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ORE 654 Applications of Ocean Acoustics Lecture 5 Intense sounds: non-linear phenomena

ORE 654 Applications of Ocean Acoustics Lecture 5 Intense sounds: non-linear phenomena. Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology University of Hawai ’ i at Manoa Fall Semester 2013. Summary.

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ORE 654 Applications of Ocean Acoustics Lecture 5 Intense sounds: non-linear phenomena

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  1. ORE 654Applications of Ocean AcousticsLecture 5Intense sounds: non-linear phenomena Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology University of Hawai’i at Manoa Fall Semester 2013 ORE 654 L5

  2. Summary • All previous discussion assumed infinitesimal amplitude • Extraordinary behavior when pressures large • Steepening of wave slopes to produce shocks • Streaming • “parametric” sources – two high frequencies beat to produce high intensity pencil beam ORE 654 L5

  3. Intense sounds: non-linear phenomena • Harmonic distortion and shock waves • Cavitation • Parametric, difference frequency, sources • Acoustic radiation pressure • Acoustic streaming • Explosives as sound sources • Any non-linear system – with high/finite amplitude – harmonics and subharmonics; sum and difference frequencies, from basic physics or less-than-ideal systems • Perturbations in, say density, no longer small, need to include higher order terms ORE 654 L5

  4. Speed of sound pressure • Sound speed is function of ambient pressure • Hooke’s Law - linear Stress (pressure change) and strain (density change) - slope determines c • Bulk modulus of elasticity E • Slope - sound speed - is a function of ρA • Slope and speed increase with ρA density ORE 654 L5

  5. Equation of state • To calculate c, need analytic relation for equation of state • This form of equation (adiabatic – no heat exchange) is well known for gases; Γ is the ratio of specific heat at constant pressure to specific heat at constant volume; K depends on units and atomic mass of gas • For water, p includes not only external pressure, but an internal cohesive pressure of ~ 3000 atm. Γ and K not ratio of specific heats but must be determined empirically ORE 654 L5

  6. Include higher order term • Differentiate to get sound speed at ambient density, and therefore c • Expand density in Taylor series and substitute • Approximate by first two terms of binomial expansion • Thus c reduces to cA if density changes very small ORE 654 L5

  7. Alternative form for equation of state • Latter is empirical, can try a different curve fit – power series (Beyer, 1975) • Equate with previous (equivalent), slightly different interpretation • B/A – “parameter of nonlinearity” used in high intensity studies • Water B/A = 5.0 at 20°C and 4.6 at 10°C • Air Γ = 1.4, B/A = 0.40 ORE 654 L5

  8. Steepening of waves • Expression for u from earlier (acoustic Mach number definition et al.) • Signal speed u + c • Faster where pressure is locally high and vice versa ORE 654 L5

  9. Harmonic distortion and shock waves • Signal speed u+c • Faster than cA where pressure is locally high and vice versa • Different signal speeds at different parts of wave • Advance of crests relative to troughs • Sawtooth / repeated shock • Loss of energy from fundamental to harmonics • Then high frequencies dissipate due to absorption Distance from source increasing ORE 654 L5

  10. Wave growth • On axis particle travels • Δx1 =cA Δt • Peak of wave travels Δx2 in same time interval • ΔX crest advance relative to axis • Assume distorted wave is fundamental + 2nd harmonic • The peak has zero slope • For small kx (i.e., kΔX), sin(kΔX) ≈ kΔX and cos(kΔX) ≈ 1 • Relative strength of 2nd harmonic P in terms of fundamental P ORE 654 L5

  11. Wave growth2nd harmonic pressure • Use basic definition of c2 = (Δp/Δρ) and say P1 ~ Δp • For plane waves 2nd harmonic growth ~ • square of fundamental P1 and • number of wavelengths progressed by fundamental kX • This for unattenuating plane wave • Spherical waves diverge, will require greater initial amplitudes to achieve same degree of distortion ORE 654 L5

