- Email: [email protected]
Field and Wave Equations
Chapter 4
Field and Wave Equations This Chapter is concerned with the equations used to describe the fields that are measured in an imaging system and their relationship with the material variables with which these fields interact. The field equations determine the physical characteristics and behaviour of a particular type of field. Two types of fields are important in imaging: the electromagnetic field and the acoustic field. The primary purpose of this Chapter is to introduce and discuss the electromagnetic and acoustic field equations which are employed in later chapters (Chapters 7 - 11 for example). From these results, we derive equations (wave equations or equations of propagation) which describe the propagation of different wavefields through various types of materials or media.
4.1
The Langevin Equation
The propagation of a wavefield can be modelled by various different wave equations depending upon the type of field, the supporting material and its physical state. In general, however, if the supporting material is assumed to be a linear medium, the scalar field u(r, t) obeys a partial integro-differential equation of the form jj(l)u(r, t) = -s(r, t) where jj(l)
= b(O) + L
For a vector field u(r, t),
jj(l)u(r, t)
=
-s(r, t)
This is the Langevin equation where jj(O) and L are linear operators: jj(O) is associated with the homogeneous portion of the medium, L is, in general, an integro-differential scattering operator, and s is a source function which describes the emission of the incident field from a given source. The operator L models the interaction of the incident field with the differential or local 85
CHAPTER 4. FIELD AND WAVE EQUATIONS
86
scattering from material inhomogeneities. The operator form
tr»
has the general
2
A
D
(1) _
A
=D
(1) (
8 8 \7,\7 , ... ,1'8t'8t 2 '
)
2.
'' '
•
The source function s is, in general, given by s=p®f where f is the probe field, p is a filter weighting function for the emitted field and ® is taken to denote the convolution over three-dimensional space and time. From this general formalism, one can consider a variety of scalar wave equations governing the propagation of different types of We' vefields .upported by different isotropic media. Two cases arise that are based on: (i) a rigorous derivation of the Langevin from a set of fundamental field equations; (ii) the proposition of a Langevin equation (a phenomenological model) whose characteristics (e.g. the phase velocity) are confirmed experimentally. The wave equation that is used to model a wavefield determines the underlying physical model for a (scatter) imaging system. This includes aspects such as the resolution of the image (e.g. the wavelength or bandwidth of the probe), the accuracy of the spatial mapping of the scatter generating parameters in an image (e.g. the level of distortion as determined by the propagation model) and image fuzziness. A fuzzy image is an image which, although attempting to display a specific scatter generating parameter, fails to achieve this because the scattered wavefield that has been measured and processed is corrupted by some other interaction that has not been included on the original model (i.e. the wave equation). Thus, all scatter-imaging techniques are highly model dependent since the reconstruction algorithm is determined by the wave equation which characterizes the medium, in particular, the model associated with the operator L. An inappropriate choice of wave equation results in image fuzziness. Scatter imaging demands appropriate modelling of the scattering dynamics, even if the computations are approximate. An inexact model will lead to a fuzzy image whereas an approximate computation may lead to poor resolution. Distortion, due to poor propagator models which are compounded in the operator b(O) and its associated (free-space) Green function, is a common artifact in many imaging systems and poor physical modelling manifests some form of distortion in most imaging methods. A general criticism, common to many imaging systems, is that emphasis is often placed on a significant amount of computation for image reconstruction and processing. This can provide good, or at least, enhanced resolution but at the expense of developing accurate models for the propagation of a wavefield through the medium that generates the scattered field from which an image is generated and interpreted. This leads to images which are well resolved but may be badly distorted and fuzzy. In this chapter, we start by considering Maxwell's equations which provide a unified theoretical framework for the interaction of electromagnetic waves with matter. In the case of the macroscopic form of these equations, we introduce material parameters such as the permittivity, the permeability and the conductivity. The elastic and acoustic field equations are then studied. An elastic
4.2. MAXWELL'S EQUATIONS
87
field describes the characteristics of 'mechanical radiation' when a material is solid and incompressible. The acoustic field equations relate the acoustic field variables (such as the pressure and velocity fields) to material parameters including the density, compressibility and viscosity. These are used to model the propagation and scattering of an acoustic wavefield when the material is a compressible solid, liquid or gas. In each case, the material is taken to be isotropic. The field equations presented provide the fundamental basis for modelling electromagnetic and acoustic imaging systems. In each case, it is shown how the field equations can (under certain conditions and approximations) be decoupled to provide a governing inhomogeneous wave equation whose complexity increases according to the number of material parameters that are considered. General methods of solving such equations are then addressed in Part II using the Green function method which is discussed in the following Chapter.
4.2
Maxwell's Equations
We shall now consider Maxwell's equations and study the electromagnetic wavefields and wave equations that arise from these equations. We first consider these equations in their microscopic form (for individual charged particles) and go on to consider the macroscopic form of Maxwell's equations (for the case when there are many particles per cubic wavelength) and briefly study the propagation of monochromatic electromagnetic waves in homogeneous media. The macroscopic form of Maxwell's equations is then used to construct inhomogeneous wave equations in a form that are suitable for applying the solutions methods discussed in Chapter 5. The motions of electrons (and other charged particles) give rise to electric and magnetic fields. These fields are described by the following equations which are a complete mathematical descriptions for the physical laws quoted 1. Coulomb's law
\7. e
= 471"p
(4.1)
Faraday's law of induction 1 ab c at
\7 x e = - - -
(4.2)
No free magnetic monopoles exist (4.3)
\7·b=O Modified (by Maxwell) Ampere's law
\7 x b = ~
ae + 471" j
cat
1 For
CGS units.
c
(4.4)
88
CHAPTER 4. FIELD AND WAVE EQUATIONS
where e is the electric field, b is the magnetic field, j is the current density, p is the charge density and c ':::' 3 X 108 ms " ! is the speed of light. These microscopic Maxwell's equations are used to predict the pointwise electric e and magnetic b fields given the charge and current d -nsities I p and j respectively). By including a modification to Ampere's law, l.e. the inclusion of the 'displacement current' term 8ej8(ct), Maxwell (see Figure 4.1) provided a unification of electricity and magnetism compounded in the equations above.
Figure 4.1: James Clerk Maxwell
4.2.1
Linearity of Maxwell's Equations
Maxwell's equations are linear because if
and then
Pl
+ P2,
jl
+ j2
--t
ei
+ e2,
bi
+ b2
where --t means 'produces'. This is because the operators 'V., 'Vx and the time derivatives are all linear operators.
4.2.2
Solution to Maxwell's Equations
The solution to these equations is based on exploiting the properties of vector calculus and, in particular, identities involving the curl.
4.2. MAXWELL'S EQUATIONS
89
Taking the curl of equation (4.2), we have
ab
1
V'xV'xe=--V'xc at and using the identity (see Chapter 1)
V' x V' x e = V'(V' . e) - V'2e together with equations (4.1) and (4.4), we get
1a 47r .) V'(47rp) - V'2e = - - (1-ae - +-J cat cat c or, after rearranging, (4.5) Taking the curl of equation (4.4), using the identity above, equations (4.2) and (4.3) and rearranging the result gives (4.6) Equations (4.5) and (4.6) are inhomogeneous wave equations for e and b. These equations are related or coupled to the vector field j (which is related to b). If we define a region of free space where p = and j = 0, then both e and b satisfy the equation 1 a2 f V'2f - c2 at 2 = 0.
°
This is the homogeneous wave equation. One possible solution of this equation (in Cartesian coordinates) is
fx = p(z - ct);
fy = 0,
fz =
°
which describes a wave or distribution p moving along z at velocity c. Thus, we have shown that in free space when
V'. e
=
0,
1 ab
V'xe=--cat'
V'. b
=
0,
V'xb=~ae. c at
Maxwell's equations describe the propagation of an electric and magnetic (or electromagnetic field) in terms of a wave traveling at the speed of light (see front cover). After developing the origins of the vector calculus, Maxwell derived the wave equations for an electromagnetic field in a paper entitled A Dynamical Theory of the Electromagnetic Field, first published in 1865 and arguably one of the greatest intellectual achievements in the history of physics.
CHAPTER 4. FIELD AND WAVE EQUATIONS
90
4.3
General Solution to Maxwell's scopic) Equations
(Micro-
The solution to Maxwell's equation in free space is specific to the charge density and current density being zero. We now investigate a method of solution for the general case. The basic method of solving Maxwell's equations (i.e. finding e and b given p and j) involves the following: (i) Expressing e and b in terms of two other fields U and A. (ii) Obtaining two separate equations for U and A. (iii) Solving these equations for U and A from which e and b can then be computed. For any vector field A
\7. \7 x A = O. Hence, if we write
b=\7xA
(4.7)
then equation (4.3) remains unchanged. Equation (4.2) can then be written as 1
&
\7xe=---\7xA c &t
or
l&A) = \7 x ( e+ ~8t
o.
The field A is called the Magnetic Vector Potential. For any scalar field U
\7 x \7U
0
=
and thus equation (4.2) is satisfied if we write
l&A ±\7U=e+-c &t or
l&A
e = -\7U - - c &t
(4.8)
where the minus sign is taken by convention. U is called the Electric Scalar Potential. Substituting equation (4.8) into Maxwell's equation (4.1) gives
\7. ( \7U or 2
1
1 &A) +- = -41lp C
o
&t
\7 U + --\7. A c &t
= -4rrp.
(4.9)
4.4. THE MACROSCOPIC MAXWELL'S EQUATIONS
91
Substituting equations (4.7) and (4.8) into Maxwell's equation (4.4) gives
a(
aA) =-J
1 1 V'xV'xA+-V'U+--
cat
cat
41T •
c
Finally, using the identity
V' x V' x A = V'(V' . A) - V'2 A we can write
a- -A V ' ( V'·A+-1 au) =--J 41T • V'2 A -12 2 2
c
at
c at
c
(4.10)
If we could solve equations (4.9) and (4.10) above for U and A then e and b
could be computed. The problem here, is that equations (4.9) and (4.10) are coupled. They can be decoupled by applying a technique known as a 'gauge transformation' called the Lorentz gauge transformation, after Lorentz who was among the first to consider it as an approach to solving these equations. The idea is based on noting that equations (4.7) and (4.8) are unchanged if we let A~A+V'X
and
lax
U~U--
c at since V'x V'X = O. If this gauge function X is taken to satisfy the homogeneous wave equation
then
1au
V'.A+~8t=O
(4.11)
which is called the Lorentz condition. With equation (4.11), equations (4.9) and (4.10) become (4.12) and
2
V'2 A _ ~ a A = _ 41Tj c2
at 2
c
respectively. These equations are non-coupled inhomogeneous wave equations whose solution (using the Green function) will be considered in the following Chapter.
