CHAPTER II: Introduction
- 50 -
CHAPTER III: English Summary
Chapter III English Summary
The style of this dissertation is such that each section, of every chapter, forms a unit that
can possibly be understood independent from the other parts. The reason is that people who
are interested in one certain section, do not necessarily like to read the whole work.
The current chapter presents a short summary of the remaining chapters. The summary
mainly focuses on listing the results, rather than presenting the entire study itself.
Chapter IV Inhomogeneous waves and bounded beams
Section IV.A : The history and properties of ultrasonic inhomogeneous
waves
• Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency
Control (Imp. Fact. 1.595 ;SCI-index, Engineering – electrical & electronic, rank:46/205)
• Invited Oral presentation at: plenary session, ‘VII International Conference for Young Researchers
on Wave Electronics and its Applications in Information and Telecommunication Systems’, St
Petersburg, Russia , 12-15 September 2004
• Oral presentation at ‘75th Anniversary Celebration of the Acoustical Society of America (147th
meeting of the Acoustical Society of America), Sheraton New York Hotel and Towers, New York,
New York, USA, 24-28 May 2004
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 Europe. This development was boosted after it was
shown that inhomogeneous waves form a natural stimulus for surface waves. Nevertheless,
the idea of inhomogeneous waves remained a mathematical ‘artifact’, even though
experiments showed how these waves can be generated and showed that their behavior
corresponds to theory. In this chapter, a historical overview is presented, together with an
overview of the properties of inhomogeneous waves. This overview is likely to form the
final breakthrough of inhomogeneous waves into the world of acoustics. Furthermore, for
the first time in history, it is shown how the complex Lame parameters can be fully
expressed in terms of intrinsic acoustic parameters such as damping and sound velocity.
Section IV.B : The principle of a chopped series equilibrium to determine
the expansion coefficients in the inhomogeneous waves
decomposition of a bounded beam
• Nico F. Declercq, Joris Degrieck, Oswald Leroy, "The Principle of a Chopped Series Equilibrium to
Determine the Expansion Coefficients in the Inhomogeneous Waves Decomposition of a Bounded
Beam", Acta Acustica United with Acustica 89, 1038-1040, 2003. (Imp. Fact. 0.346; SCI-index,
Acoustics, rank:21 /28)
• Oral presentation at ‘the 8th Western Pacific Acoustics Conference (Wespac8)’, Melbourne,
Australia, 7-9 April, 2003.
- 51 -
CHAPTER III: English Summary
Fig. III.1 (identical to Fig IV.B_2): 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
- 52 -
CHAPTER III: English Summary
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 Taylor series representation of
the exponential functions, applying the optimization procedure for these chopped series
and attributing the obtained coefficients to the original exponential functions. It is shown
that there exists an optimum series length for that purpose. The result is an improvement of
the optimization of the approximation of bounded beams by means of inhomogeneous
waves. The principle of this technique can possibly be very important in other fields of
numerical analysis as well.
Section IV.C : A useful analytical description of the coefficients in an
inhomogeneous wave decomposition of symmetrical bounded
beam
• Nico F. Declercq, Joris Degrieck, Oswald Leroy, "A useful analytical description of the coefficients
in an Inhomogeneous Wave Decomposition of a symmetrical bounded beam", Ultrasonics 43(4),
279-282, 2005 (Imp. Fact. 0.844; SCI-index, Acoustics, rank:11 /28)
• Oral presentation at ‘the 8th Western Pacific Acoustics Conference (Wespac8)’, Melbourne,
Australia, 7-9 April, 2003.
Section IV.B 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:
()
()
()
2
1
2
, expexp
,0
2
2
N
x zAi
nn n n
v
nN
ω
ϕδβ
⎛⎞
⎛
⎜⎟
⎜
=+ +
∑
=−
⎝⎠
zβ
Then, the analytical expression for the expansion coefficients is given by:
() ()
!!
(1) (1)
!!! !!!
0
Nm
mm
r
AI
mnnn mrrr
mn r
=− −
∑∑
−−
=
with
() ()
1
exp exp 1
xx
I rfx
r
p p
+∞
⎡⎤
⎛⎞
=−++
∫
⎢⎥
⎜⎟
⎝⎠
⎣⎦
−∞
dx
- 53 -
CHAPTER III: English Summary
Section IV.D : The Laplace transform to describe bounded inhomogeneous
waves
• Nico F. Declercq, Joris Degrieck, Oswald Leroy, "The Laplace transform to describe bounded
inhomogeneous waves", J. Acoust. Soc. Am. 116(1), 51-60, 2004. (Imp. Fact. 1.310; SCI-index,
Acoustics, rank:7 /28)
• Oral presentation at ‘the 145th Meeting of the Acoustical Society of America’, Nashville
Convention Center, Nashville, Tennessee, USA, 28 April - 2 May 2003.
Fig. III.2 (identical to Fig. IIIV.D_5): 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 Laplace transform, the physical connection
between infinite inhomogeneous waves and their bounded counterparts, is unveiled. It is
shown that only one of the inhomogeneous waves within the superposition, is responsible
for the global behavior of bounded inhomogeneous waves, whereas the other waves are
merely responsible for the edge formation.