  12. Wave growth - numbers • For water B/A = 5, β = 3.5 • For air B/A = 0.4 and β = 1.2 • Strongest factor is denominator – factor 16,000 larger for water • Net effect – for same fundamental pressure, 2nd harmonic grows to same magnitude in a distance 1/5000 as far in air as in water, or conversely, 5000 times as far in water ORE 654 L5

  13. Wave growth – how far to formation of shock wave? • Say shock wave formed when 2nd harmonic is half the fundamental • Distance for this case? • Sawtooth wave real situation; with diverging spherical wave happens at larger range • Distance ~ 1/Mach, 1/non-linearity, ~ wavelength • Must take into account absorption (not here) • Saturation limits sound energy that can be input • More energy – more harmonics – more loss • Higher intensity axial beam attenuated more, beam broadens ORE 654 L5

  14. Wave growth - saturation • Saturation limits sound energy that can be input • If linear, lines/curves 45° • Lines at right asymptotic limits • For 10 m case, actual level is about 6 dB less than linear at ~550 kPa (~235 dB re 1 μPa) ORE 654 L5

  15. Donald Ross Cavitation • In rarefaction/tension phase, pressure can go “negative” and the medium ruptures • Small bubbles always present near sea surface are the nuclei for rupture initiation • From Bernoulli effects/propeller blades (mixture of dipole and monopole) • Life processes (snapping shrimp) • Increases chemical activity • Erode metals, plastics, stones (kidney), … • Light production – sonoluminescence • Very high pressure, 30,000 K • Picosecond light pulses ORE 654 L5 Brian Pollack

  16. Cavitation - 2 • As sound levels rise, bubble resonance, harmonics generated • Bubbles generate subharmonics if driven near 2 x resonance • 5% harmonic distortion for signals > 0.1 atm • If peak p > 1 atm (105 Pa = 220 dB re 1 μPa) • Negative pressure is trigger for sharp increase in distortion and broadband noise, if CW f < 10 Hz • Function of f, duration, repetition, nuclei • If drive too hard, generate bubbles that decrease far-field sound propagation • Gaseous cavitation - streaming bubble clouds jet away from generation site (relative amounts of gas and water vapor ~ constant) • Vaporous cavitation – collapse of single bubbles - radiates shock waves of broadband noise ORE 654 L5

  17. Cavitation - 3 • Nuclei often bubbles trapped in cracks/crevices of solid particles • Grow by “rectified” diffusion • Start with small bubbles < 1 μm • More gas diffuses into the bubble during expansion than out during contraction when surface area smaller (more time is spent large than small) • At a critical radius, will grow explosively • Threshold definition – distortion/harmonics and/or broad band noise • Above 10 kHz, steep increase in amount of CW pressure amplitude to produce cavitation • Large differences for “pure” water and tap or seawater ORE 654 L5

  18. Cavitation - 4 • Ocean-going transducer – regions on face that exceed cavitation pressure limits, combined with near surface bubbles • “hot spots” – p > nominal • Can have greater source levels at depth, high ambient pressure effectively inhibits cavitation • Fewer cavitaiton nuclei • Smaller nuclei • Streaming moves bubbles to new locations ORE 654 L5

  19. Cavitationpulse duration and duty cycle • Pulse duration < 100 ms, average acoustic intensity required for 10 % distortion is >> than CW • At low duty cycle can drive harder cavitation 10%  Duty cycle ORE 654 L5

  20. Parametric, difference frequency, sonars • If two distinct intense sound beams are co-axial at different frequencies, non-linearity creates sum and difference frequencies • Each beam modulates the other • E.g., 500 and 600 kHz produce 100 kHz and 1100 kHz • “parametric” or virtual sources distributed along intense portion of interacting beams ORE 654 L5

  21. Parametric, difference frequency, sonars - 2 • Difference frequency – lower frequency (less absorption) • very narrow beam, ~same width as for the primaries (but with smaller transducer) • Acts as if a highly directional end-fire array (“virtual end-fire array”) • Bandwidth of difference frequency very large ORE 654 L5

  22. Parametric, difference frequency, sonars - 3 • Two signals (ignore x or observing location kx = nπ) • Instead of amplitude at a point being simple sum, amplitude of p1 will be modulated by p2 • Last term – non-linear interaction, strength m(P1,P2) • Non-linear interaction has produced sum and difference frequencies ORE 654 L5