4.4
The Macroscopic Maxwell's Equations
The microscopic form of Maxwell's equations tells us how individual charged particles and electromagnetic fields interact. When there are many particles
CHAPTER 4. FIELD AND WAVE EQUATIONS
92
per cubic wavelength, the electromagnetic radiation 'sees' only a macroscopic average. The medium is then described by its dielectric parameters: the permittivity E, the magnetic permeability f.l and the conductivity a, Simple averaging of the quantities over a small volume V, e.g.
E(r,t)
J
=~
~
e(r',t)d3r', B(r,t) =
v
J
b(r',t)dV
v
leads to the following, but not very useful, macroscopic form of Maxwell's equations: 1 sn \7. E = 41Tpmacro, \7 x E = --~, c ut
\7. B
= 0, \7 x B =
However, both Pmacro and jmacro bound electrons, i.e. we can write
1 aE
-~
41T.
+ -Jmacro'
c ut c can be split into two terms due to free and
Pmacro
= Pbound + Pfree
jmacro
= jbound + jfree'
and By bound, we mean that the electrons are bound to the nucleus to constitute an atom. If we introduce an electric polarization P of the medium to represent the average dipole moment per unit volume given by
P = -Nes where s is the average vector between bound electrons and nuclei, e is the charge of an electron and N is the average number of electrons per unit volume, then we can define the charge density of bound electrons as pbound
= -\7. P
and the current density of bound electrons in the form
.
Jbound =
ap + c\7 x M
7ft
°
where M is the magnetization vector. At optical frequencies, M = (in the absence of a strong applied magnetic field). Further, we now define the following: (i) the displacement vector given by D
= E + 41TP;
(ii) the magnetic field strength given by H = B - 41TM. From these definitions, we obtain a useful macroscopic form of Maxwell's equations given by 1 aB \7 . D = 41TPfree, \7 x E = --~ c ut
\7 . B = 0, \7 x H
1
sn
41T.
=-~+-
Jfree
c dt c These equations are valid for media which are: (i) non-isotropic; (ii) inhomogeneous.
4.5. EM WAVES IN A HOMOGENEOUS MEDIUM
4.5
93
EM Waves in a Homogeneous Medium
Having derived Maxwell's equation in macroscopic form, let us now consider the type of solutions they provide for a specific case. Suppose we illuminate a homogeneous material with monochromatic radiation of angular frequency w. What are the possible solutions of Maxwell's equations in the material? i.e. what waves exist in the medium?
4.5.1
Linear Medium
Assume that all the macroscopic vectors oscillate sinusoidally at angular frequency w (this is true, in general, only for high frequency, weak fields). Define vector amplitudes E(r, t) = E(r, w) exp( -iwt), B(r, t) = B(r, w) exp( -iwt) and so on 2 , so that Maxwell's equations can be written in the form \7. D
= 4Jrp
(4.13)
\7 x E
= iWB
(4.14)
c
\7·B=O
(4.15)
iw 4Jr. (4.16) c c where p and j are taken to be the free charge density and the free current density, respectively. Let P = XeE, M = XmH and j = aE \7 x H
= --D + - J
where Xe is the electric susceptibility, Xm is the magnetic susceptibility and a is the conductivity, each of which may be tensors. Note that, in general, this linearity may not occur and P could be of the form
Here al, a2, ... are constant coefficients which would introduce a nonlinear optical material and nonlinear optical effects for example. Note that the effect of introducing monochromatic radiation (Le. a wavefield oscillating at one single frequency w) is to replace the time derivatives in Maxwell's equations with iw which significantly helps in the algebra required to derive the solutions that follow.
4.5.2
Isotropic Medium
Let Xe, Xm and a be complex scalars (not tensors) and let us define the following: (i) the dielectric constant given by
E
= 1 + 4JrXe;
2Strictly speaking E(r,w), B(r,w), etc. should be given a different notation but, in the context of the equations that follow, it is implied that all dependent variables are functions of rand wand not rand t.
CHAPTER 4. FIELD AND WAVE EQUATIONS
94
(ii) the magnetic permeability given by fl = 1 + 41TXm so that we can write D=EE
(4.17)
= flH.
(4.18)
and
B
Taking the divergence of equation (4.16) and noting that 'V. 'V x H have -iw'V . D + 41T'V . j = 0.
= 0,
we
Hence, from equation (4.13) we get p
= -~'V.j = -~'V. (aE). w
(4.19)
w
Substituting equations (4.17), (4.18) and (4.19) into Maxwell's equation (4.13)(4.16), we obtain the following time independent equations for the complex vector amplitudes: 'V .
(E + i4:a) E = 0,
'V . (flH)
'V x E
= i~fl H,
iw (..
+ i41TfT) ~ E.
= 0, 'V x H = - -;;
E
These equations apply to a linear, isotropic but inhomogeneous medium, i.e.
E,
fl and a may be functions of position. Note that, for any vector X and scalar
a, 'V. (aX)
4.5.3
= 'Va· X + a'V . X i- a'V . X,
= 0.
Homogeneous Medium
For a homogeneous medium (where of equations reduces to 'V . E 'V . H
4.5.4
unless 'Va
=
E,
= 0,
fl and a are constants, the previous set
=
'V x E
0, 'V x H
iw (
= - -;;
iWfl
-H, c E
i41Ta) + ---::;-
E.
Plane Wave Solutions
Let
E
= Eo exp(ike . r)
H
= H o exp(ike . r)
where k, is the complex wave number. Noting that 'V. [Cexp(ik e· r)]
= ike' Cexp(ik e· r)
and 'V x [Cexp(ike· r)] = ike
X
Cexp(ike· r)
4.5. EM WAVES IN A HOMOGENEOUS MEDIUM
95
we obtain
k., . Eo
= 0, k c ' H o = 0,
k., x Eo
=
j.LW - Ho,
(4.20) (4.21)
c
( i4JrO") k, x H o = - W ~ E + -:;- Eo.
(4.22)
Equations (4.20) are referred to as the transversality conditions. Substituting equation (4.21) into equation (4.22) yields
W ( E+ i4JrO") -ckc x (k., x Eo) = - - Eo. j.LW c W Using the identity
A x (B x C) = (A. C)B - (A· B)C we can write this result in the form
or, since k., . Eo
= 0,
as (4.23)
where
and 2
nc
i4Jrj.L0"
= Ej.L+--. W
Here, n c is called the complex refractive index. Let ti;
= n + iii
k.,
= k+ ia
and where n is the refractive index, Ii is the extinction index, k is the wavenumber and a is the attenuation vector. Substituting these expressions into equation (4.23) and equating the real and imaginary parts gives
k2
-
a2
= k5(n 2
-
2
1i )
(4.24)
and
k· a = k5nli. Thus, plane wave solutions exist of the form
E
= Eo exp(ikc . r);
H
= H o exp(ikc . r)
where, from equation (4.23),
I k., 1= ncko, k, . Eo = 0 and
1
H o = -k k., x Eo· j.L 0
(4.25)
96
4.5.5
CHAPTER 4. FIELD AND WAVE EQUATIONS
Non-absorbing Media (K;
=
0)
Equations (4.24) and (4.25) reduce to
and k·a=O. Two kinds of waves are possible: (i) Real vector waves where a = 0, k, = k,
1k 1= kon and
E(r,w) = Eo(r,w) exp(ik· r) or E(r, t) = Eo(r, t) exp[i(k. r - wt)]. This is like a free space plane wave. The velocity of propagation is w/ k cf n. and the wavelength, is A/n. Both amplitude and phase are constant and perpendicular to k, i.e. the wave is homogeneous. Since k . Eo = 0, the real and imaginary parts of E are perpendicular to k. H is also perpendicular to k and Re[E] is perpendicular to Re[H]. (ii) Complex wave vector where k is perpendicular to a so that k· a = 0 and E(r, t) = Eo(r, t) exp( -a' r) exp[i(k· r - wt)] which propagates along k with velocity w/k < cfn, The amplitude is constant over planes perpendicular to a and the phase is constant over planes perpendicular to k - the wave is homogeneous.
4.5.6
Absorbing Media (K; > 0, k· a
i= 0)
(i) Homogeneous wave: k and a are in the same direction, k = nk o, a = Kko and E(r, t) = Eo(r, w) exp(-Kko . r) exp[i(nko . r - wt)]. This wave propagates along k o at velocity cfn, with wavelength Ao/n and decreases exponentially along the direction of propagation. Both amplitude and phase are constant and perpendicular to k o and both E and H are perpendicular to ko. Re[E] is not perpendicular to Re[H]. (ii) Inhomogeneous wave: k and a are not in the same direction. There is constant phase perpendicular to k and constant amplitude perpendicular to a. Since a has a component along k, there is a decrease of amplitude along k.
4.6. EM WAVES IN AN INHOMOGENEOUS MEDIUM
4.6
97
EM Waves in an Inhomogeneous Medium
In the previous Section, we considered the EM waves that can occur in a homogeneous material that is linear and isotropic by studying Maxwell's equations for monochromatic propagation. We now turn our attention to developing wave equations for a medium that is linear, isotropic and inhomogeneous using Maxwell's equations in the forrn''
\7. EE = P,
(4.26)
= 0,
(4.27)
\7 'IlH
\7 x E and
\7 x H
=
aH
at '
(4.28)
+j.
(4.29)
-lI-
r:
= E~~
where E(r, t) is the electric field (volts/metre), H(r, t) is the magnetic field (amperes/metre), j(r, t) is the current density (amperes/metre"), p(r, t) is the charge density (charge/metre''}, E(r) is the permittivity (farads/metre) and Il(r) is the permeability (henries/metre). The values of E and 11 in a vacuum (denoted by EO and 110, respectively) are EO = 8.854 X 10- 12 farads/metre and 110 = 47l' X 10- 7 henries/metre. In electromagnetic imaging problems there are two important physical models to consider, based on whether a material is either conductive or non-conductive.
4.6.1
Conductive Materials
In this case, the medium is assumed to be a good conductor. A current is induced which depends on the magnitude of the electric field and the conductivity a (siemens/metre) of the material from which the object is composed. The relationship between the electric field and the current density is given by Ohm's law (4.30) j =aE A good conductor is one where a is large. By taking the divergence of equation (4.29) and noting that \7 . (\7 x H) = 0 we obtain (using equation (4.26) for constant E)
Bp
a
at + ~p =
O.
The solution to this equation is
p(t) = Po exp( -at/E), where Po = p(t = 0). This solution shows that the charge density decays exponentially with time. Typical values of E are r - 10- 12 - 10- 10 farads/metre. Hence, provided a is 3For 51 units.
98
CHAPTER 4. FIELD AND WAVE EQUATIONS
not too small, the dissipation of charge is very rapid. It is therefore physically reasonable to set the charge density to zero and, for problems involving the interaction of electromagnetic waves with good conductors, equation (4.26) can be approximated by \7. fE = 0 (4.31) and equation (4.29) becomes BE
= fat + o-E.
