Inhomogeneous waves
The problem of the square root of a negative number, already
existed at the beginning of the Christian calendar, see for example Stereometrica, by Heron of Alexandria. Around 800
years later, the idea of the existence of a solution was crushed by the Indian
mathematician Mahavira, who stated As
in the nature of things, a negative is not a square, it has no square root.
Girolamo Cardano was the
mathematician who first discovered a solution, in 1545, though he thought his
discovery was fictitious and useless. The further development and spreading of
the idea of complex numbers, was the result of brilliant scientists such as
Caspar Wessel, Rene Descartes, Gottfried Wilhelm von Leibniz, Leonard Euler and
Carl Friedrich Gauss.
In the theory of waves, and in particular the theory of acoustics and
ultrasonics, a complex number has been introduced after the discovery by
Leonhard Euler (in Introductio in analysin
infinitorum, 1748) that an exponential function,
containing an imaginary argument, is analytically equal to the combination of a
cosine in the real space and a sine in the imaginary space, hence this notion
enables the addition of an imaginary twin to each real wave phenomenon and
analyze the problem in the complex space. After this analysis, it is possible
to extract the real solution from the complex result. The merit of this
procedure is the fact that all mathematical expressions in the theory of waves,
are contracted and simplified after the transformation into the complex space.
An acoustic wave is oscillatory in time and in space. From mechanical
considerations, it is possible to obtain the wave equation, relating the
temporal properties to the spatial properties of sound. The simplest solitary
solution of this equation, is the so called
homogeneous plane wave. It is characterized by a harmonic function like the one
obtained by Euler, possessing a real argument. This involves a real wave vector
and a real frequency. Even though pure plane waves are the simplest solitary
solutions of the wave equation, they are able to generate more complicated
sound fields by means of a superposition in the framework of the Fourier
theory. Nevertheless, there are more solitary solutions possible. Most of them
will probably never be discovered.
One,
very useful, solitary solution is the inhomogeneous plane wave. It is a wave
that contains temporal and spatial harmonic as well as spatial inharmonic
amplitude variations. Being a solution of the wave equation, it fulfills strict
relationships between the harmonic and the inharmonic amplitude variations,
through the so called dispersion relation. If Euler’s notation is followed,
inhomogeneous plane waves can easily be described as pure homogeneous plane
waves, having a complex wave vector.
If also the temporal variation contains inharmonic contributions, a complex
frequency is involved and the wave is called complex harmonic and more
specific complex harmonic homogeneous or complex harmonic
inhomogeneous, depending on the lack, respectively presence of an imaginary
part of the wave vector.
Inhomogeneous waves are interesting phenomena, because their propagation
properties differ from pure homogeneous plane waves. Also their polarization is
different. Nevertheless, such kind of waves have always been considered
mathematical artifacts that are only generated experimentally (along
interfaces) under certain conditions during scattering.
It was not until Claeys and Leroy discovered that inhomogeneous waves are natural
building blocks of bounded beams, resulting in a physical connection between
the Schoch effect and the generation of Rayleigh
waves on smooth surfaces, that the world of acoustics realized the physical
importance of inhomogeneous waves. Inhomogeneous waves are present inside
bounded beams and they are the origin of the stimulation of surface waves.
Later, inhomogeneous waves have been generated experimentally and their
excellent ability to stimulate surface waves, has been
validated.
Because Oswald Leroy has been my advisor (together with Joris Degrieck) when I
was a PhD student, a study of inhomogeneous waves has been almost naturally a
mandatory requirement before studying additional topics. While studying this
wonderful subject, I have written a historical overview of the concept of
inhomogeneous waves. It can be found in IEEE-UFFC and in my list of
publications.
The decomposition of bounded beams into inhomogeneous waves is mathematically
well established, apart from numerical instabilities, for which there have not
yet been found ultimate solutions. Two enhancements of the stability of the
decomposition have been developed lately.
As mentioned earlier, inhomogeneous waves have been generated experimentally
and it has been shown that their behavior corresponds to what is mathematically
predicted. Nevertheless, the experimentally generated inhomogeneous waves only
correspond to mathematical inhomogeneous waves within a limited spatial
interval. This is, of course, due to the finite dimensions of transducers that
generate such waves. An unavoidable question that immediately arises is, of
course, "how is it possible that such bounded inhomogeneous waves behave
like infinite inhomogeneous waves? It is possible to simulate the
behavior of bounded inhomogeneous wave by approaching them as a summation of
infinite plane waves, in the framework of the Fourier theory. But, this is
merely a simulation, not a scientific answer to the posed question.
