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Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology

ORE 654 Applications of Ocean Acoustics Lecture 7a Scattering of plane and spherical waves from spheres. Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology University of Hawai ’ i at Manoa Fall Semester 2014. Scattering.

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Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology

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  1. ORE 654Applications of Ocean AcousticsLecture 7aScattering of plane and spherical waves from spheres Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology University of Hawai’i at Manoa Fall Semester 2014 ORE 654 L5

  2. 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 ORE 654 L5

  3. Plane and spherical waves • If a particle size is < first Fresnel zone, then effectively ensonified • Spherical waves ~ plane waves ORE 654 L5

  4. Plane and spherical waves • TX – gated ping • Scattered, spherical from center • Real – interfering waves from complicated surface • Can separate incident and scattered outside penumbra (facilitated by suitable pulse) ORE 654 L5

  5. Incident and scattered p(t) • 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) ORE 654 L5

  6. Scattering length • 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 too ORE 654 L5

  7. Differential Scattering cross-section • 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” Alpha particle tracks. Charged particle debris from two gold-ion beams colliding - wikipedia ORE 654 L5

  8. Backscatter • Transmitter acts as receiver (θ = 180°) • “mono-static”, • backscattering cross-section • (will concentrate on this, and total integrated scatter) ORE 654 L5

  9. Total cross-sections for scattering, absorption and extinction • 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 ORE 654 L5

  10. Target strength TS • dB measure of scatter • For backscatter (monostatic) • In terms of cross section, length • Note – usually dependent on incident angle too ORE 654 L5

  11. Sonar equation with TS • Assumes monostatic • Could have bi-static, then TLs different ORE 654 L5

  12. Sonar equation with TS – example • Fish detected • R = 1 km • f = 20 kHz • SL = 220 dB re 1 μPa • SPL = +80 dB re 1 μPa • TS? • L? ORE 654 L5

  13. Kirchhoff approximation - geometric • 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) ORE 654 L5

  14. A plane facet • 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) ORE 654 L5

  15. Sphere – scatter • 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 ORE 654 L5

  16. Sphere – geometric 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 ORE 654 L5

  17. Sphere – geometric scatter - 2 • 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)1/2 ORE 654 L5

  18. Sphere – geometric scatter - 3 • ka >> 1 • Large a radius and/or small wavelength (high frequency) • Agrees with exact solution Geometric Rayleigh Mie ORE 654 L5

  19. 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 ORE 654 L5

  20. 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 - add ORE 654 L5

  21. Rayleigh scatter - 2 • Simplest: Small object, fixed, incompressible, no waves in interior • Monopole scatter because incompressible • Dipole because fixed (wave field goes by) ORE 654 L5

  22. 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 R ORE 654 L5

  23. 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 ORE 654 L5

  24. 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 ORE 654 L5

  25. 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 ORE 654 L5

  26. Rayleigh scatter – scattered pressure • Scattered = monopole + dipole • kR >> 1 >> ka • Reference 1 m • ka can be as large a 0.5 ORE 654 L5

  27. 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) ORE 654 L5

  28. 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). ORE 654 L5

  29. 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 ORE 654 L5

  30. Scatter from a fluid sphere • Represent marine animals • For fish: • L is 1 – 2 orders of magnitude smaller than for rigid sphere (0.28) ORE 654 L5

  31. 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) ORE 654 L5

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