- 54 -
CHAPTER III: English Summary
Section IV.E : The representation of 3D Gaussian beams by means of
inhomogeneous waves
• Nico F. Declercq, Joris Degrieck, Oswald Leroy, "The inhomogeneous wave decomposition of 3D
Gaussian-like bounded beams", Ultrasonics 42, 273-276, 2004. (Imp. Fact. 0.844; SCI-index,
Acoustics, rank:11 /28)
• poster presentation at ‘Ultrasonics International 2003’, Granada, Spain, 30 June- 3 July 2003
Fig. III.3 (identical to Fig. IV.E_4): The profile of a 3D quasi-Gaussian beam, approached by means of a
superposition of inhomogeneous waves. The tails that appear are comparable to those in Fig. III.1.
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.
- 55 -
CHAPTER III: English Summary
Section IV.F : Focal length control of complex harmonic and comlex pulsed
ultrasonic bounded beams
• Accepted for publication in J. Appl. Phys. (Imp. Fact. 2.281; SCI-index, Physics-Applied,
rank:13/76)
• Poster presentation at ‘VII International Conference for Young Researchers on Wave Electronics
and its Applications in Information and Telecommunication Systems’, St Petersburg, Russia , 12-
15 September 2004
Fig. III.4 (identical to Fig. IV.F_5): Example of a square profile focused bounded beam, propagating
from the right hand side to the left hand side. The white oval spot is the focus.
The ability of ultrasound, on the one hand to coagulate thrombocytes and on the other
hand to transform into heat, makes it an established tool for medical applications. In the
case of focused ultrasound, the generated effects can be induced locally. At the moment,
focused beams are commonly formed by means of phased array technology because of
flexibility, especially when tuning the focal distance. This technology however is
extremely expensive, which is an important and ‘pity’ disadvantage in developing
countries. Principally, the focal distance of a single element focusing transducer, must also
be tunable, though by changing the frequency, which most often results in a severe
diminishing of the generated sonic amplitude, an effect resulting from the inability of most
transducers to generate vibrations different from their first harmonic or odd multiples.
In this section, it is shown that it is possible to tune the focal distance of a single
element focusing transducer, by changing the input amplitude in a distinct manner. The
theory behind the effect is the complex harmonic wave theory. It is shown that the effect is
generated, not only in the theoretical case of infinite signals, but also in the realistic
situation of short signals. The result of this research may be important in the fabrication of
- 56 -
CHAPTER III: English Summary
affordable medical equipment.
Section IV.G : On the existence and the excitation of a new kind of leaky
surface waves
• Oral presentation at ‘Tenth International Congress on Sound and Vibration’, Stockholm, Sweden, 7-
10 July 2003
By formulating the continuity conditions at a solid-liquid interface, it is possible to
obtain the characteristic equation of surface waves. This equation relates the material
parameters to the velocity of surface waves. Practically, the solution corresponds to a
complex pole of the reflection coefficient of incident sound on the interface. A complex
velocity then corresponds to leaky surface waves. Nevertheless, the result is also a function
of the appropriate sign choice for the normal components of the wave vectors. By applying
novel knowledge concerning the sign choice, obtained from experiments by Marc
Deschamps, it is possible to established the well-known existence of leaky Rayleigh
waves, a kind which radiates energy into the liquid, but also predict a novel type of surface
waves, that leaks energy both into the liquid and into the solid.
Chapter V The interaction of sound with continuously
varying layers
Section V.A : The interaction of inhomogeneous waves and Gaussian
beams with mud in between a hard solid and an ideal liquid
• Nico F. Declercq, Oswald Leroy, Joris Degrieck, Jeroen Vandeputte,
"
The interaction of
inhomogeneous waves and Gaussian beams with mud in between a hard solid and an ideal liquid",
Acta Acustica United with Acustica 90, 819-829, 2004 (Imp. Fact. 0.346; SCI-index, Acoustics,
rank:21 /28)
In the shipping and dredging industry, it is of primordial importance to estimate the
nautical depth in rivers and harbors. The bottom is often covered by a layer of mud that is
not homogeneous and consists of a fluid-like upper layer, a solid-like lower layer and a
transition zone in between. The transition zone is the real nautical bottom. When taking the
critical decision to enter a ship into a harbor or to remain anchored, knowledge of the exact
nautical depth is crucial. Classical echo-sounding is not capable of detecting the nautical
bottom, because it is based on normal incident sound and does not generated shear waves
in the mud. The nautical bottom is characterized by an abrupt change of the shear
parameters but not the compressional parameters. Therefore only the transition between
water and fluid-like mud is detectable by means of classical echo-sounding, as well as the
transition between the solid-like mud layer and the hard or sandy underground.
The present section describes a newly developed model that describes the interaction of
sound in a continuous system of layers in mud. The propagation of bounded beams is
simulated and it is shown that oblique incident bounded beams produce a reflected sound
pattern that is susceptible to the position of the nautical depth. Therefore a method is
discovered that is indirectly capable of detecting the position of the nautical depth.
- 57 -
Ricevi informazioni sui nostri servizi, sulle offerte e non perdere news e consigli su università e lavoro.