  23. Parametric, difference frequency, sonars - 4 • These sum and difference frequencies generated at all points of intense interaction along beam • Analogous to a line array of sources - end-fire ORE 654 L5

  24. Parametric, difference frequency, sonars - 5 • More detailed analysis • Re-cast wave equation for secondary or “scattered” pressure, with non-linear source term • Westervelt’s wave equation for non-linear secondary tones • Assume two primaries with attenuation (they die off quickly) • Get get 2nd (and higher) harmonics and sum and differences • Difference frequency generated whenever P1P2 large, contribution from beam near source largest • Once generated – launched, “on its own” (Huygen’s wavelets along beam) • Primary and sum frequencies die off ORE 654 L5

  25. Parametric, difference frequency, sonars – 6 • Narrow beam pattern (high directional resolution with small transducer) • Beamwidth relatively insensitive to changes in difference frequency • No side lobes in Dd (secure acoustic comms) • Inherent broad bandwidth • Projector cavitation not a problem ORE 654 L5

  26. Parametric, difference frequency, sonars - 7 • Increase in bandwidth • BW of a primary typically ±5% • f1 = 418 ± 21 kHz; f2 = 482 ± 24 kHz • fave = 450 ± 22.5 kHz • fdiff = 64 kHz ± 22.5 kHz - ±35% ORE 654 L5

  27. Parametric source example • Given 2 f’s, what is beam width? • What size piston to produce the LF? • What size piston needed? • What is the reduction in source radius for same beamwidth? ORE 654 L5

  28. Parametric source efficiency • Same power W and pressure P • Power radiated through primary beam area S0 • On-axis rms Pd (using beam width) • Average intensity ~ P2d ORE 654 L5

  29. Parametric source example • For preceding parameters, P1=100W • Efficiency = 0.7% • Increase efficiency: • Increase difference frequency • Increase power • Decrease beamwidth (lower primary frequency with constant beam area – larger transducer) • Only Power increase without sacrificing advantages – limit by saturation effects and beam broadening and cavitation • Level can be increased by non-linear oscillation of bubbles, but some loss in radiation directionality • Used for sub-bottom profiling ORE 654 L5

  30. Tritech SeaKing Parametric SBP Sub-Bottom Profiler • Primary frequency 200 kHz • Primary beamwidth 4 degrees • Low frequency 20 kHz • Low frequency beamwidth 4.5 degrees • Pulse length 100 μseconds • Range resolution of HF Dependent upon rangescale (10-100mm) • Range resolution of LF Dependent upon rangescale (60 μseconds@30m) • Power requirements 24VDC @ 410mA (Nominal for DST model) • Transducer 200 mm diameter • Weight in air 6.3kg • Weight in water 2.7kg • Maximum operational depth 4000m ORE 654 L5

  31. Acoustic streaming • Non-linear – harmonic distortion and shocks • Can also cause unidirectional flow • “quartz wind” outward jetting or drift of water in front of transducer • Strongest on axis, distances of meters • Can be >1000 acoustic velocity • Eddy/recirculate/3-D ORE 654 L5

  32. Movie -1 ORE 654 L5

  33. Movie - 2 ORE 654 L5

  34. Acoustic streaming – radiation pressure • Langevin radiation pressure = average momentum carried through unit area in unit time = time average of momentum per unit volume ρAu by particle velocity u (U = rms velocity) • This also = average energy density in beam = <ε> = average intensity / cA ORE 654 L5

  35. Acoustic streaming – momentum transfer • Spatial change in momentum – absorption / dissipation • Newton’s 2nd law – rate of change of momentum per unit area is force per unit area or change in pressure ΔPAacross slab dx • Loss of acoustic pressure in path dx creates pressure gradient causing flow • Or, loss of momentum in acoustic beam made up with gain in momentum of fluid mass – conservation of momentum • Bubbles on a transducer face – cavitation, asymmetric toroid produces destructive jet into wall, but superimposed on mean flow pattern ORE 654 L5