\7 x H
Note that, in imaging problems, the material may not necessarily be conductive throughout but may be a varying dielectric with distributed conductive elements. For example, in imaging the surface of the Earth using microwave radiation (Synthetic Aperture Radar), the electromagnetic scattering model is based on a 'ground truth' that is predominantly a dielectric (dry ground surfaces and dry vegetation for example) with distributed conductors (e.g. metallic objects on a dry ground surface, the sea and to a lesser extent rivers and lakes).
4.6.2
Non-conductive Dielectrics
In this case, it is assumed that the conductivity of the medium is negligible and no current can flow, and hence
j=O and equation (4.29) is just \7 x H
BE
= fat.
Also, if the conductivity is zero then p = Po and if po = 0 then equation (4.26) becomes \7. fE = O. The issues of when a material is a conductor or a dielectric is compounded in the relative importance of the terms j and f(BEjBt) in equation (4.29). Let us consider the electric and magnetic fields to be monochromatic waves, so that equation (4.29) becomes (with j = o-E) \7 x H(r,w) = (iwf+o-)E(r,w).
The relative importance of the terms on the right hand side of equation (4.29) is then determined by the magnitudes of 0- and WE. If 0-
-» Wf
1
then conduction currents dominate and the medium is a conductor. If 0-
-
Wf
«1
then displacement currents dominate and the material behaves as a dielectric. When ~ rv 1 Wf
4.6. EM WAVES IN AN INHOMOGENEOUS MEDIUM
99
the material is a quasi-conductor; some types of semi-conductor fall into this category. Note that the ratio a/wE is frequency dependent and that, consequently, a conductor at one frequency may be a dielectric at another. For example, copper has a conductivity of 5.8 x 10 7 siemens/metre and E ~ 9 X 10- 12 farads/metre so that W
WE
16
Up to a frequency of 10 Hz (the frequency of ultraviolet light) a/wE » 1, and copper is a conductor. At a frequency of 10 20Hz (the frequency of X-rays), however, a ju»: « 1 and copper behaves as a dielectric. This is why X-rays travel distances of many wavelengths in copper. An insulator has a conductivity in the order of 10- 1:' siemens/metre and a permittivity of the order of 10- 11 farads/metre, which gives WE -
rv
104 w
a so the conduction current is negligible at all frequencies.
4.6.3
EM Wave Equation
In many electromagnetic imaging systems, the field that is measured is the electric field. It is therefore appropriate to use a wave equation which describes the behaviour of the electric field. This can be obtained by decoupling Maxwell's equations for the magnetic field H. Starting with equation (4.28), we divide through by f-L and take the curl of the resulting equation. This gives
'V x
(~'V x E) f-L
=
-~'V x H. 8t
By taking the derivative with respect to time t of equation (4.29) and using Ohm's law - equation (4.30) - we obtain
8 8 2E 8E -('V x H) = E +a-. 8t 8t 2 8t From the previous equation we can then write
'V x
(~'V x E) f-L
2E
= -E 8 _ a 8E 8t 2 8t
Expanding the first term, multiplying through by f-L and noting that
we get
8 2E 8E 'V x 'V x E+Ef-L 8t 2 +af-L7it = ('Vlnf-L) x 'V x E. Expanding equation (4.31) we have
E'V·E+E·'VE=O
(4.32)
CHAPTER 4. FIELD AND WAVE EQUATIONS
100 or
\7 . E
=
-E· \7 In E.
Hence, using the vector identity
we obtain the following wave equation for the electric field
This equation is inhomogeneous in E, fJ and (1'. Solutions to this equation provide information on the behaviour of the electric field in a fluctuating conductive dielectric environment. In electromagnetic imaging problems, interest focuses on the behaviour of the scattered EM wavefield generated by variations in the material parameters E, fJ and (1'. In this context, E, fJ and (1' are sometimes referred to as the electromagnetic scatter generating parameters. In electromagnetic imaging, the problem is to reconstruct these parameters by measuring certain properties of the scattered electric field. This is a three parameter inverse problem which requires us to first solve for the electric field E given E, fJ and (1'.
4.6.4
Inhomogeneous EM Wave Equations
In order to solve the wave equation derived in the last section using the most appropriate analytical methods for imaging science (i.e. Green function solutions which are discussed in the following Chapter), it must be re-cast in the form of the Langevin equation
where L is an inhomogeneous differential operator. Starting with equation (4.32), by adding 8 2E 1 EO - - - \7 x \7 x E 8t 2 fJo
to both sides of this equation and re-arranging, we can write
where IE
E - EO
= - - and EO
Ii-'
fJ - fJo
= ---. fJ
We can then use the result (valid for p rv 0) \7 x \7 x E = - \72E + \7 (\7 . E)
=
-\7 2E - \7(E· \7lnE)
4.7. ELASTIC FIELD EQUATIONS
101
so that the above wave equation can be written as
Finally, introducing the Fourier transform
J (X)
E(r, t)
=
-1
271"
~ w) exp(iwt)duJ, E(r,
-(X)
we can write the above wave equation in the time independent form
where 1 - w Co = - k --, - and Zo = /-loCo. Co VEo/-l O
The parameter Zo is the free space wave impedance and is approximately equal to 376.6 ohms. The constant Co is the velocity at which electromagnetic waves propagate in a perfect vacuum. In electromagnetic imaging, the fundamental problem is to obtain images of the parameters "Ie, "II-' and the conductivity a.
4.7
Elastic Field Equations
The propagation and scattering of acoustic radiation in a non-compressible solid is characterized by the elastic field equations. For a stationary isotropic material, the governing equation of motion for an elastic field is given by
where s(r, t) is the displacement vector (metres), per) is the material density (mass per unit volume) and T(r, t) is the material stress tensor (force per unit volume). The material stress tensor is determined by two parameters known as the first and second elastic Lame parameters (0: and (3, respectively), whose dimensions are force x time/volume, and is given by T(r, t) where I (=
=
o:(r)I\7 . s(r, t)
xx + yy + zz)
+ (3(r) [\7s(r, t) + s(r, t)\7]
is the unit dyad and [\7s
+ s\7] =
\7s
+ \7s t
where \7s t denotes the transposition of the dyadic field \7s.
102
CHAPTER 4. FIElD AND WAVE EQUATIONS
4.8
Inhomogeneous Elastic Wave Equation
Consider an inhomogeneous elastic material embedded in a homogeneous medium with material constants Po, 0:0 and (30' By adding
cPs
Po 8t 2
-
\7 . [0:0 1\7 . s
+ (3o(\7s + s\7)]
to both sides of the equation
and using the results \72s
=
\7(\7. s) - \7 x \7 x s,
\7s
+ s\7 = 2\7s + 1 x
\7 x sand \7·1 == \7
we obtain
where 'Y,., 'Y/3 and 'Yp are functions of the elastic field defined by 'Yo< () r
=
o:(r)-o:o () (3 , 'Y/3 r 0:0 + 2 0
and the constants CL and respectively, given by 2
cL
=
0:0
CR
(3(r)-(3o d () (3 an 'Yp r 0:0 + 2 0
=
p(r)-po Po
are the longitudinal and rotational wave speeds
+ 2(30 Po
=
an
d
2
(30
cR=-,
Po
CL
> CR·
The displacement vector s consists of both longitudinal and rotational components. We can write this vector in terms of two potentials u and w as s = \7u + \7 x w.
Here, u is the longitudinal displacement potential for which \7 x \7u = O. The first term is the longitudinal or compression wave and the second term describes the shear wave. In general, an elastic wavefield is composed of both compression and shear waves. In order to simplify the pro' .lem and work with a singlecomponent wave equation, we consider the case when the shear component is negligible. In an imaging system in which the shear component is not measured, this is a reasonable assumption. However, this case assumes that mode coupling between the compression and shear waves, which can affect the shear wave component that is measured, is negligible. This is a reasonable assumption when the elastic material is weakly inhomogeneous. Thus, substituting the equation s = \7u into the above equation for s, we have
4.9. ACOUSTIC FIELD EQUATIONS
103
Applying the divergence operator to each term, then gives
1 a +'V2 'Ya'V 2 U + 2" -a 2'V . 'Yp'VU + 'V'V'Y{3 C t 2
: 'V'VU
L
where
e(r, t)
=
'V2u (r , t)
and: denotes the tensor product. The wavefield e is the elastic dilatation. The quantitative imaging problem in this case is, on the basis of this elastic wave equation, is to obtain a reconstruction of the parameters 'Yp, 'Ya and 'Y{3 from measurements of the field e(r, t). As with the electromagnetic case, this is a three parameter problem. Finally, introducing the Fourier transforms
J 00
e(r, t)
=
E(r, w) exp( -iwt)dt
-00
and
J 00
u(r,t)
=
U(r,w)exp(-iwt)dt,
-00
we arrive at the inhomogeneous wave equation
4.9
Acoustic Field Equations
The acoustic field equations are obtained by linearizing the hydrodynamic equations of motion and may be written in the form
'V·v
ap
K-
(4.33)
av - 'V . T
(4.34)
=
'Vp = P at
at
where v(r, t) is the velocity field (length/time), p(r, t) is the pressure field (force/area), T(r, t) is the material stress tensor (force/volume), p(r) is the density (mass/volume) and K(r) is the compressibility (area/force). Compared with the elastic field equation, we now have another material parameter, namely, the compressibility. It is assumed that the material to which these equations comply is adiabatic. The first equation comes from the law of
CHAPTER 4. FIELD AND WAVE EQUATIONS
104
conservation of mass and the second equation is a consequence of the law of conservation of momentum. For compressional waves alone, the material stress tensor is given by T = 10:\7 . v + 2(3\7v
\7u, u
where v is taken to be given by being the longitudinal velocity potential. The parameters 0: and (3 are related to the bulk <: and shear 77 viscosities of a material by the equations 2 0: = <: --"7 3 and (3=77 When the viscosity is zero (i.e. when <: = "7 = 0) we are left with the following acoustic field equations: \7 . v = r: ap
(4.35)
at av \7p = p at
(4.36)
As in previous cases, it is assumed that p, n; 0: and (3 are both isotropic and time invariant.
4.9.1
Acoustic Wave Equation
By decoupling the field equations (4.35) and (4.36) for v, a scalar wave equation for the pressure p can be obtained. This is accomplished by dividing equation (4.36) by p and taking the divergence of each term, giving
~\7. at v = \7. (~\7P) p Differentiating equation (4.35) with respect to time t then gives
a
at \7 . v
a2 p
= Ii at 2 •
Hence, we can write \7. ( -1 \7p ) p or, after expanding the first term, \72p - lip
a2p at 2
Ii-
=0
(4.37)
aat2p = \7lnp· \7p. 2
This wave equation is based on a physical model where it is assumed that the propagation and scattering of acoustic waves is only due to variations in the compressibility and density. When fluctuations in the bulk and shear viscosities are present, an additional source of scattering occurs. In this case the appropriate wave equation is obtained by decoupling equations (4.33) and (4.34) for p, giving 1
\7 ( --;;.,\7. v
)
= p
aat2 v 2
-
ata [\7(0:\7. v) + 2\7· ((3\7v)].
