Signal Loss. A1 A L[Neper] = ln or L[dB] = 20log 1. Proportional loss of signal amplitude with increasing propagation distance: = α d

Transcription

1 Part 6 ATTENUATION

2 Signal Loss Loss of signal amplitude: A1 A L[Neper] = ln or L[dB] = 0log 1 A A A 1 is the amplitude without loss A is the amplitude with loss Proportional loss of signal amplitude with increasing propagation distance: L = α d d is the propagation distance α is the attenuation coefficient Major classes of attenuation: absorption (viscosity, relaxation, heat conduction, elastic hysteresis, etc) scattering (inhomogeneities), also causes incoherent material noise α=α +α absorption scattering For example: α [db/m] 0. f [MHz] water α Plexiglas [db/m] 100 f [MHz] f denotes frequency

3 Scattering Induced Attenuation Single-scatterer: Ps = γ I Ps scattered power by the inhomogeneity γ is the scattering cross section of the average scatterer I intensity of the incident wave Single-scattering approximation: Let us consider a volume of given cross section A and length d. The coherent acoustic power P = AI transmitted through this region decreases by an amount of N dp = Ps = Psi = N γ I = n γ Ad I = n γd P i = 1 N number of scatterers in volume Ad n the number density of the scatterers dp = P n γ d P = Pe n d i γ i ½ n d u = u e γ α = ½ n γ lim γ = 0 ω 0 lim γ A s ω

4 General Considerations on Scattering Similarity: f 1, λ1 f, λ D 1 D d 1 d Scaling: Scattering Loss: d D λ f1 ξ= = = = d1 D1 λ1 f L = L 1 α 1 L1/ d = 1 =ξ α L/ d α1λ1 αλ = 1 Normalized Attenuation: D α n = λα( D, λ ) = αn( ) λ

5 Power relationship: αn ( D, f ) Di f i α ( D, f) = adi f i + a is a constant determined mainly by the "degree" (relative deviation from the host medium) of the inhomogeneity and the nature of the interaction (e. g., shear or longitudinal wave, etc.) 1 Polycrystalline material: D is the grain size a is a function of anisotropy Low-frequency (Rayleigh) region: α 3 4 Rayleigh ( D, f) = ar D f Intermediate (stochastic) region: α ( D, f) = stochastic a D f s High-frequency (geometrical) region: α ( D, f) = geometrical a D g 1 Surface wave attenuation on a slightly rough surface: α 4 5 roughness( h, f) = ar h f h is the rms roughness a r is a function of the rms roughness-to-autocorrelation length ratio

6 Scattering Induced Attenuation in Polycrystalline Materials Low-Frequency (Rayleigh) Region ( λ >> D ) Incident Wave u i Scatterer Scattered Wave u s r us0 e ( ) s(, ) s0 () i kr ω u r θ = u F θ t r / r is the amplitude of the wave at a distance r from the scatterer F( θ ) is the directivity function ( θ denotes the polar angle) Linear superposition: us0 Δ Vui u i is the incident wave amplitude Δ is the relative change of the elastic properties V is the scatterer volume

7 For example, for cubic crystals: Δ = C44 C11 C1 1 The total scattered power from a single scatterer: Δ V u P i s us ds ds Δ V I i S S r Ii u i denotes the intensity of the incident wave α = ½ n γ γ Δ V is the scattering cross section n = V 1 and V D3 α Rayleigh Δ V α Rayleigh = a R D f Δ D f a R is a constant that is proportional to Δ Intermediate (Stochastic) Region ( λ D ) θ refraction Incident Ray Refracted Ray θ divergence

8 In the case of weak anisotropy: θrefraction Δ θ λ/ D divergence Geometrical region: θdivergence θ refraction i. e., above a frequency where λ/ D Δ Stochastic region (forward scattering): weak random phase perturbation Φ ( x, y) multiplies the coherent (average) wave < eiφ ( x, y) > ei<φ ( x, y) > e ½ <Φ ( x, y) > Loss of the coherent wave: L ½ ϕ ϕ =<Φ ( x, y) > ϕ Δ Df L Δ D f α stochastic = as D f Δ D f a s is a constant that is proportional to Δ

9 High-Frequency (Geometrical) Region ( λ << D ) d Incident Plane Wave Transmitted Plane Wave αgeometrical D 1 Summary: Regime Functional Dependence Rayleigh ( D <<λ) α Δ D3 f 4 stochastic ( DΔ λ D) α Δ D f geometrical ( λ DΔ) α D 1

10 Grain Scattering Induced Attenuation In Polycrystalline Iron (100 μm grain diameter) log{attenuation Coefficient [db/cm]} Rayleigh region stochastic region geometrical region longitudinal shear log{frequency [MHz]}

11 Experimental Grain Scattering Induced Attenuation longitudinal wave in SAE 100 steel Attenuation Coefficient [db/cm] µm 48 µm 38 µm 31 µm 18 µm 10 µm Frequency [MHz] Complicating effects: preferred orientation between neighboring grains (e. g., prior austenite grain structure, columnar grain structure, severe plastic flow, etc.) shape of the grains can also be very different from the ideal uniaxial shape (e. g., needle-like alpha (hexagonal) grains in titanium)

