ORE 654 Applications of Ocean Acoustics Lecture 6c Scattering 1 Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology University of Hawai’i at Manoa Fall Semester /11/20151ORE 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 11/11/2015ORE 654 L52

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

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

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) 11/11/2015ORE 654 L55 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 too 11/11/2015ORE 654 L56 Scattering length

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” 11/11/2015ORE 654 L57 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) 11/11/2015ORE 654 L58 Backscatter

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 11/11/2015ORE 654 L59 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 11/11/2015ORE 654 L510 Target strength TS

Assumes monostatic Could have bi-static, then TLs different 11/11/2015ORE 654 L511 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? 11/11/2015ORE 654 L512 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) 11/11/2015ORE 654 L513 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 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) 11/11/2015ORE 654 L514 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 11/11/2015ORE 654 L515 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 11/11/2015ORE 654 L516 Sphere – geometric scatter

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 11/11/2015ORE 654 L517 Sphere – geometric scatter - 2

ka >> 1 Large a radius and/or small wavelength (high frequency) Agrees with exact solution 11/11/2015ORE 654 L518 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 11/11/2015ORE 654 L519

Rayleigh scatter Small sphere ka << 1 Scatter all diffraction 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) If ρ 1 ≠ρ 0, scattered p ~ cosθ Two separate effects - add 11/11/2015ORE 654 L520

Rayleigh scatter - 2 Small object, fixed, incompressible, now waves in interior Monopole scatter because incompressible Dipole because fixed (wave field goes by) 11/11/2015ORE 654 L521

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 11/11/2015ORE 654 L522

Rayleigh scatter - 4 Boundary conditions velocity and displacement at surface = 0 At R=a, u and dP/dR = 0 U scattered at R=a 11/11/2015ORE 654 L523

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 11/11/2015ORE 654 L524

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 11/11/2015ORE 654 L525

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

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 11/11/2015ORE 654 L527

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 11/11/2015ORE 654 L528

Scatter from a fluid sphere Represent marine animals For fish: L is 1 – 2 orders of magnitude smaller than for rigid sphere 11/11/2015ORE 654 L529