(4.38)
4.9. ACOUSTIC FIELD EQUATIONS
4.9.2
105
Inhomogeneous Acoustic Wave Equations
As with electromagnetic problems, in acoustic imaging, we need to develop inhomogeneous wave equations of the form
or depending on whether we need to solve for the pressure wavefield p or the velocity wavefield v, respectively.
Non-viscous Medium Consider the case when the viscosity of the material is zero. By adding E}2p 1 "'0- -\7. \7p 2
8t
po
to both sides of equation (4.37), we can write
where r; - "'0
'YK = - - - ,
'Yp
P - Po =-
and Co
p
"'0
1 = -.
VPo"'o
Here, Co is the velocity at which acoustic waves propagate in a homogeneous material with a density Po and compressibility "'0. If we then introduce the Fourier transform
J 00
p(r, t)
=
~ 21f
P(r, w) exp(iwt)dt
-00
then the above wave equation becomes
where k = w / Co. The inverse scattering problem posed by this equation involves reconstructing 'YK and 'Yp by measuring appropriate properties of the scattered pressure field. This is a two parameter reconstruction problem.
Viscous Medium The problem becomes a little more complicated when we consider an inhomogeneous viscous material. In this case, by adding
8 -[\7(0:0\7· v)
8t
+ 2\7· (,8o\7v)J -
2
8 v - \7 ( -\7. 1 Pov) 8t 2 "'0
CHAPTER 4. FIELD AND WAVE EQUATIONS
106
to both sides of equation (4.38) and noting that for compression waves only
since v
= V'u
= 0, we obtain the
and V' x V'u
wave equation
(4.39) where '" -
'YI<
'Y;3
f3 - f30 f3' ao + 2 0
=
The parameter form
"'0
= -",-,
TO
Co
=
P - Po
-----p;;-'
=
'Yp
'Yo:
1 ~ and
=
TO
v Po"'o
a - ao ao + 2f30
= "'o(ao + 2(30).
is known as the relaxation time and may be written in the TO
+ 2f30
ao
=
poc6
and the quantity ao
4
+ 2f30 = (0 + 31/0
is the compressional viscosity. Introducing the Fourier transform
J
.
00
v(r, t) = -1 21T
V(r, w) exp(zwt)dt,
-00
equation (4.39) can be written as (1 + iWT)V' 2V
+ k 2V =
-iWT[V'b'\ V'. V)
-k2'Yp V
+ 2V'·
+ V'b"
V' . V)
b;3V'V)],
If we then use the relationship WT
=
kf!
where I! is the relaxation length (= T/ CO) and divide through by 1 + ikf!, we obtain (V'2 + e)V
=
-e'Yp V
1
+ 1 + iU V'bl< V' . V)
ikf!
-1
+ ikf![V'b'\ V'. V) + 2V'· b;3V'V)]
where
~=
k
vI + ikf!
4.9. ACOUSTIC FIELD EQUATIONS
107
Finally, we can obtain a scalar wave equation by substituting V for V'u and taking the divergence of each term to obtain
where
and
The field U is the acoustic dilatation. It represents the fractional change in the volume of a material due to the disturbance of an acoustic wave. The imaging problem posed by the wave equation above involves finding inverse solutions for the four acoustic scatter generating parameters 'Yp, 'YI<' 'Yo. and 'Y(3' This is a four parameter reconstruction problem. Any spatial or temporal inhomogeneity in the acoustic properties of a medium will scatter acoustic radiation. Thus, fluctuations in the density, compressibility, shear and bulk viscosity will produce scattering of different directional behaviours and can therefore, in principle, be characterized by their own directional scattering properties. A small density fluctuation (with regard to the wavelength of the field) within an otherwise homogeneous medium will exhibit a differential displacement amplitude when exposed to the oscillatory forces within an acoustic field. This motion is essentially equivalent to an oscillating particle in an otherwise stationary medium. Hence, a density fluctuation is basically a dipole source of sound emission (or scattering), exhibiting a nonisotropic directivity pattern with an intensity that depends on a differential force, i.e, the acoustic pressure gradient V'p. A fluctuation in compressibility will expand and contract quite differently to its surroundings when exposed to an acoustic field and is equivalent to a pulsating object within an otherwise stationary medium. Thus, a compressibility fluctuation is essentially a monopole source of sound emission which gives rise to isotropic scattering. This isotropic directivity pattern has an intensity dependence related to the acceleration of the source boundaries by the acoustic pressure, and is thus determined by the second time derivative 8 2pj8t2 . An additional source of acoustic scattering occurs with any variation in the viscosity of the medium which is a measure of the absorption of a wave by the medium. A small fluctuation in the viscosity produces a local change in the amplitude of the wavefront of a passing acoustic disturbance. This is equivalent to a small vibrating membrane within an otherwise stationary medium. Thus, a viscosity fluctuation represents a sound source having a relatively complicated directivity pattern (multi-pole scattering) but predominantly along the forward and backward directions of the acoustic wave vector with an intensity related to the vibrational velocity.
CHAPTER 4. FIELD AND WAVE EQUATIONS
108
4.10
Discussion
This Chapter has been concerned with investigating the field equations for the electromagnetic, elastic and acoustic fields. In each case, it has been shown how these equations can be reduced or decoupled to provide a linear inhomogeneous scalar wave equation (a Langevin equation) of the form
where L is an inhomogeneous linear differential operator which involves the 'scatter generating parameter' sets (')'e, 'YM' a) for the electromagnetic case (where the material may be composed of 'good conductors'), (')'p, 'Yo, 'Y(3) for the elastic case and (')'p, 'Y", 'Y"" 'Y(3) for the case of an acoustic field. The difference between the last two cases is determined by whether or not the material is compressible. Note that, in the case of acoustics. the field equations are a result of linearizing the hydrodynamic equations of motion whereas, in electromagnetism, the field equations are linear. Thus in acoustics, it is possible to incur a number of nonlinear effects that are not considered here. In electromagnetism, nonlinear behaviour can occur as a result of the polarization vector having a nonlinear relationship with the electric field vector. Also note that the wave equations derived here for the elastic and acoustic fields assume that the shear wave component is negligible. This is not always the case, particularly when elastic wavefields propagate through solids (seismic waves for example). The wave equations derived here are the result of trying to find a balance between developing a physical model that is relatively complete but 'simple' enough for the 'forward problem' (solving for the wavefield u given L) and the 'inverse problem' (solving for the material parameter sets given u) to become tractable using the analytical methods discussed in the following chapter - such is the nature of physics!
4.11
Summary of Important Results
The Langevin equation b(1)u(r, t) = -s(r,~)
where b(1)
= b(O) + L
D(O) is the homogeneous differential operator and erator.
L is the inhomogeneous op-
Maxwell's (microscopic) equations in eGS units "V·e=p "V·b=O "V x e
lab c
= ---
at
4.11. SUMMARY OF IMPORTANT RESULTS
109
18e j \7xb=--+c 8t c where e(r, t) is the (microscopic) electric field generated by a point charge, b(r, t) is the (microscopic) magnetic field, p(r, t) is the charge density, j(r, t) is the current density and c is the speed of light. Maxwell's (macroscopic) equations in SI units \7. EE = P \7. MH = 0 \7 x E 't"7
v x
H
8H
=
-Mat 8E
.
= Eat +J
where E(r, t) is the (macroscopic) electric field (volts/metre), H(r, t) is the (macroscopic) magnetic field (amperes/metre), j(r, t) is the current density (amperes/metre"), p(r, t) is the charge density (charge/metre''), E(r) is the permittivity (farads/metre) and M(r) is the permeability (henries/metre). Ohm's law j
= (TE
where (T is the conductivity in siemens/metre. Charge decay rate
p(t) = Po exp( -(Tt/E) Elastic field equation
82s P8t 2
=
\7. T
where s(r, t) is the displacement vector (metres), p(r) is the material density (mass per unit volume) and T(r, t) is the material stress tensor (force per unit volume). Acoustic field equations 8p \7. v = r.-
8t
\7p
8v
= p 8t -
\7. T
where v(r, t) is the velocity field (length/time), p(r, t) is the pressure field (force/area), T(r, t) is the material stress tensor (force/volume), p(r) is the density (mass/volume) and x(r) is the compressibility (area/force).
CHAPTER 4. FIELD AND WAVE EQUATIONS
110
Inhomogeneous wave equation for EM waves in a conductive dielectric
where
LE =
ikzoo-E + 'V(E· 'VInE) + 'V x
k2 ')'f E -
E - EO ')'f=--' EO
J-L
C'Yf.l'V x E);
J-Lo
')'f.l=--
J-L
E(r,w) - Electric field (Fourier transform of) E -
Permittivity
J-L - Permeability 0- -
Conductivity
Inhomogeneous wave equation for a compressional elastic wavefield in a non-compressible solid
where
Lo = C'Ya + 2
2')'(3)'V 2'V 2U + 2'VC'Ya + 2')'(3) . 'V'V2 u W
2
2
+'V ')'a'V U + 2"" 'V. ')'p'VU + 'V'V')'(3 : 'V'VU, CL
P- Po Po
')'p=--,
')'a
=
A - AO AO
+ 2J-Lo
and ')'(3
=
J-L - J-Lo AO
+ 2J-Lo
.
e(r,w) = 'V2 u (r , w) - elastic dilatation (Fourier transform of) P - Density
a, (3 - First and second Lame parameters, respectively Inhomogeneous wave equation for an acoustic pressure field propagating in a non-viscous compressible medium
where
P - Po
')'p=--.
P
per, w) - Pressure field (Fourier transform of) K, -
Compressibility
111
4.12. FURTHER READING
P - Density Inhomogeneous wave equation for a compressional acoustic wavefield propagating in a viscous compressible material
where
P - Po Po
'Yp= - - ,
'Ya
=
a - ao ao + 2f3o' 'Y(3
=
f3 - f30 ao + 2f3o
and
u(r, w) - Longitudinal velocity potential (Fourier transform of) U = \72 u - Acoustic dilatation K -
Compressibility
P - Density a, f3 - First and second Lame parameters, respectively f - Relaxation length
4.12
Further Reading
• Webster A G, Partial Differential Equations of Mathematical Physics, Stechert, 1933 • Stratton J A, Electromagnetic Thoery, McGraw-Hill, 1941. • Morse P M and Feshbach H, Methods of Theoretical Physics, McGrawHill, 1953. Atkin R H, Theoretical Electromagnetism, Heinemann, 1962. • Morse PM and Ingard K U, Theoretical Acoustics, McGraw-Hill, 1968. • Butkov E, Mathematical Physics, Addison-Wesley, 1973.
112
CHAPTER 4. FIELD AND WAVE EQUATIONS • Bleaney B I and Bleaney B, Electricity and Magnetism, Oxford University Press, 1976.
• Sander K F and Reed GAL, Transmission and Propagation of Electromagnetic Waves, 1986. • Evans G A, Blackledge, J M and Yardley P, Analytical Solutions to Partial Differential Equations, Springer, 1999.
