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Chapter 2:

Acoustic Propagation Outline • Tools: – Newton’s second law – Hooke’s law • Basics: – Strain, stress – Longitudinal, shear – Elastic moduli • propagation – Wave equations – , – Impedance matching • Applicatoin: Elasticity imaging • à Mechanical . – 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 wave. • Physical quantities to describe a sound wave: displacement, strain and . • 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

(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

• Newton’s second law • Displacement à • 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)