ORE 654 Applications of Ocean Acoustics Lecture 7a Scattering of plane and spherical waves from...
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Transcript of ORE 654 Applications of Ocean Acoustics Lecture 7a Scattering of plane and spherical waves from...
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
04/18/23 1ORE 654 L5
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
04/18/23 ORE 654 L5 2
Plane and spherical waves
• If a particle size is < first Fresnel zone, then effectively ensonified
• Spherical waves ~ plane waves
04/18/23 ORE 654 L5 3
• 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”
04/18/23 ORE 654 L5 7
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)
04/18/23 ORE 654 L5 8
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
04/18/23 ORE 654 L5 9
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
04/18/23 ORE 654 L5 10
Target strength TS
• Assumes monostatic• Could have bi-static, then TLs different
04/18/23 ORE 654 L5 11
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?
04/18/23 ORE 654 L5 12
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)
04/18/23 ORE 654 L5 13
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)
04/18/23 ORE 654 L5 14
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
04/18/23 ORE 654 L5 15
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
04/18/23 ORE 654 L5 16
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
04/18/23 ORE 654 L5 18
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
04/18/23 ORE 654 L5 19
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)
04/18/23 ORE 654 L5 21
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
04/18/23 ORE 654 L5 22
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
04/18/23 ORE 654 L5 23
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
04/18/23 ORE 654 L5 24
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
04/18/23 ORE 654 L5 25
Rayleigh scatter – scattered pressure• Scattered = monopole +
dipole• kR >> 1 >> ka• Reference 1 m• ka can be as large a 0.5
04/18/23 ORE 654 L5 26
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)
04/18/23 ORE 654 L5 27
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).
04/18/23 ORE 654 L5 28
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
04/18/23 ORE 654 L5 29
Scatter from a fluid sphere• Represent marine animals• For fish:• L is 1 – 2 orders of
magnitude smaller than for rigid sphere (0.28)
04/18/23 ORE 654 L5 30
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)
04/18/23 ORE 654 L5 31