Field and Wave Equations This Chapter is concerned with the equations used to describe the fields that are measured in an imaging system and their relationship with the material variables with which these fields interact. The field equations determine the physical characteristics and behaviour of a particular type of field. Two types of fields are important in imaging: the electromagnetic field and the acoustic field. The primary purpose of this Chapter is to introduce and discuss the electromagnetic and acoustic field equations which are employed in later chapters (Chapters 7 - 11 for example). From these results, we derive equations (wave equations or equations of propagation) which describe the propagation of different wavefields through various types of materials or media.
4.1
The Langevin Equation
The propagation of a wavefield can be modelled by various different wave equations depending upon the type of field, the supporting material and its physical state. In general, however, if the supporting material is assumed to be a linear medium, the scalar field u(r, t) obeys a partial integro-differential equation of the form jj(l)u(r, t) = -s(r, t) where jj(l)
= b(O) + L
For a vector field u(r, t),
jj(l)u(r, t)
=
-s(r, t)
This is the Langevin equation where jj(O) and L are linear operators: jj(O) is associated with the homogeneous portion of the medium, L is, in general, an integro-differential scattering operator, and s is a source function which describes the emission of the incident field from a given source. The operator L models the interaction of the incident field with the differential or local 85
CHAPTER 4. FIELD AND WAVE EQUATIONS
86
scattering from material inhomogeneities. The operator form
tr»
has the general
2
A
D
(1) _
A
=D
(1) (
8 8 \7,\7 , ... ,1'8t'8t 2 '
)
2.
'' '
•
The source function s is, in general, given by s=p®f where f is the probe field, p is a filter weighting function for the emitted field and ® is taken to denote the convolution over three-dimensional space and time. From this general formalism, one can consider a variety of scalar wave equations governing the propagation of different types of We' vefields .upported by different isotropic media. Two cases arise that are based on: (i) a rigorous derivation of the Langevin from a set of fundamental field equations; (ii) the proposition of a Langevin equation (a phenomenological model) whose characteristics (e.g. the phase velocity) are confirmed experimentally. The wave equation that is used to model a wavefield determines the underlying physical model for a (scatter) imaging system. This includes aspects such as the resolution of the image (e.g. the wavelength or bandwidth of the probe), the accuracy of the spatial mapping of the scatter generating parameters in an image (e.g. the level of distortion as determined by the propagation model) and image fuzziness. A fuzzy image is an image which, although attempting to display a specific scatter generating parameter, fails to achieve this because the scattered wavefield that has been measured and processed is corrupted by some other interaction that has not been included on the original model (i.e. the wave equation). Thus, all scatter-imaging techniques are highly model dependent since the reconstruction algorithm is determined by the wave equation which characterizes the medium, in particular, the model associated with the operator L. An inappropriate choice of wave equation results in image fuzziness. Scatter imaging demands appropriate modelling of the scattering dynamics, even if the computations are approximate. An inexact model will lead to a fuzzy image whereas an approximate computation may lead to poor resolution. Distortion, due to poor propagator models which are compounded in the operator b(O) and its associated (free-space) Green function, is a common artifact in many imaging systems and poor physical modelling manifests some form of distortion in most imaging methods. A general criticism, common to many imaging systems, is that emphasis is often placed on a significant amount of computation for image reconstruction and processing. This can provide good, or at least, enhanced resolution but at the expense of developing accurate models for the propagation of a wavefield through the medium that generates the scattered field from which an image is generated and interpreted. This leads to images which are well resolved but may be badly distorted and fuzzy. In this chapter, we start by considering Maxwell's equations which provide a unified theoretical framework for the interaction of electromagnetic waves with matter. In the case of the macroscopic form of these equations, we introduce material parameters such as the permittivity, the permeability and the conductivity. The elastic and acoustic field equations are then studied. An elastic
4.2. MAXWELL'S EQUATIONS
87
field describes the characteristics of 'mechanical radiation' when a material is solid and incompressible. The acoustic field equations relate the acoustic field variables (such as the pressure and velocity fields) to material parameters including the density, compressibility and viscosity. These are used to model the propagation and scattering of an acoustic wavefield when the material is a compressible solid, liquid or gas. In each case, the material is taken to be isotropic. The field equations presented provide the fundamental basis for modelling electromagnetic and acoustic imaging systems. In each case, it is shown how the field equations can (under certain conditions and approximations) be decoupled to provide a governing inhomogeneous wave equation whose complexity increases according to the number of material parameters that are considered. General methods of solving such equations are then addressed in Part II using the Green function method which is discussed in the following Chapter.
4.2
Maxwell's Equations
We shall now consider Maxwell's equations and study the electromagnetic wavefields and wave equations that arise from these equations. We first consider these equations in their microscopic form (for individual charged particles) and go on to consider the macroscopic form of Maxwell's equations (for the case when there are many particles per cubic wavelength) and briefly study the propagation of monochromatic electromagnetic waves in homogeneous media. The macroscopic form of Maxwell's equations is then used to construct inhomogeneous wave equations in a form that are suitable for applying the solutions methods discussed in Chapter 5. The motions of electrons (and other charged particles) give rise to electric and magnetic fields. These fields are described by the following equations which are a complete mathematical descriptions for the physical laws quoted 1. Coulomb's law
\7. e
= 471"p
(4.1)
Faraday's law of induction 1 ab c at
\7 x e = - - -
(4.2)
No free magnetic monopoles exist (4.3)
\7·b=O Modified (by Maxwell) Ampere's law
\7 x b = ~
ae + 471" j
cat
1 For
CGS units.
c
(4.4)
88
CHAPTER 4. FIELD AND WAVE EQUATIONS
where e is the electric field, b is the magnetic field, j is the current density, p is the charge density and c ':::' 3 X 108 ms " ! is the speed of light. These microscopic Maxwell's equations are used to predict the pointwise electric e and magnetic b fields given the charge and current d -nsities I p and j respectively). By including a modification to Ampere's law, l.e. the inclusion of the 'displacement current' term 8ej8(ct), Maxwell (see Figure 4.1) provided a unification of electricity and magnetism compounded in the equations above.
Figure 4.1: James Clerk Maxwell
4.2.1
Linearity of Maxwell's Equations
Maxwell's equations are linear because if
and then
Pl
+ P2,
jl
+ j2
--t
ei
+ e2,
bi
+ b2
where --t means 'produces'. This is because the operators 'V., 'Vx and the time derivatives are all linear operators.
4.2.2
Solution to Maxwell's Equations
The solution to these equations is based on exploiting the properties of vector calculus and, in particular, identities involving the curl.
4.2. MAXWELL'S EQUATIONS
89
Taking the curl of equation (4.2), we have
ab
1
V'xV'xe=--V'xc at and using the identity (see Chapter 1)
V' x V' x e = V'(V' . e) - V'2e together with equations (4.1) and (4.4), we get
1a 47r .) V'(47rp) - V'2e = - - (1-ae - +-J cat cat c or, after rearranging, (4.5) Taking the curl of equation (4.4), using the identity above, equations (4.2) and (4.3) and rearranging the result gives (4.6) Equations (4.5) and (4.6) are inhomogeneous wave equations for e and b. These equations are related or coupled to the vector field j (which is related to b). If we define a region of free space where p = and j = 0, then both e and b satisfy the equation 1 a2 f V'2f - c2 at 2 = 0.
°
This is the homogeneous wave equation. One possible solution of this equation (in Cartesian coordinates) is
fx = p(z - ct);
fy = 0,
fz =
°
which describes a wave or distribution p moving along z at velocity c. Thus, we have shown that in free space when
V'. e
=
0,
1 ab
V'xe=--cat'
V'. b
=
0,
V'xb=~ae. c at
Maxwell's equations describe the propagation of an electric and magnetic (or electromagnetic field) in terms of a wave traveling at the speed of light (see front cover). After developing the origins of the vector calculus, Maxwell derived the wave equations for an electromagnetic field in a paper entitled A Dynamical Theory of the Electromagnetic Field, first published in 1865 and arguably one of the greatest intellectual achievements in the history of physics.
CHAPTER 4. FIELD AND WAVE EQUATIONS
90
4.3
General Solution to Maxwell's scopic) Equations
(Micro-
The solution to Maxwell's equation in free space is specific to the charge density and current density being zero. We now investigate a method of solution for the general case. The basic method of solving Maxwell's equations (i.e. finding e and b given p and j) involves the following: (i) Expressing e and b in terms of two other fields U and A. (ii) Obtaining two separate equations for U and A. (iii) Solving these equations for U and A from which e and b can then be computed. For any vector field A
\7. \7 x A = O. Hence, if we write
b=\7xA
(4.7)
then equation (4.3) remains unchanged. Equation (4.2) can then be written as 1
&
\7xe=---\7xA c &t
or
l&A) = \7 x ( e+ ~8t
o.
The field A is called the Magnetic Vector Potential. For any scalar field U
\7 x \7U
0
=
and thus equation (4.2) is satisfied if we write
l&A ±\7U=e+-c &t or
l&A
e = -\7U - - c &t
(4.8)
where the minus sign is taken by convention. U is called the Electric Scalar Potential. Substituting equation (4.8) into Maxwell's equation (4.1) gives
\7. ( \7U or 2
1
1 &A) +- = -41lp C
o
&t
\7 U + --\7. A c &t
= -4rrp.
(4.9)
4.4. THE MACROSCOPIC MAXWELL'S EQUATIONS
91
Substituting equations (4.7) and (4.8) into Maxwell's equation (4.4) gives
a(
aA) =-J
1 1 V'xV'xA+-V'U+--
cat
cat
41T •
c
Finally, using the identity
V' x V' x A = V'(V' . A) - V'2 A we can write
a- -A V ' ( V'·A+-1 au) =--J 41T • V'2 A -12 2 2
c
at
c at
c
(4.10)
If we could solve equations (4.9) and (4.10) above for U and A then e and b
could be computed. The problem here, is that equations (4.9) and (4.10) are coupled. They can be decoupled by applying a technique known as a 'gauge transformation' called the Lorentz gauge transformation, after Lorentz who was among the first to consider it as an approach to solving these equations. The idea is based on noting that equations (4.7) and (4.8) are unchanged if we let A~A+V'X
and
lax
U~U--
c at since V'x V'X = O. If this gauge function X is taken to satisfy the homogeneous wave equation
then
1au
V'.A+~8t=O
(4.11)
which is called the Lorentz condition. With equation (4.11), equations (4.9) and (4.10) become (4.12) and
2
V'2 A _ ~ a A = _ 41Tj c2
at 2
c
respectively. These equations are non-coupled inhomogeneous wave equations whose solution (using the Green function) will be considered in the following Chapter.
4.4
The Macroscopic Maxwell's Equations
The microscopic form of Maxwell's equations tells us how individual charged particles and electromagnetic fields interact. When there are many particles
CHAPTER 4. FIELD AND WAVE EQUATIONS
92
per cubic wavelength, the electromagnetic radiation 'sees' only a macroscopic average. The medium is then described by its dielectric parameters: the permittivity E, the magnetic permeability f.l and the conductivity a, Simple averaging of the quantities over a small volume V, e.g.
