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ORE 654Applications of Ocean Acoustics
Lecture 7b
Scattering by Bubbles
Bruce HoweOcean and Resources Engineering
School of Ocean and Earth Science and TechnologyUniversity of Hawai’i at Manoa
Fall Semester 2014
04/21/23 1ORE 654
Bubbles- Outline
• Background and History• Scattering from a spherical gas bubble• Single pulsating bubbles• Multiple-bubble effects
04/21/23 ORE 654 2
Background
• Urick and Hoover 1957 – much of scatter came from below the surface – presumably bubbles created by breaking waves
• Blanchard and Woodcock (1957) – salt nuclei in the marine boundary layer generated by breaking waves affecting thunderstorm and cloud formation
• LaFond and Dill 1957 – surface slicks – internal waves, active convergence zones
04/21/23 ORE 654 L5 3
Background - 2• Previous thinking – any bubbles would dissolve by
diffusion or rise quickly to the surface. • Flaws:
– Ocean bubbles are not clean – dirty surface inhibits diffusion, or they exist in crevices that inhibit diffusion
– Ocean currents create friction drag that counters buoyancy– Bubbles constantly replaced by source mechanisms –
spilling/breaking waves, rainfall, continental aerosols dropping into sea, generated by photosynthesis or other living matter, decomposing, or from gas hydrates
• Many, many bubbles near sea surface – affect sound propagation, aid study of near surface ocean
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Background - 3• Glotov et al. 1962 – lab flume• Medwin 1970 reported
– Barnhouse et al., 1964 – photographs of bubbles in quiescent sea
– Buxcey, 1964, first measurements, using acoustics• Medwin 1977 – function of depth, time of
day/night, wind, slicks, etc – using acoustics• Tim Leighton, 1994 – The Acoustic Bubble
04/21/23 ORE 654 L5 5
Background - 4• Marine snow detritus makes it difficult to for simple photography to
identify small bubbles < 40 um• Laser holograohy (O’Hern et al, 1985) showed 105-106 /m3 radius 15 μm
near surface, more when rough – verified previous 2 decades of similarly determined acoustic results
• Peak density at 10 – 15 μm. • Invert using linear acoustic methods for bubble identification and count
• Acoustic backscatter• Increased attenuation• Differential sound speed• Doppler shift• Non-linear behavior• Also passive listening under waves or rain to hear new bubbles being
formed• Near surface attenuation can be as high as 60 dB, sound speed ~10s m/s
less than nominal c based on T,S,p
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Bubbles
• Different from rigid sphere• Impedance ρcwater / ρcair = 3600
• Can resonate – absorption cross section ~400 times the geometrical cross section
• Change compressibility of seawater and therefore c(f) – medium now dispersive – meters/second during storms
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Scattering directivity• Ideal spherical bubble
directivity• Essentially
omnidirectional for ka << 1, large λ
• Note large L at ka = 0.0136
• As ka grows above one, more forward scatter, more backscatter, less to sides
• Cf “Mie scattering” for EM waves – Maxwell’s equations, all ka
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Backscattering length and cross-section• Compare sea-level
gas bubble with rigid sphere
• Resonance at ka = 0.0136 (a/λ ~ 0.0022)
• Does not include significant damping due to thermal and viscous stresses – decrease height and broadens resonance peak
04/21/23 ORE 654 L5 9
Single bubble resonance• Resonance peak at ka = 0.0136 known since 1930s research on
sounds of running water• For such small ka, same pressure at all points on bubble• Most effective mode of oscillation is breathing/monopole • Equivalent to a mass-spring system, lumping acoustical
parameters• Equivalent mass due to inertia of water immediately
surrounding bubble that moves with ~same radial displacement• Spring stiffness f(compressibility of bubble, surface tension (for
small bubbles))• Bubble radius a = spring reference position; da = ξ linear
displacement of spring• Later treat loss of energy due to re-radiation, thermal and
viscous losses
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Spring
• Newton’s F=ma• F = restoring spring
force• Assume
displacement harmonic
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Bubble stiffness
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• Need restoring force pressure x area
• Volume, surface area• Absolute pressure in bubble ~
static ambient• Small pressure changes• Adiabatic ideal gas (no change
in entropy/temperature (ratio of specific heats)
• Chain rule• Surface area x pressure = F =
spring restoring force (Hooke’s Law)
Bubble mass
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• Need equivalent mass to use in ma, the inertial force
• Most mass is liquid next to bubble, not gas/vapor in bubble itself
• Calculate – inertial force ~ pressure of re-radiated sound
• Later verify for resonance λ >> a• Now assume ka << 1, scattered
pressure isotropic• Acoustic momentum equation gives
particle acceleration in terms of pressure – for spherical equation case
• Inertial force at surface F=ma• Equivalent mass• Low frequency approximation –
oscillating mass is a (shell of volume 3 x bubble volume) x water density
