Team of Brazil Problem 05 Bright waves reporter: Bárbara Cruvinel Santiago

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago2 Team of Brazil Problem 5: Bright waves Problem 05 Bright waves Illuminate a water tank. When there are waves on the water surface, you can see bright and dark patterns on the bottom of the tank. Study the relation between the waves and the pattern.

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago3 Team of Brazil Problem 5: Bright waves Contents Introduction Caustics Image point Catastrophe theory General features Experiments Experimental data Theory validation Conclusion Analysis Comparison

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago4 Team of Brazil Problem 5: Bright waves Caustics Envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. BrightPenumbra Waves act like lenses Caustics are formed

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago5 Team of Brazil Problem 5: Bright waves Depth Critical depth: 1 st ray crosses the 2 nd Image point distance: maximum brightness About 2 or 3 cm. * From this distance on, the brightness decreases. * WALKER, J. “Shadows Cast on the Bottom of a Pool Are Not Like Other Shadows. Why?”. 1988

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago6 Team of Brazil Problem 5: Bright waves Depth n1n1 n2n2 I O N C Θ1Θ1 Θ2Θ2 α β ϒ * * Assuming small angles * Image point distance d q R p

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago7 Team of Brazil Problem 5: Bright waves Depth n1n1 n2n2 I O N C Θ1Θ1 Θ2Θ2 α β ϒ In the infinity d q R p

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago8 Team of Brazil Problem 5: Bright waves Depth C R R C R R λ Specific case of water n = 4/3. E

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago9 Team of Brazil Problem 5: Bright waves Width of the pattern Bulkhead in q. Other depths: catastrophe theory

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago10 Team of Brazil Problem 5: Bright waves Width of the pattern Fold catastrophes united by cusps. Varies the catastrophe opening.

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago11 Team of Brazil Problem 5: Bright waves Width of the pattern a = 0.1 f = 30 Hz g = 9.81 m/s² a = 1 H = 1.5 cm g = 9.81 m/s² General expression for bright pattern’s width behavior:

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago12 Team of Brazil Problem 5: Bright waves Other wave’s characteristics Influences: λ: size of the pattern A: radius T, v and f: periodic interchange between bright and dark patterns. Type of wave: circular or linear patterns. A A’

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago13 Team of Brazil Problem 5: Bright waves Interference Two punctual sources: Constructive: n = 0, 1; 2; 3... Destructive: n = 0,5; 1,5; 2,5; 3,5... Reflected water waves: Same frequency and opposite directions.

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago14 Team of Brazil Problem 5: Bright waves Interference Two punctual sources: Standing waves are produced Easier to analyze experimentally. Reflected water waves: Same frequency and opposite directions. Constructive: n = 0, 1; 2; 3... Destructive: n = 0,5; 1,5; 2,5; 3,5...

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago15 Team of Brazil Problem 5: Bright waves Material 1.Ripple tank 2.Lead weights 3.Cylinder 4.Metal support 5.Nylon Acoustic boxes Water Aquarium Computer Wave generator

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago16 Team of Brazil Problem 5: Bright waves Experimental description Experiment 1: analysis of the used source. Experiment 2: bright pattern interference analysis. Experiment 3: obtaining the coefficient a of the catastrophe theory. Experiment 4: depth variation analysis. Experiment 5: frequency variation analysis.

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago17 Team of Brazil Problem 5: Bright waves Experiment 1: analysis of the used wave source Source Interference Coefficient a Depth Frequency Wave fronts near wave’s source High energy/amplitude Different “q” Different brightness Patterns differ in geometry Wave source

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago18 Team of Brazil Problem 5: Bright waves Experiment 2: interference analysis Source Interference Coefficient a Depth Frequency

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago19 Team of Brazil Problem 5: Bright waves Experiment 2: interference analysis Source Interference Coefficient a Depth Frequency Some points of interference can clearly be seen. Change in the pattern’s geometry

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago20 Team of Brazil Problem 5: Bright waves Experiment 3: coefficient “a” Source Interference Coefficient a Depth Frequency