E(r,t)
J
=~
~
e(r',t)d3r', B(r,t) =
v
J
b(r',t)dV
v
leads to the following, but not very useful, macroscopic form of Maxwell's equations: 1 sn \7. E = 41Tpmacro, \7 x E = --~, c ut
\7. B
= 0, \7 x B =
However, both Pmacro and jmacro bound electrons, i.e. we can write
1 aE
-~
41T.
+ -Jmacro'
c ut c can be split into two terms due to free and
Pmacro
= Pbound + Pfree
jmacro
= jbound + jfree'
and By bound, we mean that the electrons are bound to the nucleus to constitute an atom. If we introduce an electric polarization P of the medium to represent the average dipole moment per unit volume given by
P = -Nes where s is the average vector between bound electrons and nuclei, e is the charge of an electron and N is the average number of electrons per unit volume, then we can define the charge density of bound electrons as pbound
= -\7. P
and the current density of bound electrons in the form
.
Jbound =
ap + c\7 x M
7ft
°
where M is the magnetization vector. At optical frequencies, M = (in the absence of a strong applied magnetic field). Further, we now define the following: (i) the displacement vector given by D
= E + 41TP;
(ii) the magnetic field strength given by H = B - 41TM. From these definitions, we obtain a useful macroscopic form of Maxwell's equations given by 1 aB \7 . D = 41TPfree, \7 x E = --~ c ut
\7 . B = 0, \7 x H
1
sn
41T.
=-~+-
Jfree
c dt c These equations are valid for media which are: (i) non-isotropic; (ii) inhomogeneous.
4.5. EM WAVES IN A HOMOGENEOUS MEDIUM
4.5
93
EM Waves in a Homogeneous Medium
Having derived Maxwell's equation in macroscopic form, let us now consider the type of solutions they provide for a specific case. Suppose we illuminate a homogeneous material with monochromatic radiation of angular frequency w. What are the possible solutions of Maxwell's equations in the material? i.e. what waves exist in the medium?
4.5.1
Linear Medium
Assume that all the macroscopic vectors oscillate sinusoidally at angular frequency w (this is true, in general, only for high frequency, weak fields). Define vector amplitudes E(r, t) = E(r, w) exp( -iwt), B(r, t) = B(r, w) exp( -iwt) and so on 2 , so that Maxwell's equations can be written in the form \7. D
= 4Jrp
(4.13)
\7 x E
= iWB
(4.14)
c
\7·B=O
(4.15)
iw 4Jr. (4.16) c c where p and j are taken to be the free charge density and the free current density, respectively. Let P = XeE, M = XmH and j = aE \7 x H
= --D + - J
where Xe is the electric susceptibility, Xm is the magnetic susceptibility and a is the conductivity, each of which may be tensors. Note that, in general, this linearity may not occur and P could be of the form
Here al, a2, ... are constant coefficients which would introduce a nonlinear optical material and nonlinear optical effects for example. Note that the effect of introducing monochromatic radiation (Le. a wavefield oscillating at one single frequency w) is to replace the time derivatives in Maxwell's equations with iw which significantly helps in the algebra required to derive the solutions that follow.
4.5.2
Isotropic Medium
Let Xe, Xm and a be complex scalars (not tensors) and let us define the following: (i) the dielectric constant given by
E
= 1 + 4JrXe;
2Strictly speaking E(r,w), B(r,w), etc. should be given a different notation but, in the context of the equations that follow, it is implied that all dependent variables are functions of rand wand not rand t.
CHAPTER 4. FIELD AND WAVE EQUATIONS
94
(ii) the magnetic permeability given by fl = 1 + 41TXm so that we can write D=EE
(4.17)
= flH.
(4.18)
and
B
Taking the divergence of equation (4.16) and noting that 'V. 'V x H have -iw'V . D + 41T'V . j = 0.
= 0,
we
Hence, from equation (4.13) we get p
= -~'V.j = -~'V. (aE). w
(4.19)
w
Substituting equations (4.17), (4.18) and (4.19) into Maxwell's equation (4.13)(4.16), we obtain the following time independent equations for the complex vector amplitudes: 'V .
(E + i4:a) E = 0,
'V . (flH)
'V x E
= i~fl H,
iw (..
+ i41TfT) ~ E.
= 0, 'V x H = - -;;
E
These equations apply to a linear, isotropic but inhomogeneous medium, i.e.
E,
fl and a may be functions of position. Note that, for any vector X and scalar
a, 'V. (aX)
4.5.3
= 'Va· X + a'V . X i- a'V . X,
= 0.
Homogeneous Medium
For a homogeneous medium (where of equations reduces to 'V . E 'V . H
4.5.4
unless 'Va
=
E,
= 0,
fl and a are constants, the previous set
=
'V x E
0, 'V x H
iw (
= - -;;
iWfl
-H, c E
i41Ta) + ---::;-
E.
Plane Wave Solutions
Let
E
= Eo exp(ike . r)
H
= H o exp(ike . r)
where k, is the complex wave number. Noting that 'V. [Cexp(ik e· r)]
= ike' Cexp(ik e· r)
and 'V x [Cexp(ike· r)] = ike
X
Cexp(ike· r)
4.5. EM WAVES IN A HOMOGENEOUS MEDIUM
95
we obtain
k., . Eo
= 0, k c ' H o = 0,
k., x Eo
=
j.LW - Ho,
(4.20) (4.21)
c
( i4JrO") k, x H o = - W ~ E + -:;- Eo.
(4.22)
Equations (4.20) are referred to as the transversality conditions. Substituting equation (4.21) into equation (4.22) yields
W ( E+ i4JrO") -ckc x (k., x Eo) = - - Eo. j.LW c W Using the identity
A x (B x C) = (A. C)B - (A· B)C we can write this result in the form
or, since k., . Eo
= 0,
as (4.23)
where
and 2
nc
i4Jrj.L0"
= Ej.L+--. W
Here, n c is called the complex refractive index. Let ti;
= n + iii
k.,
= k+ ia
and where n is the refractive index, Ii is the extinction index, k is the wavenumber and a is the attenuation vector. Substituting these expressions into equation (4.23) and equating the real and imaginary parts gives
k2
-
a2
= k5(n 2
-
2
1i )
(4.24)
and
k· a = k5nli. Thus, plane wave solutions exist of the form
E
= Eo exp(ikc . r);
H
= H o exp(ikc . r)
where, from equation (4.23),
I k., 1= ncko, k, . Eo = 0 and
1
H o = -k k., x Eo· j.L 0
(4.25)
96
4.5.5
CHAPTER 4. FIELD AND WAVE EQUATIONS
Non-absorbing Media (K;
=
0)
Equations (4.24) and (4.25) reduce to
and k·a=O. Two kinds of waves are possible: (i) Real vector waves where a = 0, k, = k,
1k 1= kon and
E(r,w) = Eo(r,w) exp(ik· r) or E(r, t) = Eo(r, t) exp[i(k. r - wt)]. This is like a free space plane wave. The velocity of propagation is w/ k cf n. and the wavelength, is A/n. Both amplitude and phase are constant and perpendicular to k, i.e. the wave is homogeneous. Since k . Eo = 0, the real and imaginary parts of E are perpendicular to k. H is also perpendicular to k and Re[E] is perpendicular to Re[H]. (ii) Complex wave vector where k is perpendicular to a so that k· a = 0 and E(r, t) = Eo(r, t) exp( -a' r) exp[i(k· r - wt)] which propagates along k with velocity w/k < cfn, The amplitude is constant over planes perpendicular to a and the phase is constant over planes perpendicular to k - the wave is homogeneous.
4.5.6
Absorbing Media (K; > 0, k· a
i= 0)
(i) Homogeneous wave: k and a are in the same direction, k = nk o, a = Kko and E(r, t) = Eo(r, w) exp(-Kko . r) exp[i(nko . r - wt)]. This wave propagates along k o at velocity cfn, with wavelength Ao/n and decreases exponentially along the direction of propagation. Both amplitude and phase are constant and perpendicular to k o and both E and H are perpendicular to ko. Re[E] is not perpendicular to Re[H]. (ii) Inhomogeneous wave: k and a are not in the same direction. There is constant phase perpendicular to k and constant amplitude perpendicular to a. Since a has a component along k, there is a decrease of amplitude along k.
4.6. EM WAVES IN AN INHOMOGENEOUS MEDIUM
4.6
97
EM Waves in an Inhomogeneous Medium
In the previous Section, we considered the EM waves that can occur in a homogeneous material that is linear and isotropic by studying Maxwell's equations for monochromatic propagation. We now turn our attention to developing wave equations for a medium that is linear, isotropic and inhomogeneous using Maxwell's equations in the forrn''
\7. EE = P,
(4.26)
= 0,
(4.27)
\7 'IlH
\7 x E and
\7 x H
=
aH
at '
(4.28)
+j.
(4.29)
-lI-
r:
= E~~
where E(r, t) is the electric field (volts/metre), H(r, t) is the magnetic field (amperes/metre), j(r, t) is the current density (amperes/metre"), p(r, t) is the charge density (charge/metre''}, E(r) is the permittivity (farads/metre) and Il(r) is the permeability (henries/metre). The values of E and 11 in a vacuum (denoted by EO and 110, respectively) are EO = 8.854 X 10- 12 farads/metre and 110 = 47l' X 10- 7 henries/metre. In electromagnetic imaging problems there are two important physical models to consider, based on whether a material is either conductive or non-conductive.
4.6.1
Conductive Materials
In this case, the medium is assumed to be a good conductor. A current is induced which depends on the magnitude of the electric field and the conductivity a (siemens/metre) of the material from which the object is composed. The relationship between the electric field and the current density is given by Ohm's law (4.30) j =aE A good conductor is one where a is large. By taking the divergence of equation (4.29) and noting that \7 . (\7 x H) = 0 we obtain (using equation (4.26) for constant E)
Bp
a
at + ~p =
O.
The solution to this equation is
p(t) = Po exp( -at/E), where Po = p(t = 0). This solution shows that the charge density decays exponentially with time. Typical values of E are r - 10- 12 - 10- 10 farads/metre. Hence, provided a is 3For 51 units.
98
CHAPTER 4. FIELD AND WAVE EQUATIONS
not too small, the dissipation of charge is very rapid. It is therefore physically reasonable to set the charge density to zero and, for problems involving the interaction of electromagnetic waves with good conductors, equation (4.26) can be approximated by \7. fE = 0 (4.31) and equation (4.29) becomes BE
= fat + o-E.
\7 x H
Note that, in imaging problems, the material may not necessarily be conductive throughout but may be a varying dielectric with distributed conductive elements. For example, in imaging the surface of the Earth using microwave radiation (Synthetic Aperture Radar), the electromagnetic scattering model is based on a 'ground truth' that is predominantly a dielectric (dry ground surfaces and dry vegetation for example) with distributed conductors (e.g. metallic objects on a dry ground surface, the sea and to a lesser extent rivers and lakes).
