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Monochromatic solution in Cartesian coordinates, plane wave:
→ →
−
→
s(−
x , t) = A exp{j(ωt − k · −
x )}
• c is the speed of propagation.
• s is a general scalar field (electromagnetics: electric or magnetic field,
acoustics: sound pressure ...).
2
2
2
2
∂
s
s
s
∂
s
∂
∂
1
→
−
∇s = 2+ 2+ 2 = 2 2
∂x
∂y
∂z
c ∂t
This is the equation in array signal processing:
Wave equation
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Wave equation and arbitrary
solutions
• The wave equation is linear
• Solution may be a sum of complex
exponentials
• Almost any signal may be expressed as a
sum of complex exponentials using
Fourier theory
• Therefore any signal, no matter its shape,
may be a solution to the wave equation and the shape will be preserved as it
propagates
• Propagating waves are therefore ideal
carriers of information
• Modified by the boundary conditions - to
determine which components that are
excited
• Propagation is determined by the
deviations of the medium from ideal
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• Frequency and wavelength: c = λ · ω/2π = λ/f
• Dispersion relation: ω = c · k
→
−
→
−
→
−
• Wavenumber vector: k = ω α , | k | = 2π/λ
− −
→
• Propagating sinusoidal plane wave: sin(ωt − k · →
x)
→
−
→
→
• Slowness vector: −
α = k /ω, |−
α | = 1/c
→
→
• Propagating plane wave: s(t − −
α ·−
x)
Plane waves
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Wave equation in spherical
coordinates
wavefront
φ: azimuth
θ: elevation
z
k
sφ,θ
φ
y
θ
xn
x
Transducer array
Assumption:
Solution exhibits spherical symmetry
1 ∂ 2 ∂s
1 ∂ 2s
(r
)= 2 2
2
r ∂r ∂r
c ∂t
Monochromatic solution, spherical wave:
A
s(r, t) = exp{j(ωt − kr)}
r
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• Non-linearity: c = c(s(t))
Generation of harmonics, shock waves
→
• Refraction: c = c(−
x)
Snell’s law: sin θi/ci = sin θt/ct
J&D 2, Sep 2000 SH/5
• Attenuation: c = c< + jc=
Wavenumber is no longer real, imaginary part gives attenuation.
Waveform changes with distance
• Dispersion: c = c(ω)
Group and phase velocity, dispersion equation: ω = f (k) 6= c · k.
Some frequencies may be evanescent ( = non-propagating)
• Doppler effect:
→
Source, sensor or both are moving: ω 0 ≈ ω(1 + −
α ·−
v→
rel )
Non-ideal media
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ω 2τ
≈−
2c
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Ref: Kinsler & Frey, Fundamentals of Acoustics
Typical values: Air: τ = 1.7 · 10−10 sec, water: τ = 6 · 10−13 sec.
k=0
For ωτ 1, the wavenumber can be simplified to k<0 ≈ k and
J&D 2, Sep 2000 SH/6
Complex velocity of sound: c0 = c(1 + jωτ )1/2, where τ is a relaxation time
required for (1) viscous stresses, (2) heat conduction, or (3) molecular
changes to occur.
Losses in fluid acoustic media
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κ−1
ω κτ
τ /2 = k − j
2c
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Typical: Tissue: κ ≈ 1, water, air: κ ≈ 2
k ≈ k − jkω
0
Since (ω κ−1τ )2 1 this can be approximated to
0
ω
ω (1 − jω κ−1τ )1/2
=
k =
c 1 + (ω κ−1τ )2
c(1 + jω κ−1τ )1/2
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Complex velocity of sound: c0 = c(1 + jω κ−1τ )1/2, wavenumber is
Lossy fluid acoustic media (Generalization)
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=e
κ
− ω2cτ x
= 10
κ
− αω
20 x
DEPARTMENT OF INFORMATICS
Typical value for attenuation in liver for medical ultrasound is
κ = 1, . . . , 1.3 and α = 0.35, . . . , 0.9 dB/MHz/cm at 1 MHz.
e
k= ·x
where k= has unit Nepers/m. To convert to dB/m, solve:
s(x, t) = A exp{j(ωt − k< · x)} exp{k= · x}
Propagation (scalar case):
Acoustic losses
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• Salt water: B/A = 5.2, blood and tissue: B/A = 6, . . . , 10.
— Eventually a shock wave is formed.
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— More and more energy is transferred to higher harmonics as the
wave propagates.
— Waveform is distorted.
• Positive peaks propagate faster than negative peaks:
B s(t) ( 2A +1)
c(t) = c(1 +
) B
2A c
• A and B are the 1. and 2. order Taylor series coefficients for the
pressure. B/A is a measure of the non-linearity.
The velocity of sound, c, varies with the amplitude, s:
Non-linearity in acoustics
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Diffraction: deviation from
geometrical model
Rayleigh-Sommerfeld diffraction formula
from a hole with aperture A:
Z
1
exp{jkr}
→
−
→
−
s( x ) =
s( x h)
cos θdA
jλ A
r
Fresnel (paraxial) approximation:
• cos θ ≈ 1, r ≈ d, spherical surfaces ≈
quadratic surfaces.
• Nearfield approximation valid within
approx. 150 of z-axis.
• 2D convolution between field in hole and a
quadratic phase function.
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Diffraction: Nearfield-farfield
Fraunhofer approximation
• Neglect quadratic phase variation over
hole.
• Far-field approximation valid for d D2/λ
where D is max linear dimension of hole.
• s(x, y) = 2D Fourier transform of field in
hole.
• Link between the physics and the signal
processing!
Nearfield-farfield limit
Not a clear transition, several limits are used:
• Fresnel limit: dF = D2/4λ
• Rayleigh distance: dR = 2D2/λ,
max difference in path length is λ/16 or
dR = D2/λ, with max difference of λ/8 or
dR = k · D2/8 = πD2/4λ
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−∞
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Any spatiotemporal signal can be decomposed into infinitely many
plane waves as long as the Fourier transform converges.
Inverse Fourier transform:
Z ∞Z ∞
1
→
−
→ −
−
→
−
→
→
−
S( k , ω) exp{j(ωt − k · x )}d k dω
s( x , t) =
(2π)4 −∞ −∞
−∞
Four-dimensional Fourier transform:
Z ∞Z ∞
→ → −
−
→
−
→
s(−
x , t) exp{−j(ωt − k · −
x )}d→
x dt
S( k , ω) =
Wavenumber-Frequency Space
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Norsk terminologi
• Bølgeligningen
• Planbølger, sfæriske bølger
• Propagerende bølger, bølgetall
→
•−
α - sinking/sakking
• Dispersjon
• Attenuasjon eller demping
• Refraksjon
• Ikke-linearitet
• Diffraksjon; nærfelt, fjernfelt
• Gruppeantenne ( = array)
Kilde: Bl.a. J. M. Hovem:
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