating. An order unity decrease of the current flowing in the filaments has a strong effect on the Joule heating of the filaments, which in turn has a strong influence on the thermionic effect. As a consequence, the value of the additional resistance R was chosen to be a few times that of the ballast's impedance Z b ∼ 315 Ω. Flickering light emission from the tube was observed for R values in the range 500-1000 Ω. For R values greater than 1000 Ω, no light emission was observed. Note that similar results have also been observed when decreasing the voltage amplitude feeding the circuit with R = 0 (for instance using an autotransformer between the mains and the circuit), which also reduces the current flowing in the filaments during the pre-heating phase.

Description of flickering regimes

Several flickering regimes have been observed depending on the value of the additional resistance R and typical time series of light emission are reported in Fig. 6 (for the sake of clarity, time-series are vertically shifted). The regime shown in Fig. 6(a) consists in a random distribution of short pulses, and was observed for R values in the range 910 to 550 Ω. In this regime, the plasma is sustained only during a fraction of a single halfperiod of the mains 50 Hz cycle: a single peak of light emission is observed before fading out. As the value of R is decreased, light emission is no longer limited to short pulses, as shown in Fig. 6(b-e). The plasma may be sustained in the tube during several mains cycles, leading to a flickering regime consisting in a train of pulses of light emission. Note that the light is 50 Hz modulated during one flicker, which is highly uncomfortable to the human eye -it is shown later that the plasma is only sustained during a given polarity of the mains. Successive flickers may be excited as reported in Fig. 6(b) and(c) or may alternate with short pulses as reported in Fig. 6(e). Finally, for R values lower than 500 Ω, the flickering regime slowly drifts and finally leads to normal operation of the tube as shown in Fig. 6(f). Note that in this last regime, and for times larger than 0.2 s, the light is 100 Hz modulated as during normal operation (however, the introduction of the additional resistance R ∼ 1.5Z b reduces the light intensity by a factor around 2.5). The detailed analysis of the regime reported in Fig. 6(f) is instructive. Simultaneous time series of light intensity P lum , voltage across the tube V tube , and plasma current I tube are reported in Fig. 7. For 0 < t < 0.12 s significant light emission is only observed when the voltage across the tube is positive: the plasma is only sustained during a given polarity of the 50 Hz cycle, resulting in 50 Hz modulation of light emission. This is likely to originate in the efficient pre-heating of only one of the two filaments, say electrode 1. During the positive voltage half-wave, plasma breakdown is obtained when electrode 1 acts as the cathode, while breakdown is not reached when electrode 1 acts as the anode, i.e. during the negative voltage half-wave (not that the scale of Fig. 7(b) is not symmetric). It is important to note that, since thermionic emission exponentially increases with temperature according to Richardson's law, a very weak temperature difference between both electrodes leads to a large difference in current emitted by the thermionic effect. Weak geometric or assembly asymmetries between both electrodes are thus expected to result in strong difference in thermionic emission at onset. However, as electrode 2 is further heated by ion bombardment and current injection into the plasma, it acts as a more and more efficient cathode for 0.12 < t < 0.18 s. Finally a symmetric behaviour is recovered for times larger than 0.2 s, resulting in a 100 Hz modulation of light emission as reported in Section 2.2. The asymmetric regime may thus only be obtained in a narrow range of parameters. 3.2. Time statistics of flickers 3.2.1. How flickers onset are related to the opening of the starter. The common experience with aged and flickering fluorescent tubes links flickering light emission and sound emission from the starter. As long as no plasma is sustained in the fluorescent tube, the voltage across the starter is high enough to reach breakdown of the starter gas: as a consequence, the starter continuously alternates closed and open phases. Plasma breakdown in the tube necessarily occurs at the opening of the starter, as shown in the previous Section. However, as shown in Fig. 8(a), light emission pulses do not systematically coincide with the opening of the starter (shown as red lines in the top panel). In our experimental setup, the inefficient pre-heating of the filaments requires the opening of the starter within a short period of the 50 Hz mains to ensure plasma breakdown in the fluorescent tube. Indeed, plasma breakdown requires a high enough over-voltage across the ballast (which depends on the value of the current at the opening time), and during the positive polarity of the 50 Hz mains voltage (as described in the previous subsection). For these reasons, plasma breakdown is not observed for all opening events of the starter. The filament temperature T f ilament is reported in Fig. 8(b), and shows that a stationary regime for the filament temperature is reached, allowing for the statistical analysis of flickering regimes. The regime reported in Fig. 8 consists of single pulses, but it is important to note that, for longer flickers, the opening of the starter is not systematically associated to light emission. The time-distribution of light flickers is investigated in details in the next two subsections; for the sake of simplicity, the temporal dynamics of the opening/closing sequences of the starter is not investigated explicitly. Pulses typically last less than 5 ms and do not strongly change from one to the other, we thus focus our attention on the distribution of the waiting-times between two successive pulses. One hour-long acquisitions were acquired for several values of the additional resistance R. The photodiode signal is used to trig a fast oscilloscope (with a 5 ms time base) and the digital TTL trigger-out signal of the oscilloscope is acquired at a rate of 250 Hz. The probability distribution functions (pdf) of the waiting-times are displayed in Fig. 9(a) for three values of R and are consistent with exponential distributions. The pulses thus appear to be randomly distributed, without any correlation between occurrences. The characteristic mean waiting-time τ between two pulses times weakly evolves with R: τ decreases from 0.62 s to 0.57 s and finally 0.51 s as R decreases from 910 to 700 and finally 550 Ω respectively. It is important to note that this decreasing trend is observed since the tails of the pdf are fairly well resolved. As the value of R decreases, the current in the pre-heating phase increases, and the plasma is more likely to ignite: the increase of the filaments's temperature make them more emissive and the increase of the current increases the over-voltage L b dI/dt across the magnetic ballast. As a consequence, the mean waiting-time slowly decreases with decreasing values of R.