4.6.2
Non-conductive Dielectrics
In this case, it is assumed that the conductivity of the medium is negligible and no current can flow, and hence
j=O and equation (4.29) is just \7 x H
BE
= fat.
Also, if the conductivity is zero then p = Po and if po = 0 then equation (4.26) becomes \7. fE = O. The issues of when a material is a conductor or a dielectric is compounded in the relative importance of the terms j and f(BEjBt) in equation (4.29). Let us consider the electric and magnetic fields to be monochromatic waves, so that equation (4.29) becomes (with j = o-E) \7 x H(r,w) = (iwf+o-)E(r,w).
The relative importance of the terms on the right hand side of equation (4.29) is then determined by the magnitudes of 0- and WE. If 0-
-» Wf
1
then conduction currents dominate and the medium is a conductor. If 0-
-
Wf
«1
then displacement currents dominate and the material behaves as a dielectric. When ~ rv 1 Wf
4.6. EM WAVES IN AN INHOMOGENEOUS MEDIUM
99
the material is a quasi-conductor; some types of semi-conductor fall into this category. Note that the ratio a/wE is frequency dependent and that, consequently, a conductor at one frequency may be a dielectric at another. For example, copper has a conductivity of 5.8 x 10 7 siemens/metre and E ~ 9 X 10- 12 farads/metre so that W
WE
16
Up to a frequency of 10 Hz (the frequency of ultraviolet light) a/wE » 1, and copper is a conductor. At a frequency of 10 20Hz (the frequency of X-rays), however, a ju»: « 1 and copper behaves as a dielectric. This is why X-rays travel distances of many wavelengths in copper. An insulator has a conductivity in the order of 10- 1:' siemens/metre and a permittivity of the order of 10- 11 farads/metre, which gives WE -
rv
104 w
a so the conduction current is negligible at all frequencies.
4.6.3
EM Wave Equation
In many electromagnetic imaging systems, the field that is measured is the electric field. It is therefore appropriate to use a wave equation which describes the behaviour of the electric field. This can be obtained by decoupling Maxwell's equations for the magnetic field H. Starting with equation (4.28), we divide through by f-L and take the curl of the resulting equation. This gives
'V x
(~'V x E) f-L
=
-~'V x H. 8t
By taking the derivative with respect to time t of equation (4.29) and using Ohm's law - equation (4.30) - we obtain
8 8 2E 8E -('V x H) = E +a-. 8t 8t 2 8t From the previous equation we can then write
'V x
(~'V x E) f-L
2E
= -E 8 _ a 8E 8t 2 8t
Expanding the first term, multiplying through by f-L and noting that
we get
8 2E 8E 'V x 'V x E+Ef-L 8t 2 +af-L7it = ('Vlnf-L) x 'V x E. Expanding equation (4.31) we have
E'V·E+E·'VE=O
(4.32)
CHAPTER 4. FIELD AND WAVE EQUATIONS
100 or
\7 . E
=
-E· \7 In E.
Hence, using the vector identity
we obtain the following wave equation for the electric field
This equation is inhomogeneous in E, fJ and (1'. Solutions to this equation provide information on the behaviour of the electric field in a fluctuating conductive dielectric environment. In electromagnetic imaging problems, interest focuses on the behaviour of the scattered EM wavefield generated by variations in the material parameters E, fJ and (1'. In this context, E, fJ and (1' are sometimes referred to as the electromagnetic scatter generating parameters. In electromagnetic imaging, the problem is to reconstruct these parameters by measuring certain properties of the scattered electric field. This is a three parameter inverse problem which requires us to first solve for the electric field E given E, fJ and (1'.
4.6.4
Inhomogeneous EM Wave Equations
In order to solve the wave equation derived in the last section using the most appropriate analytical methods for imaging science (i.e. Green function solutions which are discussed in the following Chapter), it must be re-cast in the form of the Langevin equation
where L is an inhomogeneous differential operator. Starting with equation (4.32), by adding 8 2E 1 EO - - - \7 x \7 x E 8t 2 fJo
to both sides of this equation and re-arranging, we can write
where IE
E - EO
= - - and EO
Ii-'
fJ - fJo
= ---. fJ
We can then use the result (valid for p rv 0) \7 x \7 x E = - \72E + \7 (\7 . E)
=
-\7 2E - \7(E· \7lnE)
4.7. ELASTIC FIELD EQUATIONS
101
so that the above wave equation can be written as
Finally, introducing the Fourier transform
J (X)
E(r, t)
=
-1
271"
~ w) exp(iwt)duJ, E(r,
-(X)
we can write the above wave equation in the time independent form
where 1 - w Co = - k --, - and Zo = /-loCo. Co VEo/-l O
The parameter Zo is the free space wave impedance and is approximately equal to 376.6 ohms. The constant Co is the velocity at which electromagnetic waves propagate in a perfect vacuum. In electromagnetic imaging, the fundamental problem is to obtain images of the parameters "Ie, "II-' and the conductivity a.
4.7
Elastic Field Equations
The propagation and scattering of acoustic radiation in a non-compressible solid is characterized by the elastic field equations. For a stationary isotropic material, the governing equation of motion for an elastic field is given by
where s(r, t) is the displacement vector (metres), per) is the material density (mass per unit volume) and T(r, t) is the material stress tensor (force per unit volume). The material stress tensor is determined by two parameters known as the first and second elastic Lame parameters (0: and (3, respectively), whose dimensions are force x time/volume, and is given by T(r, t) where I (=
=
o:(r)I\7 . s(r, t)
xx + yy + zz)
+ (3(r) [\7s(r, t) + s(r, t)\7]
is the unit dyad and [\7s
+ s\7] =
\7s
+ \7s t
where \7s t denotes the transposition of the dyadic field \7s.
102
CHAPTER 4. FIElD AND WAVE EQUATIONS
4.8
Inhomogeneous Elastic Wave Equation
Consider an inhomogeneous elastic material embedded in a homogeneous medium with material constants Po, 0:0 and (30' By adding
cPs
Po 8t 2
-
\7 . [0:0 1\7 . s
+ (3o(\7s + s\7)]
to both sides of the equation
and using the results \72s
=
\7(\7. s) - \7 x \7 x s,
\7s
+ s\7 = 2\7s + 1 x
\7 x sand \7·1 == \7
we obtain
where 'Y,., 'Y/3 and 'Yp are functions of the elastic field defined by 'Yo< () r
=
o:(r)-o:o () (3 , 'Y/3 r 0:0 + 2 0
and the constants CL and respectively, given by 2
cL
=
0:0
CR
(3(r)-(3o d () (3 an 'Yp r 0:0 + 2 0
=
p(r)-po Po
are the longitudinal and rotational wave speeds
+ 2(30 Po
=
an
d
2
(30
cR=-,
Po
CL
> CR·
The displacement vector s consists of both longitudinal and rotational components. We can write this vector in terms of two potentials u and w as s = \7u + \7 x w.
Here, u is the longitudinal displacement potential for which \7 x \7u = O. The first term is the longitudinal or compression wave and the second term describes the shear wave. In general, an elastic wavefield is composed of both compression and shear waves. In order to simplify the pro' .lem and work with a singlecomponent wave equation, we consider the case when the shear component is negligible. In an imaging system in which the shear component is not measured, this is a reasonable assumption. However, this case assumes that mode coupling between the compression and shear waves, which can affect the shear wave component that is measured, is negligible. This is a reasonable assumption when the elastic material is weakly inhomogeneous. Thus, substituting the equation s = \7u into the above equation for s, we have
4.9. ACOUSTIC FIELD EQUATIONS
103
Applying the divergence operator to each term, then gives
1 a +'V2 'Ya'V 2 U + 2" -a 2'V . 'Yp'VU + 'V'V'Y{3 C t 2
: 'V'VU
L
where
e(r, t)
=
'V2u (r , t)
and: denotes the tensor product. The wavefield e is the elastic dilatation. The quantitative imaging problem in this case is, on the basis of this elastic wave equation, is to obtain a reconstruction of the parameters 'Yp, 'Ya and 'Y{3 from measurements of the field e(r, t). As with the electromagnetic case, this is a three parameter problem. Finally, introducing the Fourier transforms
J 00
e(r, t)
=
E(r, w) exp( -iwt)dt
-00
and
J 00
u(r,t)
=
U(r,w)exp(-iwt)dt,
-00
we arrive at the inhomogeneous wave equation
4.9
Acoustic Field Equations
The acoustic field equations are obtained by linearizing the hydrodynamic equations of motion and may be written in the form
'V·v
ap
K-
(4.33)
av - 'V . T
(4.34)
=
'Vp = P at
at
where v(r, t) is the velocity field (length/time), p(r, t) is the pressure field (force/area), T(r, t) is the material stress tensor (force/volume), p(r) is the density (mass/volume) and K(r) is the compressibility (area/force). Compared with the elastic field equation, we now have another material parameter, namely, the compressibility. It is assumed that the material to which these equations comply is adiabatic. The first equation comes from the law of
CHAPTER 4. FIELD AND WAVE EQUATIONS
104
conservation of mass and the second equation is a consequence of the law of conservation of momentum. For compressional waves alone, the material stress tensor is given by T = 10:\7 . v + 2(3\7v
\7u, u
where v is taken to be given by being the longitudinal velocity potential. The parameters 0: and (3 are related to the bulk <: and shear 77 viscosities of a material by the equations 2 0: = <: --"7 3 and (3=77 When the viscosity is zero (i.e. when <: = "7 = 0) we are left with the following acoustic field equations: \7 . v = r: ap
(4.35)
at av \7p = p at
(4.36)
As in previous cases, it is assumed that p, n; 0: and (3 are both isotropic and time invariant.
4.9.1
Acoustic Wave Equation
By decoupling the field equations (4.35) and (4.36) for v, a scalar wave equation for the pressure p can be obtained. This is accomplished by dividing equation (4.36) by p and taking the divergence of each term, giving
~\7. at v = \7. (~\7P) p Differentiating equation (4.35) with respect to time t then gives
a
at \7 . v
a2 p
= Ii at 2 •
Hence, we can write \7. ( -1 \7p ) p or, after expanding the first term, \72p - lip
a2p at 2
Ii-
=0
(4.37)
aat2p = \7lnp· \7p. 2
This wave equation is based on a physical model where it is assumed that the propagation and scattering of acoustic waves is only due to variations in the compressibility and density. When fluctuations in the bulk and shear viscosities are present, an additional source of scattering occurs. In this case the appropriate wave equation is obtained by decoupling equations (4.33) and (4.34) for p, giving 1
\7 ( --;;.,\7. v
)
= p
aat2 v 2
-
ata [\7(0:\7. v) + 2\7· ((3\7v)].
