ORE 654Applications of Ocean Acoustics

Lecture 7aScattering of plane and spherical waves

from spheres

Bruce HoweOcean and Resources Engineering

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

Fall Semester 2014

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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)04/18/23 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)04/18/23 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 too04/18/23 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

• Two equivalent definitions:– 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

• Mie Scattering – ka ~ 1

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Sphere – scatter

• Rigid, perfect reflector• 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)1/204/18/23 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

Ray

leig

h

Mie

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• Two conditions cause scatter:

– 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) (~ sphere moving)

• If ρ1≠ρ0, scattered p ~ cosθ• Two separate effects - add04/18/23 ORE 654 L5 20

Rayleigh scatter - 2

• Simplest: Small object, fixed, incompressible, no 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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R

Rayleigh scatter - 4• Boundary conditions

velocity and displacement at surface = 0

• At R=a, u and dP/dR = 0

• U scattered at R=a• ka small ex ≈ 1 + x

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Rayleigh scatter - monopole• Volume flow, integral of

radial velocity over surface of the sphere m3/s

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

monopole• Using kR >> 1 >> ka

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Rayleigh scatter - dipole• Volume flow, integral of

radial velocity over surface of the sphere

• First term ~ oscillating flow in z direction

• Previous expression for dipole in terms of monopole

• Again, 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 – small 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

– Omnidirectional (e dominates)

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Rayleigh scattering comments

• If e = 1, same elasticity as water, first term (monopole) is zero – has zero isotropic scatter

• Zero dipole scatter when density is same as water g = 1

• Terms add/cancel depending on relative magnitude of e and g

• If ka << 1 and e>1 and g>1, backscatter is very small – rigid sphere (e>>1, g>>1).

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Rayleigh scatter – small 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 (0.28)

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Scattering from SphereRF – Mie theory

• Mie scattering ka ~ 1• Discrete (coupled) dipole

scatterer• Maxwell’s equations –

electromagnetism• Monostatic radar cross

section for metal sphere• X axis – number of

wavelengths in a circumference – kR

• Y axis – RCS relative to projected area of sphere

• F4 in low frequency – Rayleigh (lambda > 2πR)

• =1 in high frequency (optical) limit (λ << R)

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