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Chapter 2: Acoustic Wave Propagation Outline • Tools: – Newton’s second law – Hooke’s law • Basics: – Strain, stress – Longitudinal, shear – Elastic moduli • Acoustic wave propagation – Wave equations – Reflection, refraction – Impedance matching • Applicatoin: Elasticity imaging • Acoustics à Mechanical waves. – Newton’s second law – Strain-displacement equations – Constitutive equations: Hooke’s law • Elastic medium: – Elastostatics: Stress-strain relationships – Elastodynamics: Wave equations Basics of Acoustic Waves • A medium is required for a sound wave. • Physical quantities to describe a sound wave: displacement, strain and pressure. • Longitudinal (compressional) vs. shear (transverse). Basics of Acoustic Waves • Longitudinal Wave: y displacement w w+δw zo z’ z L Direction of propagation rarefaction compression Basics of Acoustic Waves • Shear Wave (No Volume Change): y y v+δv v z L Direction of propagation z Basics of Acoustic Waves • Longitudinal Wave: Basics of Acoustic Waves • Shear Wave: Basics of Acoustic Waves • Water Wave: Basics of Acoustic Waves • Rayleigh Wave: Displacement and Strain • Displacement: movement of a particular point. • Strain: – Displacement variations as a function of position. – Fractional change in length. – Deformation. – Can be extended to volume change. Displacement and Strain • Compressional strain: ∂w δw = L ≡ SL ∂z • Shear strain: ∂w S≡ ∂z ∂v S≡ ∂z Stress (Pressure) • Force per unit area applied to the object. • Net force applied to a unit volume: !T / ! z L applied stress -T internal stress T -(T+δT) compressional stress applied stress -T T+δT internal stress T -(T+δT) z T+δT z shear stress Hooke’s Law • T=cS, where c is the elastic constant. • Tensor representation: Tensor notation Reduced notation xx 1 2 yy zz yz=zy zx=xz xy=yx 3 4 5 6 Hooke’s Law (General Form) ⎡T1 ⎤ ⎡c11 c12 c13 c14 c15 c16 ⎤ ⎡ S1 ⎤ ⎢T ⎥ ⎢ c c c c c c ⎥ ⎢ S ⎥ ⎢ 2 ⎥ ⎢ 21 22 23 24 25 26 ⎥ ⎢ 2 ⎥ ⎢T3 ⎥ ⎢ c31 c32 c33 c34 c35 c36 ⎥ ⎢ S 3 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎢T4 ⎥ ⎢ c 41 c 42 c 43 c 44 c 45 c 46 ⎥ ⎢ S 4 ⎥ ⎢T5 ⎥ ⎢ c c c c c c ⎥ ⎢ S 5 ⎥ ⎢ ⎥ ⎢ 51 52 53 54 55 56 ⎥ ⎢ ⎥ ⎢⎣T6 ⎥⎦ ⎢⎣ c61 c62 c63 c64 c65 c66 ⎥⎦ ⎢⎣ S 6 ⎥⎦ • Stress tensor symmetry: no rotation. • Strain tensor symmetry: by definition. Hooke’s Law (Isotropy) ⎡T1 ⎤ ⎡c11 c12 c12 ⎢T ⎥ ⎢c c c ⎢ 2 ⎥ ⎢ 12 11 12 ⎢T3 ⎥ ⎢c12 c12 c11 ⎢ ⎥ = ⎢ ⎢T4 ⎥ ⎢0 0 0 ⎢T5 ⎥ ⎢0 0 0 ⎢ ⎥ ⎢ ⎢⎣T6 ⎥⎦ ⎢⎣0 0 0 0 0 0 ⎤ ⎡ S1 ⎤ 0 0 0 ⎥⎥ ⎢⎢ S 2 ⎥⎥ 0 0 0 ⎥ ⎢ S 3 ⎥ ⎥ ⎢ ⎥ c 44 0 0 ⎥ ⎢ S 4 ⎥ 0 c 44 0 ⎥ ⎢ S 5 ⎥ ⎥ ⎢ ⎥ 0 0 c 44 ⎥⎦ ⎢⎣ S 6 ⎥⎦ c11 = c12 + 2c44 = λ + 2µ • Lamé constants: λ and µ (shear modulus). Common Elastic Constants • Young’s modulus (elastic modulus, E): T zz = ( λ + 2µ ) S zz + λ ( S xx + S yy ) = λ ( S xx + S yy + S zz ) + 2µS zz ≡ λΔ + 2µS zz ( △: dilation ) Tzz µ (3λ + 2µ ) E≡ = S zz λ+µ E ≈ 3µ for liquid and soft tissues Common Elastic Constants • Bulk modulus (reciprocal of compressibility, B): p p B≡− =− δV V Δ (T xx + T yy + T zz ) p≡− = −B ⋅ Δ 3 3λ + 2µ B= 3 Common Elastic Constants • Poisson ratio (negative of the ratio of the transverse compression to the longitudinal compression, σ): S yy λ σ ≡− = S zz 2 ( λ + µ ) • σ approaches to 0.5 for liquid and soft tissues. Acoustic Wave Equation Acoustic Wave Equation • • • • Newton’s second law Displacement à Particle velocity Particle velocity à Pressure Impedance = Pressure/Particle Velocity Acoustic Wave Equations • Electrical and mechanical analogy: R v(t) L C electrical rm damping km spring mass mechanical Acoustic Wave Equations d 2q dq q L 2 + R + = v(t ) dt C dt d 2w dw m 2 + rm + k m w = f (t ) dt dt Electrical Mechanical q charge w displacement i=dq/dt current U=dw/dt particle velocity V voltage f force (stress, pressure) L inductance m mass 1/C 1/capacitance km stiffness R resistance rm damping Acoustic Wave Equations w(zo, t) p(zo, t) w(zo+δz, t) p(zo+δz, t) zo