Furthermore, it is a connection between homogeneous plane waves and
bounded inhomogeneous waves. Besides, simulating a phenomenon is quite
different from understanding it. The real answer is found in one of my papers,
where the mathematical link between infinite inhomogeneous plane waves and
bounded inhomogeneous waves is found through the
In the past, the study of bounded beams in terms of inhomogeneous waves, has
always been limited to the 2D case, because it was impossible to expand a 3D
bounded beam into inhomogeneous waves. This problem has been solved and the
concept is outlined in a paper in Ultrasonics.
Also a study of inhomogeneous waves has been presented in mud layers, in
diffraction phenomena and in piezoelectric media. Most of these additional
studies have not been included in the historical survey in IEEE-UFFC, as
mentioned above.
Inhomogeneous waves and
bounded beam
The
history and properties of ultrasonic inhomogeneous waves
Inhomogeneous waves are generalized plane
waves. They are described as classical plane waves, except that the wave
parameters, such as the wave vector, are complex valued. The Americans have
been the first to publish features of this kind of waves, but the theory was
ultimately developed in
The
principle of a chopped series equilibrium to determine the expansion
coefficients in the inhomogeneous waves decomposition of a bounded beam
Fig.: The horizontal axis is the distance to
the center of the beam, divided by the Gausian beam
width, whereas the vertical axis is the amplitude. This figure shows an extreme
situation of a badly conditioned optimization. The upper part presents the
exact gaussian profile as a dashed line and the
numerically approached profile, applying classical methods, as a solid line.
The approximation is quite different within the body of the profile and results
in exponentially growing ‘tails’ outside the profile where a zero value is
expected. The lower part of this figure shows the same result, though applying
the chopped series equilibrium principle. Note that the result is much
improved!
Contrary to the classical Fourier method, decomposing a bounded beam into pure plane waves, having different amplitudes, phases and directions, in the inhomogeneous wave method, a bounded beam is decomposed into inhomogeneous waves, having different amplitudes, phases, but having equal propagation directions. During the historical development of the inhomogeneous wave theory, it has been revealed that the method is correct within a limited distance along the propagation direction and within a limited range along the width of the bounded beam. The limited range of validity along the propagation direction is not crucial, because inhomogeneous wave are primarily considered in the case when a bounded beam interacts with a plane interface at relatively small angles. Nevertheless, the limitation along the width of the bounded beam is very important, because whenever strong beam shifts or beam profile deformations are induced in the case of surface wave generation, the effect can possibly occur in areas where the bounded beam, approximated by means of a superposition of inhomogeneous waves, is badly conditioned. The primary reason for the inhomogeneous wave method being badly conditioned along the width of the bounded beam, is the fact that the optimization is performed by means of exponential functions. This is a delicate question, because a very small numerical error results in a very large difference between the numerical approximation and the exact profile, at significant distances from the origin.
This section presents a technique, based
on a chopped
A
useful analytical description of the coefficients in an inhomogeneous wave
decomposition of symmetrical bounded beam
The previous section is primarily devoted to numerical problems in the inhomogeneous wave theory. This is due to the obvious fact that analytical expressions for the expansion coefficients in the decomposition of bounded beams into inhomogeneous waves have never been found. Section IV.C explains how an analytical expression can be obtained and presents the result.
If the spatial description of a bounded beam is given by:
Then, the analytical expression for the expansion coefficients is given by:
with
The
Fig.: The horizontal axis denotes the distance along the wave
front, whereas the vertical axis denotes the amplitude. The dashed line
corresponds to the profile of an infinite inhomogeneous wave, whereas the solid
line corresponds to the profile of a bounded inhomogeneous wave.
A
beautiful aspect of the theory of inhomogeneous waves, is the fact that several
features emerge that do not necessarily coincide with human intuition, but that
are experimentally verifiable. Nevertheless, those specific experiments are
performed by means of bounded inhomogeneous waves instead of infinite
inhomogeneous waves. The reason is, of course, the finite dimension of
transducers. Therefore, the correspondence between theory and experiment is not
obvious. The cause of this correspondence is revealed in this section where, by
means of the
The representation of 3D Gaussian beams by means of inhomogeneous waves
Fig. : The profile of a 3D quasi-Gaussian
beam, approached by means of a superposition of inhomogeneous waves.
The development of the inhomogeneous wave theory has been accompanied by the discovery that bounded beam can be represented as a superposition of inhomogeneous waves. The method of determining the expansion coefficients in the decomposition, was based on Prony’s technique, transforming an equation containing exponential functions, into a polynomial equation. After identification with Laguerre polynomials, the expansion coefficients can be determined. Nevertheless, the method has thus far been limited to beams that are bounded in only one direction. This is a serious shortcoming, because it limits application of the inhomogeneous wave theory to more realistic situations where sound beams are bounded in two directions. The current section introduces a novel method to determine the expansion coefficients, that is also applicable in the latter situation of realistic beams bounded in two directions.