1. 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 11/30/20151ORE 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 11/30/2015ORE 654 L52

3. Plane and spherical waves If a particle size is < first Fresnel zone, then effectively ensonified Spherical waves ~ plane waves 11/30/2015ORE 654 L53

4. TX – gated ping Scattered, spherical from center Real – interfering waves from complicated surface Can separate incident and scattered outside penumbra (facilitated by suitable pulse) 11/30/2015ORE 654 L54 Plane and spherical waves

5. TX – gated ping Assumed high frequency with duration t p, peak P inc Shadow = destructive interference of incident and scattered/diffracted sound If pulse short enough, can isolate the two waves in penumbra (but not shadow) 11/30/2015ORE 654 L55 Incident and scattered p(t)

6. 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 11/30/2015ORE 654 L56 Scattering length

7. Simply square scattering length to give an effective area m 2 (from particle physics scattering experiments); differential solid angle Depends on geometry and frequency Can be “bistatic” or “monostatic” 11/30/2015ORE 654 L57 Differential Scattering cross-section Alpha particle tracks. Charged particle debris from two gold-ion beams colliding - wikipedia

8. Transmitter acts as receiver (θ = 180°) “mono-static”, backscattering cross-section (will concentrate on this, and total integrated scatter) 11/30/2015ORE 654 L58 Backscatter

9. Two equivalent definitions: – Integrate over sphere – Scattered power/incident intensity (units m 2 ) 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 11/30/2015ORE 654 L59 Total cross-sections for scattering, absorption and extinction

10. dB measure of scatter For backscatter (monostatic) In terms of cross section, length Note – usually dependent on incident angle too 11/30/2015ORE 654 L510 Target strength TS

11. Assumes monostatic Could have bi-static, then TLs different 11/30/2015ORE 654 L511 Sonar equation with TS

12. Fish detected – R = 1 km – f = 20 kHz – SL = 220 dB re 1 μPa – SPL = +80 dB re 1 μPa TS? L? 11/30/2015ORE 654 L512 Sonar equation with TS – example

13. 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) 11/30/2015ORE 654 L513 Kirchhoff approximation - geometric

14. Simplest sub-element for Kirchhoff Full solution Ratio reflected pressure from a finite square to that of an infinite plane Fraunhofer – incident plane wave P bs ~ area Fresnel – facet large ~ infinite plane – oscillations from interference of spherical wave on plane facet (recall – large plate, virtual image distance R behind plate) 11/30/2015ORE 654 L514 A plane facet

15. 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 11/30/2015ORE 654 L515 Sphere – scatter

16. 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 11/30/2015ORE 654 L516 Sphere – geometric scatter

17. Geometric Scattered power gs Rays within dθ i at angle θ i are scattered within increment dθ s = 2dθ i at angle θ s = 2θ i ; polar coords at range R Incoming power = outgoing power Pressure ratio = L/R L normalized by (area circle) 1/2 11/30/2015ORE 654 L517 Sphere – geometric scatter - 2

18. ka >> 1 Large a radius and/or small wavelength (high frequency) Agrees with exact solution 11/30/2015ORE 654 L518 Sphere – geometric scatter - 3 Geometric Rayleigh Mie

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 11/30/2015ORE 654 L519

20. Rayleigh scatter Small sphere ka << 1 Scatter all diffraction Two conditions cause scatter: – If sphere bulk elasticity E 1 (=1/compressibility) E 0, opposite phase – If ρ 1 >ρ 0, inertia causes lag  dipole (again, phase reversal if opposite sense) (~ sphere moving) If ρ 1 ≠ρ 0, scattered p ~ cosθ Two separate effects - add 11/30/2015ORE 654 L520

21. Rayleigh scatter - 2 Simplest: Small object, fixed, incompressible, no waves in interior Monopole scatter because incompressible Dipole because fixed (wave field goes by) 11/30/2015ORE 654 L521

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 11/30/2015ORE 654 L522 R

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 e x ≈ 1 + x 11/30/2015ORE 654 L523

24. Rayleigh scatter - monopole Volume flow, integral of radial velocity over surface of the sphere m 3 /s (integral cosθ term = 0) Previous expression for monopole Using kR >> 1 >> ka 11/30/2015ORE 654 L524

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 11/30/2015ORE 654 L525

26. Rayleigh scatter – scattered pressure Scattered = monopole + dipole kR >> 1 >> ka Reference 1 m ka can be as large a 0.5 11/30/2015ORE 654 L526

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) 11/30/2015ORE 654 L527

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 g>1, backscatter is very small – rigid sphere (e>>1, g>>1). 11/30/2015ORE 654 L528

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 11/30/2015ORE 654 L529

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) 11/30/2015ORE 654 L530

31. Scattering from Sphere RF – 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 F 4 in low frequency – Rayleigh (lambda > 2πR) =1 in high frequency (optical) limit (λ << R) 11/30/2015ORE 654 L531