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago21 Team of Brazil Problem 5: Bright waves Experiment 3: coefficient “a” Source Interference Coefficient a Depth Frequency Caustic width (cm)a 1 st E-03 2 nd E-03 3 rd E-03 4 th E-03 5 th E-03 6 th E-03 7 th E-03 Average E-03 Standard deviation E-04 a = catastrophe theory coefficient (caustic opening)

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago22 Team of Brazil Problem 5: Bright waves Experiment 3: coefficient “a” Source Interference Coefficient a Depth Frequency Expected behavior applying “a” gotten previously: Used in comparison (next experiments) General equation applied

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago23 Team of Brazil Problem 5: Bright waves Experiment 4: depth variation Source Interference Coefficient a Depth Frequency

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago24 Team of Brazil Problem 5: Bright waves Experiment 4: depth variation Source Interference Coefficient a Depth Frequency

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago25 Team of Brazil Problem 5: Bright waves Experiment 4: depth variation Source Interference Coefficient a Depth Frequency Interference by reflection produce standing waves for easier analysis.

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago26 Team of Brazil Problem 5: Bright waves Experiment 4: depth variation Source Interference Coefficient a Depth Frequency Standard deviation = m

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago27 Team of Brazil Problem 5: Bright waves Experiment 5: frequency variation Source Interference Coefficient a Depth Frequency Decreasing width with increasing frequency

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago28 Team of Brazil Problem 5: Bright waves Experiment 5: frequency variation Source Interference Coefficient a Depth Frequency Standard deviation = 1.77∙10 -5 m

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago29 Team of Brazil Problem 5: Bright waves Conclusion The patterns are formed by caustics. a ≈ 1.5∙10 -3 (exp. 3) and characterizes the opening of the caustic. Predictions with this coefficient fitted experimental data. Behavior of the patterns in function of the depth and frequency were showed and complies with the expected behavior (exp. 4/5). depth/wavelength/amplitude/frequencydistance “q”/bright pattern’s length interferencepattern‘s length and/or geometrytype of wave/wave sourcepattern‘s geometryperiod/frequency/velocitybright/dark periodic interchange

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago30 Team of Brazil Problem 5: Bright waves References 1. SERWAY, Raymond A.; JR., John W. Jewett – Princípios de Física – Volume 4 – Óptica e Física Moderna – Thomson Learning – fs.html 3. &q=c%E1ustica&group=

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago31 Team of Brazil Problem 5: Bright waves Thank you!Thank you!

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago32 Team of Brazil Problem 5: Bright waves Depth d q R p n1n1 n2n2 I O N C Θ1Θ1 Θ2Θ2 α β ϒ

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago33 Team of Brazil Problem 5: Bright waves Depth d q R p n1n1 n2n2 I O N C Θ1Θ1 Θ2Θ2 α β ϒ

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago34 Team of Brazil Problem 5: Bright waves Depth d q R p n1n1 n2n2 I O N C Θ1Θ1 Θ2Θ2 α β ϒ No infinito

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago35 Team of Brazil Problem 5: Bright waves Differences when the tank is lit by its bottom One more refraction. Role of water lenses is inverted. Convergence above the waves. The bulkhead is the lenses of the projector. The behavior in function of the depth continues the same (also shown by exp. 4).

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago36 Team of Brazil Problem 5: Bright waves Deduction of length in function of depth

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago37 Team of Brazil Problem 5: Bright waves Trigonometric approximations

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago38 Team of Brazil Problem 5: Bright waves Different fluids Refraction index: convergence is altered and, thus, q is altered. Absorbance: bright pattern intensity changes. Viscosity: wave features, as wave length, velocity, frequency etc are altered.

Team of Brazil Problem ## Title Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara GuinsbergBad Saulgau, 20 th – 29 th July, 2012 Reporter: Bárbara Cruvinel Santiago39 Team of Brazil Problem 5: Bright waves Capillary waves Are generated by surface tension effects mainly. With a high amplitude, we consider gravitaonal wave, for the high energy used. Software for our expriments: Audacity – The maximum amplitude (1 in arbitrary units) of the software was used.