ORE 654Applications of Ocean Acoustics

Lecture 7

Scattering by Bubbles

Bruce HoweOcean and Resources Engineering

School of Ocean and Earth Science and TechnologyUniversity of Hawai’i at Manoa

Fall Semester 2013

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 1956 – 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

04/21/23 ORE 654 L5 4

Background - 3• Glotov et al. 1962 – lab flume, breaking waves• 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 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

04/21/23 ORE 654 L5 6

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

04/21/23 ORE 654 L5 7

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

04/21/23 ORE 654 L5 8

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 losses04/21/23 ORE 654 L5 10

Spring

• Newton’s F=ma• F = restoring spring

force• Assume

displacement harmonic

• No forcing here, so natural frequency

04/21/23 ORE 654 L5 11

Bubble stiffness

04/21/23 ORE 654 L5 12

• 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

04/21/23 ORE 654 L5 13

• 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

04/21/23 ORE 654 L5 14

• 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

04/21/23 ORE 654 L5 15

• 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

04/21/23 ORE 654 L5 18

• Combined effects of re-readiation/scattering, thermal effects, and shear viscosity

Damping at resonance

04/21/23 ORE 654 L5 19

• 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 20

• 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 21

• 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 22

• 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

04/21/23 ORE 654 L5 23

• 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

04/21/23 ORE 654 L5 24

• Compare resonant and rigid sphere

• Shown in earlier figure

• Resonant bubble has scattering length ~105 > than for rigid

• Easy to distinguish!

04/21/23 25

Bubble Scattering at resonance

• Resonance scattering• Bubble radius

Extinction and absorption

• Extinction – = absorption (convert to heat) +

scatter– Direct – power = rate of work on

bubble (power = 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 26

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 27

Pulsations in a sound field– linear friction

• Add forcing – non-homogeneous

• 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 28

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 swim bladders)

• 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 29

Other effects

• Salinity – fR about 10% greater in salt water• Remote sensing of bubbles near sea surface -

complications– 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 30

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

04/21/23 ORE 654 L5 31

04/21/23 ORE 654 L5 32

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 33

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 34

Attenuation due to bubbles – one size• More than just molecular

absorption• Bubble absorption and

scattering• Assume here one size, widely

separated in space (> 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 35

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 near resonance• Bubble density at resonance given measured attenuation

04/21/23 ORE 654 L5 36

Attenuation example• 50 kHz• Assume size distribution ~ a-4

• Total Se = area under σen(a)da curve

04/21/23 ORE 654 L5 37

Sound speed ~ frequency and void fraction

04/21/23 ORE 654 L5 38

• 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

04/21/23 ORE 654 L5 39

• Simplify• Multiply numerator

and denominator by complex conjugate, simplify

• Define A, B

Sound speed ~ frequency and void fraction

04/21/23 ORE 654 L5 40

• 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 41

• Speed then is real part• Write in terms of void fraction U = volume

gas/volume water• kR = ωR/c is value of k0 at resonance; Y = ωR/ω

Sound speed ~ frequency and void fraction

04/21/23 ORE 654 L5 42

• 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, f < fR, Y > 1, Δc < 0

• Above resonance goes to c0

Above fR

Below fR

Sound speed ~ bubbles at high frequency

04/21/23 ORE 654 L5 43

• Above resonance, f > fR, 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 44

• 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 45

• Replace N by n(a)da and U by u(a) = n(a)da x V

• Contributions to compressibility are small, add linearly

Wood’s equation

04/21/23 ORE 654 L5 46

• 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 47

• For typical values• If U = 0.0001, c reduced 53 %

Standing wave resonator

04/21/23 ORE 654 L5 48

No bubbles

with bubbles

Bubble number distribution

04/21/23 ORE 654 L5 49

• Resonator• 12 m/s

Data – wind, void fraction, sound

speed

04/21/23 ORE 654 L5 50

• 0.7 m depth• 12 m/s wind

• 0.5 m depth• 8 m/s wind