• Shell thickness = a, radius of bubble
Breathing frequency
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• Writing Newton’s second law – external forces F = ma for radial displacement using mass and stiffness
• Assume simple harmonic displacement
• Get bubble resonant frequency
Breathing frequency - 2
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• Simple formulae• ka is indeed small• Near surface (2 m) “60 um ~ 60
kHz”• At 10 m, f=76 kHz• Good representation• Begins to fail:
• Small < 5 um (important in cavitation)
• Dirty bubbles with skin debris
• Bubbles in crevices• Bubbles near surfaces• Non-spherical bubbles
Damping
04/21/23 ORE 654 L5 16
• Surface tension forces ~ curvature, viscosity
• No longer adiabatic (small, becomes isothermal γ ~ 1, heat transfer happens quickly because large relative surface area)
• Little changes in resonant frequency but major change in damping (peak amplitude, width) – affects scattering
• Write γ ~ γ(b+id) to allow phase lag between pressure and volume, function of frequency and bubble size
• Correct with functions b and β• b = b(d/b, γ, f, gas thermal
conductivity, gas specific heat, radius, density)
• β = β(surface tension, ambient pressure, radius, specific heat, b)
• b, β somewhat mutually counteracting• Earlier, simpler equations – errors <
8% for a < 2 μm at sea level
Damping
04/21/23 ORE 654 L5 17
• Damping coefficients: re-radiation, thermal conductivity, shear viscosity
• Bubbles – different gasses (γ ~ 1 and 1.67,) may have organic skins or detritus on surface, parts of fish or plankton
• Result … total damping in i […]
Damping at resonance
04/21/23 ORE 654 L5 18
• f = fR, fR/f = 1• Amplitude limited by imaginary term that affects
amplitude and phase• Figure – damping constants at resonance• Thermal – significant > 1 kHz• Viscous > 100 kHz (note 1/a2 ~ surface tension)• Resonance radiation < 1 kHz• Approximate formula 100 Hz to 100 kHz
Damping - Q
04/21/23 ORE 654 L5 19
• Sound propagating through bubbly water attenuates and scatters
• Attenuation caused by thermal conductivity and shear viscosity absorption principally at the bubble wall, as well as reradiation (scatter) out of the beam
• For single bubble – add damping term to inertial and stiffness forces
• Without damping, no limit to amplitude growth at resonance
• Sharpness of resonant peak defined by Q
• At sea level, resonant damping constant due to reradiation is 0.0136.
Damping - example
04/21/23 ORE 654 L5 20
• Bubbles generated by breaking waves – lab experiment
• Glotov, 1962• Largest number per
unit volume were radius 60 μm
• What is resonance frequency and half power frequencies?
• Approximate fr, then correct (with b, β) from figure
• Use preceding figure to get δ
Acoustical cross-section
04/21/23 ORE 654 L5 21
• Recall total acoustical cross-section = ratio of total scattered power / incident plane wave intensity (here ka < 1)
• At resonance, magnitude is limited by damping• Greater than geometric cross-section by 4/δ2
• Can be substantially greater, 100s
Acoustical cross-section - small
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• When ka << 0.1, scattering omnidirectional• Differential in any direction = total cross section / total
spherical angle• Relative backscatter length, ka < 1
Special ka = 0.0136 bubble
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• Compare resonant and rigid sphere
• Shown in earlier figure
• Resonant bubble has scattering length ~105 > than for rigid
• Easy to distinguish!
04/21/23 24
Bubble Scattering at resonance
• Resonance scattering• Bubble radius
Extinction and absorption
• Extinction – = absorption (convert to heat) +
scatter– Direct – power = rate of work on
bubble (pressure x area x velocity) / incident intensity
• Absorption – total - extinction• At resonance, cross-sections
>> than for rigid sphere• Because of direct connection
between resonant frequency and bubble radius, makes acoustic measurements of bubbles advantageous
04/21/23 ORE 654 L5 25
Extinction and absorption - 2 • Why such a large reaction relative to
light (geometric)? • Consider specific acoustic impedance =
pressure/particle velocity• Bubble near resonance is a low
impedance “hole” compared to surrounding water
• Hole distorts sound field over large volume – causes energy flux toward bubble center from plane wave far beyond cross section
• When viewed at a large distance acoustically, absorption, scatter, and extinction appears like large body
• Figure – ka < 1• Smaller than resonance, cross-sections
~a6
• Large bubbles have large extinction and scattering cross-sections
• Total scatter and absorption cross-sections may be biased by presence of large bubbles
• Absorption measurement not contaminated by larger bubbles04/21/23 ORE 654 L5 26
Pulsations in a sound field – linear friction• Include damping force with
inertial and stiffness forces in harmonic oscillator equation
• Linear friction – directly proportional to radial velocity by mechanical resistance RM ~ δ
• Lumps together reradiation, shear viscosity, and thermal conductivity
• “can be shown”• Equate reals• Entrained mass m• Direct analogy to linear
dynamical systems
04/21/23 ORE 654 L5 27
Other effects
• Near surface at depth z = dipole with separation 2z (direct and reflected/mirror source)
