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
y displacement
w
w+δw zo z’ z
Direction of rarefaction compression L propagation Basics of Acoustic Waves
• Shear Wave (No Volume Change):
y y
v+δv v z z Direction of L propagation 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 ∂w S ≡ ∂z • Shear strain: ∂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 internal stress applied stress internal stress T -T -T T -(T+δT) T+δT -(T+δT) T+δT z z
compressional stress shear stress Hooke’s Law
• T=cS, where c is the elastic constant. • Tensor representation:
Tensor notation Reduced notation
xx 1 yy 2 zz 3 yz=zy 4 zx=xz 5 xy=yx 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 ⎥⎢S3 ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢T4 ⎥ ⎢c41 c42 c43 c44 c45 c46 ⎥⎢S4 ⎥ ⎢T ⎥ ⎢c c c c c c ⎥⎢S ⎥ ⎢ 5 ⎥ ⎢ 51 52 53 54 55 56 ⎥⎢ 5 ⎥ ⎣⎢T6 ⎦⎥ ⎣⎢c61 c62 c63 c64 c65 c66 ⎦⎥⎣⎢S6 ⎦⎥
• Stress tensor symmetry: no rotation. • Strain tensor symmetry: by definition. Hooke’s Law (Isotropy)
⎡T1 ⎤ ⎡c11 c12 c12 0 0 0 ⎤⎡S1 ⎤ ⎢T ⎥ ⎢c c c 0 0 0 ⎥⎢S ⎥ ⎢ 2 ⎥ ⎢ 12 11 12 ⎥⎢ 2 ⎥ ⎢T3 ⎥ ⎢c12 c12 c11 0 0 0 ⎥⎢S3 ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢T4 ⎥ ⎢0 0 0 c44 0 0 ⎥⎢S4 ⎥ ⎢T ⎥ ⎢0 0 0 0 c 0 ⎥⎢S ⎥ ⎢ 5 ⎥ ⎢ 44 ⎥⎢ 5 ⎥ ⎣⎢T6 ⎦⎥ ⎣⎢0 0 0 0 0 c44 ⎦⎥⎣⎢S6 ⎦⎥
c11 = c12 + 2c44 = λ + 2µ • Lamé constants: λ and µ (shear modulus). Common Elastic Constants
• Young’s modulus (elastic modulus, E):
Tzz = (λ + 2µ )S zz + λ (S xx + S yy )
= λ (S xx + S yy + S zz )+ 2µS zz
≡ λΔ + 2µS zz ( △: dilation ) T µ(3λ + 2µ) E ≡ zz = Szz 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 +T +T ) p ≡ − xx yy zz = −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) rm km L damping spring
C mass
electrical mechanical Acoustic Wave Equations
d 2q dq q d 2 w dw L + R + = v(t) m + r + k w = f (t) dt 2 dt C dt 2 m dt m
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) w(zo+δz, t) p(zo, t) p(zo+δz, t) area A
zo zo+δz 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 ∂t 2 ∂z 2 Acoustic Wave Equations
− jωz / c jωz / c w ( z ,ω ) =w1 (ω )e +w 2 (ω )e
w ( z ,t ) =w1 (t − z / c )+w 2 (t + z / c )
u ( z ,t ) ≡ ∂w ( z ,t )/ ∂t u ( z ,ω ) = jωw ( z ,ω ) − jωz / c jωz / c u ( z ,ω ) = u1 (ω )e + u2 (ω )e B ∂u(z,ω) p(z,ω) = − jω ∂z − jωz / c jωz / c = Z0 (u1(ω)e − u2 (ω)e )
Characteristic impedance: Z0 = ρc Two Common Units
• Pa (Pascal, pressure) :
1Pa = 1N / m 2 = 1Kg /( m ⋅ sec2)
• Rayl (acoustic impedance) :
1Rayl = 1Pa /( m / sec) = 1Kg /( m 2 ⋅ sec) Reflection and Refraction
-∞ +∞ Z1 Z2
0 L z
p ( z ,ω ) u (ω )e − jωz / c − u (ω )e jωz / c Z ( z , ) Z 1 2 ω ≡ = 0 − jωz / c jωz / c u ( z ,ω ) u1 (ω )e + u2 (ω )e Reflection and Transmission
• Pressure and particle velocity are continuous • Medium 2 is infinite to the right
Z 1 ( L,ω ) = Z 2
u (ω )e − jωL / c − u (ω )e jωL / c Z 1 2 Z 1 − jωL / c jωL / c = 2 u1 (ω )e + u2 (ω )e Reflection and Transmission
− jω2L / c Z 2 − Z 1 − u2 (ω ) = u1 (ω )e Z 2 + Z 1
− jωz / c Z 2 − Z 1 jω ( z −2L )/ c p ( z ,ω ) = Z 1u1 (ω )(e + e ) Z 2 + Z 1
At the boundary
− jωL / c 2Z 2 p ( L ,ω ) = Z 1u1 (ω )e Z 2 + Z 1 Reflection and Transmission
• 1D:
Z 2 − Z 1 Rc = ( reflection ) Z 2 + Z 1
2Z 2 Tc = (transmission ) Z 2 + Z 1 Reflection
Hard Boundary Soft Boundary Reflection
Low Density to High Density High Density to Low Density Reflection, Transmission and Refraction • 2D:
Z1 Z2 Z 2 cosθ i − Z 1 cosθt Rc = ( reflection ) ur Z 2 cosθ i + Z 1 cosθt
2Z 2 cosθ i Tc = (transmission ) θr=θi ut Z 2 cosθ i + Z 1 cosθt θt θi sinθ c ui • Snell’s law: i = 1 sinθt c 2 −1 • Critical angle, if c 2 ≥ c1 : θic ≡ sin (c1 /c2 ) Refraction Acoustic Impedance Matching
• Single layer
C1,λ1
Zo Z1 Z2
0 L • In medium 1: u (ω )e − jωz / c − u (ω )e jωz / c Z ( z , ) Z 1 2 ω = 1 − jωz / c jωz / c u1 (ω )e + u2 (ω )e Acoustic Impedance Matching
• Avoid reflection at the boundaries:
Z cosθ + jZ sinθ Z = Z 2 1 Z (0,ω ) = Z 0 0 1 Z 1 cosθ + jZ 2 sinθ Z ( L,ω ) = Z 2 θ = ωL / c1 = 2πL / λ1
For real Zo: λ L = (2n +1) 1 for n = 0,1,2… Z = Z Z 4 1 0 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
All Soft Bone Tissues 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 Bulk
Shear Glandular Tissue Dermis Epidermis Bone of Breast Connective Tissue Cartilage Liver Contracted Muscle Relaxed Muscle Palpable Nodules Fat 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 object with different elastic different elastic properties properties
• Static or dynamic deformation due to externally applied forces. • Measurement of internal motion. • Estimation of tissue elasticity. Static Approaches
static force object with different elastic properties • Equation of equilibrium:
3 ∂σ ij f 0, i 1,2,3 ∑ + i = = j=1 ∂x j
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 ∂x2 ∂t 2 µ = ρc2 • Doppler spectrum of a vibrating target:
s(t) = Acos(ω xt)
ω x = ω0 + Δωm cos(ω Lt)