12 Typical welded zone: Ultrasonic Grain Size Assessment Over-heated welded zone: transmission C-scan of an app. 4"-wide electric resistance welded butt joint between two 0.5"-thick steel plates at 15 MHz high-pressure side electrode interface grain coarsening plastic flow low-pressure side

13 R 0 R 1 R R 3 Experimental Aspects of Ultrasonic Attenuation Measurements main bang A 0 A 1 A time A L = 0log 0 = α d + Limp + Ldiff + Lsurf A1 Impedance Mismatch: I Medium 1 d Medium (sample) Medium 1 (3) T 1 T T 3 Reflection coefficient: R1 Z Z = 1 Z + Z1 R = R1 = R1 T = T 1 T1 = 1 R Z1 and Z are the acoustic impedances of the first and second media, respectively

14 Front surface reflection: R0 = R Multiple-reflection: R1 = RT, R 3 = R T,... R n 1 n = R T Transmission: T1 = T, T = R T,... T ( n 1) n = R T where n = 1,,... R L 0 imp = 0log = 40logT = 0log(1 R ) R1 Diffraction Correction the acoustic field of a circular piston radiator at a/λ = 10 far-field near-field z = 0a z = 10a = N Near-field / far-field transition: z = a a N = λ a denotes the radius of the transducer λ is the acoustic wavelength in the medium

15 Simplified Sound Field of a Circular Piston Radiator 1 "searchlight" model -10dB contour simplified model a ξ 0-1 θ -10 db z N Two identical transducers in a pitch-catch mode: Lommel diffraction correction: DL ( z) = pr ( z) pr ( z = 0) z is the distance between the transducers (in a pulse-echo operation with a normally aligned mirror, the distance between the transducer and the mirror is only z/) / / / D L() s = 1 e i π s[ J0( π s) + ij1( π s)] In the far-field: s = z/n lim DL ( s) i s s = π/ lim D ( z) z L = πa zλ

16 Lommel Diffraction Correction for a Circular Piston Radiator Diffraction Correction far-field asymptote z / N circular piston radiator of 0.5"-diameter and z = 0 cm separation in water Diffraction Correction [db] Frequency [MHz]

17 Refraction Correction Snell s Law: sin θ c = sin θ1 c1 fluid solid For slightly divergent beams: Pulse-echo configuration: sin θ tan θ θ c z = z 1 + z c 1 L diff 0log D L z1 DL N 1 z + dc / c N1 1 1 N 1 a a f = = λ c 1 1

18 Surface Roughness Transducer Rough Surface Flaw Incident Wave Coherent Reflection θ I θ R Liquid Incoherent Reflection x Solid Incoherent Transmission θ L Coherent Longitudinal Transmission z θ T Coherent Shear Transmission

19 Phase-Screen Approximation s( xy, ) is the surface height distribution h is the rms height Λ is the correlation length h = < s ( x, y) > C(, ξ η ) = < s(, x y)( s x ξ, y η ) > = h c(, ξ η ) transverse isotropy: ρ =ξ +η Gaussian distribution: c( ρ ) = e ρ / Λ Logarithmic distribution: small curvature: c( ρ ) = e ρ / Λ h << Λ Phase perturbation (without the common e-iωt term) 0 Φω (,, ) u( ω, xyz,, = 0 ) = u ( ω, xyz,, ) ei x y u denotes the displacement field just inside the rough solid u 0 denotes the displacement field just inside the smooth solid Φ = sxy (, )[ k cos( θ ) k cos( θ )] L,T w I L,T L,T Φ = sxy (, ) k cos( θ ) R w I kl, kt and kw are wavenumbers

20 Coherent Transmission Coefficients Reflected compressional wave: L ½ <Φ > Longitudinal transmitted wave: Shear transmitted wave: L R ( ωθ, ) = 0log R 0 R L L L T ( ωθ, ) = 0log T T ( ωθ, ) = 0log T T L0 L T0 T Phase-screen approximation: L R,L,T = [db] h ω CR,L,T C = [cos( θ )/ c ] R I w 1 - L = [cos( θ L) / L cos( θi) / w] C c c 1 - T = [cos( θ T) / T cos( θi) / w] C c c cw, cl and ct are sound velocities

21 Surface Roughness Induced Attenuation of the Reflected Ultrasonic Wave at Normal Incidence Attenuation [db] µm 5.6 µm 15. µm 1.8 µm 11.4 µm 9.9 µm 8.7 µm 5.6 µm Frequency [MHz] (solid lines are best fitting f curves)

22 Surface Roughness Induced Attenuation of the Double-Transmitted Longitudinal and Shear Waves Attenuation [db] º reflection 1º long. tr. 10º long. tr. 0º long. tr. 6º shear tr. 4º shear tr. º shear tr Frequency [MHz] (solid lines are best fitting f curves).