(4.38)
4.9. ACOUSTIC FIELD EQUATIONS
4.9.2
105
Inhomogeneous Acoustic Wave Equations
As with electromagnetic problems, in acoustic imaging, we need to develop inhomogeneous wave equations of the form
or depending on whether we need to solve for the pressure wavefield p or the velocity wavefield v, respectively.
Non-viscous Medium Consider the case when the viscosity of the material is zero. By adding E}2p 1 "'0- -\7. \7p 2
8t
po
to both sides of equation (4.37), we can write
where r; - "'0
'YK = - - - ,
'Yp
P - Po =-
and Co
p
"'0
1 = -.
VPo"'o
Here, Co is the velocity at which acoustic waves propagate in a homogeneous material with a density Po and compressibility "'0. If we then introduce the Fourier transform
J 00
p(r, t)
=
~ 21f
P(r, w) exp(iwt)dt
-00
then the above wave equation becomes
where k = w / Co. The inverse scattering problem posed by this equation involves reconstructing 'YK and 'Yp by measuring appropriate properties of the scattered pressure field. This is a two parameter reconstruction problem.
Viscous Medium The problem becomes a little more complicated when we consider an inhomogeneous viscous material. In this case, by adding
8 -[\7(0:0\7· v)
8t
+ 2\7· (,8o\7v)J -
2
8 v - \7 ( -\7. 1 Pov) 8t 2 "'0
CHAPTER 4. FIELD AND WAVE EQUATIONS
106
to both sides of equation (4.38) and noting that for compression waves only
since v
= V'u
= 0, we obtain the
and V' x V'u
wave equation
(4.39) where '" -
'YI<
'Y;3
f3 - f30 f3' ao + 2 0
=
The parameter form
"'0
= -",-,
TO
Co
=
P - Po
-----p;;-'
=
'Yp
'Yo:
1 ~ and
=
TO
v Po"'o
a - ao ao + 2f30
= "'o(ao + 2(30).
is known as the relaxation time and may be written in the TO
+ 2f30
ao
=
poc6
and the quantity ao
4
+ 2f30 = (0 + 31/0
is the compressional viscosity. Introducing the Fourier transform
J
.
00
v(r, t) = -1 21T
V(r, w) exp(zwt)dt,
-00
equation (4.39) can be written as (1 + iWT)V' 2V
+ k 2V =
-iWT[V'b'\ V'. V)
-k2'Yp V
+ 2V'·
+ V'b"
V' . V)
b;3V'V)],
If we then use the relationship WT
=
kf!
where I! is the relaxation length (= T/ CO) and divide through by 1 + ikf!, we obtain (V'2 + e)V
=
-e'Yp V
1
+ 1 + iU V'bl< V' . V)
ikf!
-1
+ ikf![V'b'\ V'. V) + 2V'· b;3V'V)]
where
~=
k
vI + ikf!
4.9. ACOUSTIC FIELD EQUATIONS
107
Finally, we can obtain a scalar wave equation by substituting V for V'u and taking the divergence of each term to obtain
where
and
The field U is the acoustic dilatation. It represents the fractional change in the volume of a material due to the disturbance of an acoustic wave. The imaging problem posed by the wave equation above involves finding inverse solutions for the four acoustic scatter generating parameters 'Yp, 'YI<' 'Yo. and 'Y(3' This is a four parameter reconstruction problem. Any spatial or temporal inhomogeneity in the acoustic properties of a medium will scatter acoustic radiation. Thus, fluctuations in the density, compressibility, shear and bulk viscosity will produce scattering of different directional behaviours and can therefore, in principle, be characterized by their own directional scattering properties. A small density fluctuation (with regard to the wavelength of the field) within an otherwise homogeneous medium will exhibit a differential displacement amplitude when exposed to the oscillatory forces within an acoustic field. This motion is essentially equivalent to an oscillating particle in an otherwise stationary medium. Hence, a density fluctuation is basically a dipole source of sound emission (or scattering), exhibiting a nonisotropic directivity pattern with an intensity that depends on a differential force, i.e, the acoustic pressure gradient V'p. A fluctuation in compressibility will expand and contract quite differently to its surroundings when exposed to an acoustic field and is equivalent to a pulsating object within an otherwise stationary medium. Thus, a compressibility fluctuation is essentially a monopole source of sound emission which gives rise to isotropic scattering. This isotropic directivity pattern has an intensity dependence related to the acceleration of the source boundaries by the acoustic pressure, and is thus determined by the second time derivative 8 2pj8t2 . An additional source of acoustic scattering occurs with any variation in the viscosity of the medium which is a measure of the absorption of a wave by the medium. A small fluctuation in the viscosity produces a local change in the amplitude of the wavefront of a passing acoustic disturbance. This is equivalent to a small vibrating membrane within an otherwise stationary medium. Thus, a viscosity fluctuation represents a sound source having a relatively complicated directivity pattern (multi-pole scattering) but predominantly along the forward and backward directions of the acoustic wave vector with an intensity related to the vibrational velocity.
CHAPTER 4. FIELD AND WAVE EQUATIONS
108
4.10
Discussion
This Chapter has been concerned with investigating the field equations for the electromagnetic, elastic and acoustic fields. In each case, it has been shown how these equations can be reduced or decoupled to provide a linear inhomogeneous scalar wave equation (a Langevin equation) of the form
where L is an inhomogeneous linear differential operator which involves the 'scatter generating parameter' sets (')'e, 'YM' a) for the electromagnetic case (where the material may be composed of 'good conductors'), (')'p, 'Yo, 'Y(3) for the elastic case and (')'p, 'Y", 'Y"" 'Y(3) for the case of an acoustic field. The difference between the last two cases is determined by whether or not the material is compressible. Note that, in the case of acoustics. the field equations are a result of linearizing the hydrodynamic equations of motion whereas, in electromagnetism, the field equations are linear. Thus in acoustics, it is possible to incur a number of nonlinear effects that are not considered here. In electromagnetism, nonlinear behaviour can occur as a result of the polarization vector having a nonlinear relationship with the electric field vector. Also note that the wave equations derived here for the elastic and acoustic fields assume that the shear wave component is negligible. This is not always the case, particularly when elastic wavefields propagate through solids (seismic waves for example). The wave equations derived here are the result of trying to find a balance between developing a physical model that is relatively complete but 'simple' enough for the 'forward problem' (solving for the wavefield u given L) and the 'inverse problem' (solving for the material parameter sets given u) to become tractable using the analytical methods discussed in the following chapter - such is the nature of physics!
4.11
Summary of Important Results
The Langevin equation b(1)u(r, t) = -s(r,~)
where b(1)
= b(O) + L
D(O) is the homogeneous differential operator and erator.
L is the inhomogeneous op-
Maxwell's (microscopic) equations in eGS units "V·e=p "V·b=O "V x e
lab c
= ---
at
4.11. SUMMARY OF IMPORTANT RESULTS
109
18e j \7xb=--+c 8t c where e(r, t) is the (microscopic) electric field generated by a point charge, b(r, t) is the (microscopic) magnetic field, p(r, t) is the charge density, j(r, t) is the current density and c is the speed of light. Maxwell's (macroscopic) equations in SI units \7. EE = P \7. MH = 0 \7 x E 't"7
v x
H
8H
=
-Mat 8E
.
= Eat +J
where E(r, t) is the (macroscopic) electric field (volts/metre), H(r, t) is the (macroscopic) magnetic field (amperes/metre), j(r, t) is the current density (amperes/metre"), p(r, t) is the charge density (charge/metre''), E(r) is the permittivity (farads/metre) and M(r) is the permeability (henries/metre). Ohm's law j
= (TE
where (T is the conductivity in siemens/metre. Charge decay rate
p(t) = Po exp( -(Tt/E) Elastic field equation
82s P8t 2
=
\7. T
where s(r, t) is the displacement vector (metres), p(r) is the material density (mass per unit volume) and T(r, t) is the material stress tensor (force per unit volume). Acoustic field equations 8p \7. v = r.-
8t
\7p
8v
= p 8t -
\7. T
where v(r, t) is the velocity field (length/time), p(r, t) is the pressure field (force/area), T(r, t) is the material stress tensor (force/volume), p(r) is the density (mass/volume) and x(r) is the compressibility (area/force).
CHAPTER 4. FIELD AND WAVE EQUATIONS
110
Inhomogeneous wave equation for EM waves in a conductive dielectric
where
LE =
ikzoo-E + 'V(E· 'VInE) + 'V x
k2 ')'f E -
E - EO ')'f=--' EO
J-L
C'Yf.l'V x E);
J-Lo
')'f.l=--
J-L
E(r,w) - Electric field (Fourier transform of) E -
Permittivity
J-L - Permeability 0- -
Conductivity
Inhomogeneous wave equation for a compressional elastic wavefield in a non-compressible solid
where
Lo = C'Ya + 2
2')'(3)'V 2'V 2U + 2'VC'Ya + 2')'(3) . 'V'V2 u W
2
2
+'V ')'a'V U + 2"" 'V. ')'p'VU + 'V'V')'(3 : 'V'VU, CL
P- Po Po
')'p=--,
')'a
=
A - AO AO
+ 2J-Lo
and ')'(3
=
J-L - J-Lo AO
+ 2J-Lo
.
e(r,w) = 'V2 u (r , w) - elastic dilatation (Fourier transform of) P - Density
a, (3 - First and second Lame parameters, respectively Inhomogeneous wave equation for an acoustic pressure field propagating in a non-viscous compressible medium
where
P - Po
')'p=--.
P
per, w) - Pressure field (Fourier transform of) K, -
Compressibility
111
4.12. FURTHER READING
P - Density Inhomogeneous wave equation for a compressional acoustic wavefield propagating in a viscous compressible material
where
P - Po Po
'Yp= - - ,
'Ya
=
a - ao ao + 2f3o' 'Y(3
=
f3 - f30 ao + 2f3o
and
u(r, w) - Longitudinal velocity potential (Fourier transform of) U = \72 u - Acoustic dilatation K -
Compressibility
P - Density a, f3 - First and second Lame parameters, respectively f - Relaxation length
4.12
Further Reading
• Webster A G, Partial Differential Equations of Mathematical Physics, Stechert, 1933 • Stratton J A, Electromagnetic Thoery, McGraw-Hill, 1941. • Morse P M and Feshbach H, Methods of Theoretical Physics, McGrawHill, 1953. Atkin R H, Theoretical Electromagnetism, Heinemann, 1962. • Morse PM and Ingard K U, Theoretical Acoustics, McGraw-Hill, 1968. • Butkov E, Mathematical Physics, Addison-Wesley, 1973.
112
CHAPTER 4. FIELD AND WAVE EQUATIONS • Bleaney B I and Bleaney B, Electricity and Magnetism, Oxford University Press, 1976.
• Sander K F and Reed GAL, Transmission and Propagation of Electromagnetic Waves, 1986. • Evans G A, Blackledge, J M and Yardley P, Analytical Solutions to Partial Differential Equations, Springer, 1999.