  36. Acoustic streaming velocity u2 • Non-linear – magnitude proportional to intensity – Eckart, 1948 • For ideal beam in a tube P(r) = P for r<a and P(r) = 0 for a0≥r≥a • a = radius of non-divergent sound beam • a0 (larger) radius of tube • Measurements -> calculate bulk viscosity μ’ from intensity and streaming velocity • Liebermann (1949) helped resolve difference between theoretical and experimental values of attenuation • For fluids in general ORE 654 L5

  37. Explosives as sound sources • TNT etc ~ 4,400 J/g = 1050 cal/g • Rapid reaction/detonation – 3000 °C, p ~ 50,000 atm • Detonation velocity 5,000 - 10,000 m/s • Gunpowder – burning – 0.3 m/s, slowly growing • Two sources of sound • The shock wave ~ half the energy, propagates at > cA • Large oscillating gas bubble / gas globe ORE 654 L5

  38. Explosives as sound sources ORE 654 L5

  39. Shock wave • Instantaneous rise in pressure Pm • Then exp decay, time constant τs s • Both scale by (w1/3/R) – w weight of explosive kg, R range m • Common SUS 0.82 kg • No attenuation exponents 1 and 0 • Absorption and non-linearity ORE 654 L5

  40. Shock front propagation; the Rankine-Hugoniot equations • Earlier wave equation – infintesimal waves • Conservation of mass • Conservation of momentum • Water only slightly compressible so density ratio ~ 1 • Shock speed U depends on average slope dp/dρ in p(ρ) • Speed of sound of peak cm depends on local incremental slope ORE 654 L5

  41. Shock front velocity • Need equation of state and conservation of momentum • Speed of shock u + c can be > c ORE 654 L5

  42. Gas globe • Contains chemical gas products and water vapor • Contains half total energy of explosion • Initial acceleration, expands, decelerates, continues past radius where internal p equals external, reaches maximum radius with internal p less than external, bubble contracts, oscillates, produces bubble pulses • Period of oscillation f(energy after shock, ambient pressure and density) • Assume spherical, need partition of non-shock energy, Y ORE 654 L5

  43. Gas globe - frequency • Assume spherical, need partition of non-shock energy, Y (Joules) • At maximum radius am, KE is zero, internal energy << PE • Assume all non-shock energy (~1/2 explosion energy) Y is PE • Period of spherical bubble in ambient (future derivation) • Substitute • Period T ~ depth and yield • Real – • not spherical (large ambient p difference) • often splits in two because of dimpling • Non-sinusoidal oscillation • Frequency changes as bubble rises ORE 654 L5

  44. Interaction with ocean surface • Reflections off surface, phase reversed • “noise” after reflection – under tension/negative pressure – cavitation, microbubbles radiate, causes reflected shock to loose energy ORE 654 L5

  45. Perth-Bermuda - 1960 • 300 lb TNT (4400 J/g) • 1000 m deep • w = 136 kg • R = 1 m • Pm = 318 MPa • τ = 0.0002 s • PA = 107 Pa (1000 m) • Y = 3x108 J • am = 1.9 m • T = 0.06 s ORE 654 L5

  46. Explosion test facility • Brett et al., An experimental facility for imaging of medium scale underwater explosions, DSTO-TR-1432, 2003 • Defense Science and Technology Organization, Australia • Near Melbourne ORE 654 L5

  47. Video • 0.5 kg • 1 ms resolution • Frame 4 – shock wave causes cavitation on camera window at 4 ms • Initial phase spherical and smooth (note sunlight) • During contraction – asymmetric, flattened base – pA(z) • Bubble rises about 0.4 m ORE 654 L5

  48. P(t) • Rapid expansion and collapse near minimum radius • Slow at maximum radius • Rmax – 1.14 m, rmin – 0.27 m in 97 ms • 6.1 m3 in 0.09 s – 6 tons • 68 m3/s • Vmax = 3.6 m/s • Hydrophone 4.5 m range • Shock wave, bubble pulse at min radius • Surface reflect at 4.5 ms, walls/floor 13-19 ms ORE 654 L5

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