Investigating Helical Spring Sound Phenomenon in Science-Fiction Movies

1. 8. Reporter: Filip Landek Sci-Fi Sound

2. Table of contents Sci – FiSound– Table of contents

3. Problem description: „Tapping a helical spring can make a sound like a “laser shot” in a science-fiction movie.” „Investigate and explain this phenomenon.” Sci – FiSound– Problem description

4. Table of contents Sci – FiSound– Table of contents

5. Bendingfreevibrations of Slinkywire Longthin beamfreebendingvibrations Euler-Bernoullitheory = 0 bending in space bending in time E – Young’s modulus ρ – density (g/cm3) I – moment of inertia S – cross-sectionarea Sci – FiSound– Theoretical model

6. Equation of acousticdispersion Dispersionrelation: bendingwaveangularvelocityn  wavenumber kn n = 0, 1, 2, …,  ω (rad/s) ω – angularvelocity (rad/s) k – wavenumber (rad/m) E – Young’s modulus ρ – materialdensity (g/cm3) I – moment of inertia S – beamcross-sectionarea k (rad/m) Sci – FiSound– Theoretical model

7. ω (rad/s) k (rad/m) v – phasevelocity f – frequency λ – wavelenght ω – angularfrequency k – wavenumber Propagation of waves in differentmediums Sci – FiSound– Theoretical model

8. Propagation of waves in differentmediums Dispersivemedium Non-dispersivemedium A(mm) A(mm) Lenght (m) Lenght (m) Sci – FiSound– Theoretical model

9. Initialdisturbanceof Slinkywire  = D´ - bendingstiffness m´ - massperunitlenght t – time a – geometrical parametar E – Young’s modulus I – moment of inertia ρ – density x - displacement w(x,t) propagatingof initialdisplacementw(x,t) -x x=0 +x Sci – FiSound– Theoretical model

10. Hypotheses • DispersivemediumHigherfrequenciestravelfasterObservable time delaytd 2. Time delaytdwill be bigger for:LongerSlinkySubsequentechoes 3. Shape of the Slinky is irrelevant Sci – FiSound– Theoretical model

11. Table of contents Sci – FiSound– Table of contents

12. Experimentsetup Polyurethanefoam (soundisolation) Metal stand Microphone Pendulum Slinkyspring Metal base Sci – FiSound– Experimentsetup

13. Experimentsetup UnstrechedSlinkyspring Slinkystand Sci – FiSound– Experimentsetup

14. Experimentalmeasurements • Qualitativeanalysis: • Case 1: Slinky (48 coils) • a) Clampedend – Clampedend • b) Clampedend – Freeend • Case 2: Roundsteelwire (19 m) • a) Straightwire • b) Hand-madehelicalspring • Quantitativeanalysis: • a) Dependencyfrequency time delay • b) Dependency time delay echo • c) Dependency time delay number of coils Sci – FiSound– Experimentsetup

15. Table of contents Sci – FiSound– Table of contents

16. Quantitativeexperimentalproof of acousticdispersion Frequency (Hz) Time (s) Soundintensity (dB) Time (s) Sci – FiSound– Analysis of experimentalresults

17. Analysisofexperimentalresults 1 a) The anatomy of typicalsoundrecordedonSlinkyclamped at bothends 1 II III IV Phase II Bendingwavespropagation Echoesmodulatedwave Dampning Acousticdisperision Phase IV Silencephase Phase III Onlylowfrequenciesremain Phase 1 Intialdisturbance Sci – FiSound– Analysis of experimentalresults

18. Analysisofexperimentalresults 1 b) The anatomy of typicalsoundrecorded after hitsfreehangingSlinky 1 VII II IV V III VI Phase II Bendingwavespropagation Echoesmodulatedwave Dampning Acousticdisperision Phases IV & VII Silencephase Phases III & V & VI Secondary (internal) disturbances Phase 1 Intialdisturbance Sci – FiSound– Analysis of experimentalresults

19. 2 b) Roundwiremadeinto a helicalspring • Soundcomparison 2 a) Straightroundwire Frequency (Hz) Frequency (Hz) Time (s) Time (s) Soundintensity (dB) Soundintensity (dB) Time (s) Time (s) Sci – FiSound– Analysis of experimentalresults

20. Dependence of time delay on frequency (Clamped-ClampedSlinky with 80 coils) td – time delay LSlinky – lenght of the Slinkywire vf – velocity of a frequency Sci – FiSound– Analysis of experimentalresults

21. Time delaybetweenhigher and lowerfrequencies in echoes td – time delay s – distance a wavehastravelled vf – velocity of a frequency ne – number of echo Sci – FiSound– Analysis of experimentalresults

22. Time delaybetweenhigher and lowerfrequencies in echoes Frequency (Hz) Time (s) Sci – FiSound– Analysis of experimentalresults