zo+δz area A z • Newton’s second law: ∂ 2 w( z, t ) A( p( z, t ) − p( z + δz, t )) = ( ρ ⋅ δz ⋅ A) ∂t 2 ∂ 2w ( z , t ) ∂ 2w ( z , t ) = (B / ρ) c = B/ρ 2 2 ∂t ∂z Acoustic Wave Equations w ( z , ω ) = w 1 ( ω ) e − jωz / c + w 2 ( ω ) e jωz / c w ( z , t ) = w 1 ( t − z / c )+ w 2 ( t + z / c ) u ( z , t ) ≡ ∂w ( z , t )/ ∂t u ( z , ω ) = jωw ( z , ω ) u ( z , ω ) = u1 ( ω ) e − jωz / c + u 2 ( ω ) e jωz / c B ∂u ( z , ω ) p( z, ω ) = − jω ∂z = Z 0 (u1 (ω )e − jωz / c − u2 (ω )e jωz / c ) Characteristic impedance : Z 0 = ρc Two Common Units • Pa (Pascal, pressure) : 1Pa = 1N / m 2 = 1Kg /( m ⋅ sec 2 ) • Rayl (acoustic impedance) : 1Rayl = 1Pa /( m / sec) = 1Kg /( m 2 ⋅ sec) Reflection and Refraction -∞ 0 Z1 Z2 L +∞ z − jωz / c jωz / c u ( ω ) e − u ( ω ) e p ( z ,ω ) 2 Z ( z ,ω ) ≡ = Z0 1 u ( z ,ω ) u1 ( ω ) e − jωz / c + u 2 ( ω ) e jωz / c Reflection and Transmission • Pressure and particle velocity are continuous • Medium 2 is infinite to the right Z 1 ( L ,ω ) = Z 2 u1 ( ω ) e − jωL / c − u 2 ( ω ) e jωL / c Z1 = Z2 − jωL / c jωL / c u1 ( ω ) e + u2 (ω ) e Reflection and Transmission − u 2 ( ω ) = u1 ( ω ) e − jω2L / c p ( z , ω ) = Z 1u1 ( ω )( e Z2 − Z1 Z2 + Z1 − jωz / c Z 2 − Z 1 jω ( z − 2L )/ c + e ) Z2 + Z1 At the boundary p ( L , ω ) = Z 1u 1 ( ω ) e − jωL / c 2Z 2 Z2 + Z1 Reflection and Transmission • 1D: Z2 − Z1 Rc = ( reflection ) Z2 + Z1 2Z 2 Tc = ( transmission ) Z2 + Z1 Reflection Hard Boundary Soft Boundary Reflection Low Density to High Density High Density to Low Density Reflection, Transmission and Refraction • 2D: Z1 Z2 ur ut θr=θi Rc = Z 2 cos θ i − Z 1 cos θ t ( reflection ) Z 2 cosθ i + Z 1 cosθ t Tc = 2Z 2 cosθ i ( transmission ) Z 2 cos θ i + Z 1 cosθ t θt θi ui • Critical angle, if c2 ≥ c1 : • Snell’s law: sin θ i c 1 = sin θt c 2 θ ic ≡ sin −1( c1 / c 2 ) Refraction Acoustic Impedance Matching • Single layer C1,λ1 Zo Z1 0 Z2 L • In medium 1: u1 ( ω ) e − jωz / c − u 2 ( ω ) e jωz / c Z ( z ,ω ) = Z 1 u1 ( ω ) e − jωz / c + u 2 ( ω ) e jωz / c Acoustic Impedance Matching • Avoid reflection at the boundaries: Z ( 0 ,ω ) = Z 0 Z ( L ,ω ) = Z 2 Z 2 cosθ + jZ 1 sin θ Z0 = Z1 Z 1 cosθ + jZ 2 sin θ θ = ωL / c 1 = 2πL / λ1 For real Zo: L = (2n + 1) λ1 4 for n = 0,1,2… Z1 = Z 0 Z 2 • Double layer is common for medical transducers. Ultrasonic Elasticity Imaging Elasticity Imaging • Image contrast is based on tissue elasticity (typically Young’s modulus or shear modulus). Liquids 10 3 10 4 Glandular Tissue of Breast Liver Relaxed Muscle Fat 10 5 10 6 10 7 All Soft Tissues 10 Dermis Epidermis Connective Tissue Cartilage Contracted Muscle Palpable Nodules 8 10 9 Bone 10 10 Bone Bulk Shear Clinical Values • Remote palpation. • Quantitative measurement of tissue elastic properties. • Differentiation of pathological processes. • Sensitive monitoring of pathological states. Clinical Examples • Tumor detection by palpation: breast, liver and prostate (limited). • Characterization of elastic vessels. • Monitoring fetal lung development. • Measurements of intraocular pressure. Basic Principles • Hooke’s law: – stress=elastic modulus*strain (σ=Cε). • General approach: – Stress estimation. – Displacement measurement. – Strain Reconstruction. General Methods static force vibration object with different elastic properties object with different elastic properties • Static or dynamic deformation due to externally applied forces. • Measurement of internal motion. • Estimation of tissue elasticity. Static Approaches static force • Equation of equilibrium: 3 ∑ j =1 ∂σ ij ∂x j object with different elastic properties + f i = 0, i = 1,2,3 where σij is the second order stress tensor, f i is the body force per unit volume in the xi direction. Dynamic Approaches • Wave equation with harmonic excitation: ∂ 2U ∂ 2U µ = ρ 2 ∂x ∂t 2 µ = ρc 2 • Doppler spectrum of a vibrating target: s (t ) = A cos(ω x t ) ω x = ω 0 + Δω m cos(ω Lt )