Download Chapter 2: Acoustic Wave Propagation

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 )