ORE 654Applications of Ocean Acoustics

Lecture 6cScattering 1

Bruce HoweOcean and Resources Engineering

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

Fall Semester 2013

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Scattering• Scattering of plane and spherical waves• Scattering from a sphere

• Observables – scattered sound pressure field• Want to infer properties of scatterers

– Compare with theory and numerical results– Ideally perform an inverse

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Plane and spherical waves

• If a particle size is < first Fresnel zone, then effectively ensonified

• Spherical waves ~ plane waves

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• TX – gated ping• Scattered, spherical from center• Real – interfering waves from complicated surface• Can separate incident and scattered outside penumbra

(facilitated by suitable pulse)04/21/23 ORE 654 L5 4

Plane and spherical waves

• TX – gated ping• Assumed high frequency with duration tp, peak Pinc• Shadow = destructive interference of incident and

scattered/diffracted sound• If pulse short enough, can isolate the two waves in penumbra

(but not shadow)04/21/23 ORE 654 L5 5

Incident and scattered p(t)

• Large distance from object 1/R and attenuation• Complex acoustical scattering length L

– Characteristic for scatterer acoustic “size” ≠ physical size– Determined by experiment (also theory for simpler)– Assume incident and scattered are separated (by time/space);

ignore phase– Finite transducer size (angular aperture) integrates over solid

angle, limit resolution– Function of incident angle too04/21/23 ORE 654 L5 6

Scattering length

• Simply square scattering length to give an effective area m2 (from particle physics scattering experiments); differential solid angle

• Depends on geometry and frequency• Can be “bistatic” or “monostatic”

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Differential Scattering cross-section

Alpha particle tracks.Charged particle debris from two gold-ion beams colliding - wikipedia

• Transmitter acts as receiver (θ = 180°)• “mono-static”, • backscattering cross-section• (will concentrate on this, and total integrated scatter)

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Backscatter

• Integrate over sphere• Scattered power/incident intensity

(units m2)• Power lost due to absorption by

object – absorption cross section• power removed from incident –

extinction cross section• extinction = scattered + absorption• if scattering isotropic (spherical

bubble), integral = 4π• a/λ << 1, spherical wave scatter• a/λ >> 1, rays• In between, more difficult

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Total cross-sections for scattering, absorption and extinction

• dB measure of scatter• For backscatter

(monostatic)• In terms of cross

section, length• Note – usually

dependent on incident angle too

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Target strength TS

• Assumes monostatic• Could have bi-static, then TLs different

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Sonar equation with TS

• Fish detected– R = 1 km– f = 20 kHz– SL = 220 dB re 1 μPa– SPL = +80 dB re 1 μPa

• TS? • L?

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Sonar equation with TS – example

• Set up as before• Pressure reflection

coefficient, R, and transmission T for plane infinte wave incident on infinite plane applies to all points on a rough surface

• Geometrical optics approximation – rays represent reflected/transmitted waves where ray strikes surface

• (fold Reflection R into L)

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Kirchhoff approximation - geometric

• Simplest sub-element for Kirchhoff

• Full solution• Ratio reflected pressure

from a finite square to that of an infinite plane

• Fraunhofer – incident plane wave Pbs ~ area

• Fresnel – facet large ~ infinite plane – oscillations from interference of spherical wave on plane facet

• (recall – large plate, virtual image distance R behind plate)

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A plane facet

• Simple model• ~ often good enough for “small” non-spherical bodies,

same volume, parameters• Scatter: Reflection, diffraction, transmission• Rigid sphere - geometric reflection (Kirchhoff) ka >> 1• Rayleigh scatter - ka << 1, diffraction around body, ~(ka)4

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Sphere – scatter

• ka >> 1 (large sphere relative to wavelength, high frequency) geometrical, Kirchhoff, specular/mirrorlike

• Use rays – angle incidence = reflection at tangent point

• Ignore diffraction (at edge)• No energy absorption (T=0)• Incoming power for

area/ring element

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Sphere – geometric scatter

• Geometric Scattered power gs

• Rays within dθi at angle θi are scattered within increment dθs = 2dθi at angle θs = 2θi; polar coords at range R

• Incoming power = outgoing power

• Pressure ratio = L/R• L normalized by (area

circle)1/204/21/23 ORE 654 L5 17

Sphere – geometric scatter - 2

• ka >> 1• Large a radius and/or

small wavelength (high frequency)

• Agrees with exact solution

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Sphere – geometric scatter - 3

Geometric scatter

Sphere – geometric scatter - 4• Scattered power not a

function of incident angle (symmetry – incident direction irrelevant)

• For ka >> 1• Total scattering cross section

= geometrical cross-sectional A

• For ka > 10, L ~ independent of f – backscattered signal ~ delayed replica of transmitted

• Rays- not accurate into shadow and penumbra

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Rayleigh scatter

• Small sphere ka << 1• Scatter all diffraction• If sphere bulk elasticity E1 (=1/compressibility)

< water value E0, body compressed/expanded – re-radiates spherical wave (monopole). If E1>E0, opposite phase

• If ρ1>ρ0, inertia causes lag dipole (again, phase reversal if opposite sense)

• If ρ1≠ρ0, scattered p ~ cosθ• Two separate effects - add04/21/23 ORE 654 L5 20

Rayleigh scatter - 2

• Small object, fixed, incompressible, now waves in interior

• Monopole scatter because incompressible• Dipole because fixed (wave field goes by)

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Rayleigh scatter - 3• Sphere so small, entire surface exposed to same

incident P (figure – ka = 0.1, circumference = 0.1λ)• Total P is sum of incident + scattered

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Rayleigh scatter - 4• Boundary conditions

velocity and displacement at surface = 0

• At R=a, u and dP/dR = 0• U scattered at R=a

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Rayleigh scatter - monopole• Volume flow, integral of

radial velocity over surface of the sphere

• (integral cosθ term = 0)• Previous expression for

monopole• kR >> 1 >> ka

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Rayleigh scatter - dipole• Volume flow, integral of

radial velocity over surface of the sphere

• Previous expression for dipole in terms of monopole

• kR >> 1 >> ka

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Rayleigh scatter – scattered pressure• Scattered = monopole +

dipole• kR >> 1 >> ka• Reference 1 m• ka can be as large a 0.5

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Rayleigh scatter – elastic fluid sphere• Scattering depends

on relative elasticity and density

• Monopole – first term• Dipole – second term• In sea, most bodies

have e and g ~ 1• Bubbles

– e and g << 1– For ka << 1 can

resonate resulting in cross sections very much larger than for rigid sphere

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Rayleigh scatter – elastic sphere - 2

• Total scattering cross-section for small fluid sphere• Light scatter in atmosphere – blue λ ~ ½ red λ so

blue (ka)4 is 16 times larger• Light yellow λ 0.5 μm so in ocean all particles have

cross-sections ~ geometric area (ka large)• Same particles have very small acoustic cross

sections, scatter sound weakly• Ocean ~transparent to sound but not light

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Scatter from a fluid sphere• Represent marine animals• For fish:• L is 1 – 2 orders of

magnitude smaller than for rigid sphere

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