MODELLING OF TRANSIENT ELECTROMAGNETIC AND ACOUSTIC WAVES IN
VERTICALLY LAYERED MEDIA WITH DEPTH-DEPENDENT ATTENUATION
Martin D. Verweij
Faculty of Information Technology and Systems
Laboratory of Electromagnetic Research
This research project deals with the development of methods for the analysis of the transient
electromagnetic (EM) and acoustic wave field in a medium that may be modelled by constitutive
parameters that change in the vertical direction. Some of these parameters describe the loss behaviour
of the medium. Both theoretical and numerical aspects are taken into account. The objectives of the
theoretical investigations are a.) to form a sound basis for the numerical work and b.) to reveal
characteristic features of the electromagnetic and acoustic wave field in a vertically layered, lossy
medium. The aim of the numerical implementations is an efficient determination of the electromagnetic
and acoustic wave field in this kind of medium. Both aspects are important in view of the development
of, e.g., efficient and accurate imaging and inversion techniques.
Introduction
The characterisation of the shallow subsurface (0.25 m - 10 m) of the Earth by means of time-domain
electromagnetic (EM) pulses is of fundamental importance in environmental monitoring, civil
engineering, and archaeology. Existing modelling programs for the subsurface are based on the scalar
high-frequency ray approximation, where attenuation due to the lossy ground is simply taken into
account by an exponential decay. At many locations, for example in The Netherlands, the losses are so
high that this approach is inadequate, and a method based on the full vectorial electromagnetic wave
field equations is required. This motivates the development of fast and accurate forward
electromagnetic modelling methods that may serve as the basis for new electromagnetic imaging
processes. Moreover, it is expected that the performance and applicability of these imaging processes
will improve significantly when a vertically layered background medium with depth-dependent losses
is employed. The full vectorial analysis of the transient electromagnetic wave field in this kind of
media may be performed in a particular efficient way by employing the integral transformation
approach. The key steps in this approach are: reducing the dimensionality of the problem by application
of suitable integral transformations with respect to those co-ordinates in which the medium is invariant;
obtaining the solution of the resulting transform domain problem; and transformation of this solution
back to the space-time domain. Certain combinations of techniques – like the combination of WKBJ
asymptotics and the Cagniard-De Hoop method of inversion – yield very fast overall methods. Often
the feasibility of a method is investigated by first applying it to the more simple acoustic case.
Results in 2002
The relations between the spatial averages of microscopic electromagnetic field quantities and the
macroscopic electromagnetic field quantities in a material medium have been investigated. This is
important when deriving the constitutive behaviour or the force density from the interactions between the
particles and the microscopic electromagnetic field present in a given medium. After these basic
interactions have been determined, spatial averaging over a representative elementary domain is
performed. To arrive at relations between quantities at the macroscopic level, the identification of the
averaged microscopic quantities in terms of macroscopic quantities is the final and crucial step. This
identification turns out to be nontrivial, since it depends on the model for the electric and magnetic dipoles
in the medium. The approach has been inspired by Lorentz, who in 1902 showed that the well-known
Maxwell equations for the macroscopic electromagnetic field in matter may be deduced from postulated
Maxwell equations for the microscopic electromagnetic field. The influence of the matter is represented
by small particles that give rise to a microscopic charge density and/or a microscopic electric current
density. As such, these electrons cause the rapid spatial changes of the microscopic field. Depending on
their properties, these particles perform a certain role: they either cause conduction, polarisation or
magnetisation. The important step in Lorentz’s article is the spatial averaging of the unobservable
microscopic quantities and the identification of their averages in terms of the observable macroscopic
quantities. He shows that with the proper identifications, one indeed arrives at the macroscopic Maxwell
equations. Following Lorentz, we have applied a spatial averaging of several microscopic Maxwell
equations, and have identified the averages of the corresponding microscopic field quantities in terms of
the macroscopic field quantities. For the magnetic field it has turned out that the identifications depend on
the microscopic field quantity and the dipole model under consideration. It has turned out that it is most
natural to apply the microscopic magnetic flux density in case of the electrical current dipole model and
the microscopic magnetic field strength in case of the magnetic charge model, since the averages of the
microscopic field quantities are then equal to the corresponding macroscopic field quantities. The above
research has been presented at the 27th General Assembly of the International Union of Radio Science,
Maastricht, The Netherlands
As a former co-supervisor of the thesis project ‘Detection and characterisation of defects in steel
constructions,’ the collaboration with Dr. M.C.M. Bakker (Civil Engineering) on ultrasonic inspection
techniques has been continued. This has resulted in a paper in which two models of ultrasonic transducers
have been validated by comparison of the modelled and measured wavefield patterns, directivities, and
time domain signals. The thesis project has continued as a STW project, in which participation in the
‘Users Group’ has taken place.
Research plan for 2003
The theory of transforming the solution of a wave problem into the solution of anything in between a
diffusion problem and a wave problem will be generalised and published. Collaboration with Dr.
M.C.M. Bakker on ultrasonic inspection techniques is continued. The task of deriving the
electromagnetic Green’s tensors for a multilayer medium, announced for 2002, will be part of the thesis
research of a M.Sc. student. Application of these tensors will be demonstrated in a nano-optical
application. The modelling of the sound field in a non-linear and dispersive acoustic medium is started
as a new research topic.
Publications and presentations in 2002
Bakker, M.C.M, M.D. Verweij; An approximation to the far field and directivity of elastic wave
transducers, J. Acoust. Soc. Am., Vol. 111, pp. 1177-1188, 2002.
Manea, T.E., M.D. Verweij, H. Blok; Electrodynamics in deformable solids for electromagnetic forming –
Deformation of thin cylindrical shells, Proceedings of the Second International Conference on
Advanced Computational Methods in Engineering, Liege, Belgium, 28-31 May 2002.
Manea, T.E., M.D. Verweij, H. Blok; The importance of the velocity term in the electromagnetic forming
process, Proceedings of the 27th General Assembly of the International Union of Radio Science,
Maastricht, The Netherlands, 17-24 August 2002.
Verweij, M.D.; Four ways of deducing Maxwell’s equations from their microscopic counterparts –
Lorentz’s theory of electrons revisited, Proceedings of the 27 th General Assembly of the International
Union of Radio Science, Maastricht, The Netherlands, 17-24 August 2002.
Supervision of M.Sc. theses
Lanen, E.P.A.van; An approximation to the elastodynamic far field in a plate loaded by an angle beam
transducer.
Huijssen, J.; Modeling of nonlinear medical ultrasound from phased array transducers.
Co-supervision of Ph.D. theses
Manea, T.E.; Electrodynamics in deformable solids for electromechanical power conversion and
forming.
Henry, F.A.G.; Fracture monitoring by borehole acoustic measurements.