Statistics of longer flickers.

Let us now investigate the statistics of the longer flickers obtained when R < 550 Ω, i.e. the regimes reported in Fig. 6(b-e) where both short pulses and longer flickers are present. For a typical configuration with R = 497 Ω nearly 700 light emitting events have been measured over 1800 s, out of which 61 % are long flickers (lasting at least two mains cycles). Note that these regimes with long flickers are extremely unstable: a slow time-evolution of the filament temperature could lead either to recover a symmetric behaviour leading to continuous light emission or to recover regimes consisting in a succession of short pulses. Our study focusses on two distinct probability distribution functions: namely the pdf of the duration of on-phases (i.e. the duration of the flicker light emission), and the pdf of the duration of waiting times (or equivalently off-phases). The distribution of waiting times, shown in Fig. 9(b), is monotonically decreasing, while the distribution of on-phases, shown in Fig. 9(c), is clearly peaked with a most probable value around 0.7 s and a long tail towards large values. Note finally that the distributions shown here should be regarded as illustrative due to the unstable behaviour in this range of parameters (the fluorescent tube was finally continuously emitting light at the end of the time series under investigation). As a consequence, the deviation of the waiting-times distribution from an exponential distribution might be due to the non-stationarity of the regime -since the physical origin of the flicker onset is the same as for the pulses, one would expect the distributions to be similar. Addressing this issue would require a careful control of the stability of the flickering regime.

Conclusion

The physical processes controlling the startup of fluorescent tubes used with a magnetic ballast and a starter were investigated experimentally, using an inexpensive and easy to reproduce experimental setup. The ability of the electrodes at each end of the fluorescent tube to emit electrons is shown to be the leading parameter controlling the voltage required across the fluorescent to reach plasma breakdown. In particular, for a given tube assembly, electron emission from thermionic emission is controlled by the temperature of the electrodes. A detailed time sequence of the startup of fluorescent tubes is reported, including detailed experimental measurements of the electrode temperature. Finally the inability of aged fluorescent tubes to undergo quick ignition is experimentally addressed. In this regime, light emission flickers and random noise emission are experienced at startup. An experimental method to simulate flickering regimes using standard tubes is introduced. This relies on reducing the current heating the electrodes when the starter is closed since the flickering regime originates in inefficient pre-heating of the electrodes which only transiently sustains the plasma in the fluorescent tube. In particular, in our experimental setup, flickers are observed when only one of the electrode reaches a temperature high enough to strongly emit electrons when negatively biased. Using this method, statistically stationary flickering regimes are observed, allowing for the statistical analysis of the time distribution of flickers.

The onset of flickers appear to be random in time, with no correlation, resulting in an exponential probability distribution function.

Figure 1 .

 1 photodiode

Figure 2 .

 2 Figure 2. (a) Temporal evolution of I tube and V tube during normal operation for one 50 Hz cycle. (b) I -V characteristics.

Figure 3 .

 3 Figure 3. Influence of filament temperature on breakdown voltage: (a) experimental setup, (b) voltage breakdown as a function of T f ilament . The vertical dashed line indicates the visual modification of the tube. See text for details.

Figure 4 .

 4 Figure 4. Time series of (a) starter current (or filament current) I starter , (b) current in the tube I tube , (c) voltage across the tube V tube and (d) light intensity P lum showing a typical tube startup time sequence of a tube initially at ambient temperature. See text for details.

Figure 5 .

 5 Figure 5. Time evolution of (a) starter current (or filament current), (b) R f ilament and (c) T f ilament during pre-heating phase, when the starter is closed. Error bars displayed as grey areas. See text for details.

Figure 6 .

 6 Figure 6. Typical time-traces of flashes observed as R decreases from 910 to 457 Ω (top to bottom). See text for details.

Figure 7 .

 7 Figure 7. Time evolution of (a) light intensity P lum , (b) voltage across the tube V tube , (c) the plasma current I tube for the regime reported in Fig. 6(f), R = 457 Ω. See text for details.

Figure 8 .

 8 Figure 8. Time evolution of (a) light intensity and (b) filament temperature T f ilament . Open starter phases identified by red lines on panel (a).

3. 2 . 2 .

 22 Statistics of single pulses. Let us first investigate the statistics of successive single pulses (i.e. the regime reported in Fig.6(a)) for R values in the range [550; 910] Ω.

Figure 9 .

 9 Figure 9. (a) Probability distribution function of waiting times τ for the successive pulses regime reported in Fig. 6(a) for three values of R. Probability distribution functions of (b) waiting times τ and (c) on-phases for the long flickers reported in Fig. 6(e).

Acknowledgments

This work was supported by the ś Normale Supérieure de Lyon and Univ Claude Bernard, Lyon, France. The authors acknowledge J. Bernard, E. Camus, L. Lachaud, Q. Marsal and L. Thorens for fruitful discussions. The authors are grateful to both local and international organizing committees of the 2017 International Physicists Tournament for having put together an exciting event.