• Non-sphericity: non-unity aspect ratio, ellipsoidal shape or other – weaker scattered radiation, higher fR, broader peak (e.g., fish swimbladders)
• Non-linearity – large shape oscillates• Longuet-Higgins – 2nd harmonic shape oscillation
resonant with pulsation frequency, produces excess dissipation/damping, continues until linear again
04/21/23 ORE 654 L5 28
Other effects
• Salinity – fR about 10% greater in salt water
• Remote sensing of bubbles near sea surface– Direct backscatter from bubble region– Bubble scatter off sea surface then to source– Insonification of bubble by sound from sea surface– Path from surface scatterer to bubble to surface to
source– Need appropriate spatial resolution
04/21/23 ORE 654 L5 29
Multi-bubble effects
• Widely spaced bubbles – acoustic cross section simple sum of cross sections of individual bubbles
• Up to void fraction 0.01• Bubble clouds – under breaking waves, then entrained
by Langmuir cells and turbulence– Significant backscatter of low frequency sound (<2 kHz)– Source of low frequency ambient sound/noise (f < 500 Hz)
– oscillates as one large pseudo bubble• If bubbles close packed (wake), can be significant
reflection at pressure release interface/face of cloud
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04/21/23 ORE 654 L5 31
Courtesy Paul Koenigs
Backscatter – a cloud of bubbles• Continuous distribution of bubble sizes• Backscattering cross-section per unit volume for all
bubbles in volume• Number of bubbles between a and a+da per unit
volume• Differential cross-section for small (ka < 1) bubbles• Given n(a), numerically calculate backscatter
04/21/23 ORE 654 L5 32
Backscatter – a cloud of bubbles - 2• Assume all
contribution is from narrow resonant peak – and values constant in this narrow frequency range/size distribution, i.e., n(a)da and δ constant
• q small• Errors < few per cent
for ka < 0.1
04/21/23 ORE 654 L5 33
Attenuation due to bubbles – one size• More than just molecular
absorption• Bubble absorption and
scattering• Assume here one size, widely
separated (> extinction cross section length, or > λ)
• Power absorbed and scattered = incident intensity x extinction cross section
• Intensity change over distance
• Intensity Level IL• Spatial attenuation rate dB/m
04/21/23 ORE 654 L5 34
Attenuation due to many bubbles• Bubble size distribution• Extinction cross section per unit volume, Se ~ σeN• Attenuation rate due to mixture of bubbles• Invert for n(a)da: assume only contribution to Se near resonance
and bubble density and damping constant• Bubble density at resonance given measured attenuation
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Attenuation example• 50 kHz• Assume size distribution ~ a-4
• Total Se = area under σen(a)da curve
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Sound speed ~ frequency and void fraction
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• Sound speed2 ~ elasticity ~ 1/compressibility
• Small effect on density for practical purposes
• Compressibility = sum of bubble free and bubbles themselves
• dV = surface area x displacement
• Insonified by plane wave• Substitute for displacement
Sound speed ~ frequency and void fraction
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• Simplify• Multiply numerator
and denominator by complex conjugate, simplify
• Define A, B
Sound speed ~ frequency and void fraction
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• New expression for speed of sound
• Wavenumber, Taylor expansion
• Imaginary part = attenuation αb
• Real part = propagation of constant phase surfaces at dispersive phase ω/kRe
Sound speed ~ frequency and void fraction
04/21/23 ORE 654 L5 40
• Speed then is real part• Write in terms of void fraction U, N number/m3
• kR = ωR/c is value of k0 at resonance; Y = ωR/ω
Sound speed ~ frequency and void fraction
04/21/23 ORE 654 L5 41
• Plot fractional sound speed change for two bubble densities (same bubble size, just N different)
• At resonance fR/f = Y = 1, c = c0
• Below resonance, Y > 1, Δc < 0,
• Above resonance, Y < 1, Δc > 0, asymptotic to c0 – bubbles at high frequency have no effect
Above fR
Below fR
Sound speed ~ bubbles at high frequency
04/21/23 ORE 654 L5 42
• Above resonance, Y < 1, Δc > 0, asymptotic to c0 – bubbles at high frequency have no effect on sound phase speed
• Sound velocimeters use high frequency – MHz – to minimize impact of bubbles (still can be dropout), cA = c0
Sound speed ~ bubbles at low frequency
04/21/23 ORE 654 L5 43
• Below resonance Δc < 0, asymptotic to clf
• kRa is a constant for a given gas at a given depth (0.0136 for air at z=0m)
• Depends only on void fraction U
• Valid for U < 10-5, and ka < 1
Multiple radii bubbles
04/21/23 ORE 654 L5 44
• Replace N by n(a)da and U by u(a) = n(a)da x V
• Contributions to compressibility are small, add linearly
• Caution for U > 10-4 (linearized wavenumber approximation) – alternative forms
Wood’s equation
04/21/23 ORE 654 L5 45
• Valid for any U• If n(a)da not known• Measure low frequency asymptotic sound speed to get U• In terms of bulk modulii of elasticity of air and water• Average density ρA and average elasticity EA
• For U < 10-5, ~independent of size distribution
Wood’s equation
04/21/23 ORE 654 L5 46
• For typical values• If U = 0.0001, c reduced 53 %