23. Dependency of frequencydelayon the number of coils Sci – FiSound– Analysis of experimentalresults

24. Table of contents Sci – FiSound– Table of contents

25. Conclusions Theoretical model: • Free flexural vibrations of a long thin beam (Euler-Bernoullitheory) • Propagation of initial disturbance • Acoustic dispersion Experimental results: • Qualitative confirmation of the theory • Delay timebetween higher and lower frequencies • Quantitative analysis close congruence to the computer simulation • Dependency time delay  no. of coilslinear Sci – FiSound– Conclusions and references

26. References [1] P. Gash: Fundamental Slinky Oscillation Frequency using a Center-of-Mass Model [2] V. Henč-Bartolić, P.Kulušić: Waves and optics, School book, Zagreb, 3rd edition (in Croatian), 2004 [3] A. Nilsson,B. Liu: Vibro-Acoustics, Vol.1, Springer-Verlag GmbH, Berlin Heidelberg, 2015 [4] F. S. Crafword: Slinky whistlers, Am. J. Phys. 55(2), February 1987, p.130-134 [5] F. S. Crafword: Waves, Berkeley Physics Course, Vol.3, Berkely, 1968 [6] W. C. Elmore, M.A. Heald: Physicsofwaves, McGraw-Hill Book Company, New York, [7] J. G. Guyader: Vibration in continuous media, ISTE Ltd, London, 2002 [8] G. C. King: Vibrationsandwaves, John Wiley & Sons Ltd, London, 2009 [9] Th. D. Rossing,N. H., Fletcher: Principles of vibration and sounds, Springer-Verlag New York, lnc., 2004 [10] L.E. Kinsle et.all: Fundamentals of Acoustics,4th ed., John Willey & Sons, Inc, New York, 2000 [11] M. Géradin, D.J. Rixen: Mechanical Vibrations: Theory and Application to Structural Dynamics, 3rd ed., John Wiley & Sons, Ltd, Chichester,2015 [12] C.Y. Wang , C.M. Wang: StructuralVibration- Exact Solutions forStrings, Membranes,Beams, and Plates, CRC PressTaylor & FrancisGroup, Boca Raton, 2014 [13] A. Brandt: Noise and vibration analysis : signal analysis and experimental procedures, John Wiley & Sons Ltd, Chichester, 2011 [14]F. S. Crawford, Jr. : Waves – Berkeley PhysicsCourseVolume 3,EducationDevelopment Centre, 1965 [15] L. E. Kinsler, A. R. Frey, A. B. Coppens, and A. V. Sanders, Fundamentals of Acoustics, John Wiley NewYork, 2000 Sci – FiSound– Conclusions and references

27. Thankyou for yourattention!

29. FreehangingSlinkyparametars Coildisplacement = Centre of mass

30. Waveequationderivation = 0 1) II)

31. Ad.1. For lowfrequencyvibration, whenthethickness(h) ofbeam’s cross-section is smallerthanthevibrationswavelengthn (e.g. h = 0.0025 m  kn < 420) or thedispersionrelationandthephasevelocityrelationhavethefolowingforms [3,4,7]: n = 0, 1, 2, …,  rS… the radius of gyrationofbeamcross-section (m) cS… thephasevelocityof a particularpointin a beammaterial (m/s) kn … thewavenumberofnthbendingwave I … therotationalinertia moment of a beam’s cross-sectionsurface S … areaof a beamcross-sectionsurface (m2) n… thentheigencircularfrequencyof a bendingbeam(rad/s) Lowfrequencybendingmovementof a beamcross-sections [7]

32. Bending at high and lowfrequencies Lowfrequencyvibrations Highfrequencyvibrations ω – angularfrequency E – Young’s modulus ρ – density I – moment of inertia S – cross-sectionarea k - wavenumber

33. Ad. 2. For highfrequencywaves, thebeamdeflection is completelydeterminedbytransversalandlongitudinalwavesandthedispersionandphasevelocityrelationsshowednondispersivebehaviourofthebeamcross-section [11]. or n = 0, 1, 2, …,  Due to dispersioneffectthelowerfrequencies had beenrecorded, andheard, withdelayed time afterhighfrequencies. Highfrequencytransverseandquasi-longitudinalmovementof a beamcross-sections [3,7]

34. Dependency of frequencydelayon the number of coils Sci – FiSound– Analysis of experimentalresults

35. Mathematicallymodellingofwave damping andemittedsound  .. thewavelossfactor In the acoustic consideration the Slinky wire is modeled as continuouslinesoundsourceundertransversaloscillations. Each segment ofline (x) is anunbaffledsimplesourcewhichgeneratethe increment of sound pressurepressurelevel (SPL) in theair [10]. exp p(r,,t) … soundpressure (Pa); j = U0,n … the amplitude ofthewavevelocity 0 … thedensityofair (1.2 kg/m3) ca … thevelocityofsoundinair (343 m/s) The far field acoustic field at point p(r,,t) produced by line source of length L and radius a[10] For bothmodeledcasesthe time functiongn(t) is buildedfrom a harmonicandvanishingwavesubfunctions [3]: