









 theory of sound  analytical 


Description of bounded beams

Most research groups only use the Fourier method to describe bounded sound beams. This method is very robust. However there are situations where it is better to use other methods. Therefore we have further developed the method based on the superposition of infinite inhomogeneous waves. Several numerical improvements have been found. Also the description of bounded beams in 3D has been established within our group. The figure to your right shows the amplitude profile of a gaussian beam having two different beam widths.



liquid characterization

In the beverage industry and in the dock industry, it is often necessary to determine the kind of liquid that is contained in a small can or in a huge container. For that reason we have developed a model (combined with successful experiments) which determines the liquid by means of studying the reflected bounded beam pattern when leaky Lamb waves are generated in the container skin. This method is therefore applicable for small cans as well as for huge containers, because we do not rely on sound/echo techniques. The following figure shows the incident beam profile (black) and also the beam profile when the liquid is water (blue) or oil (red).



polar scan simulations

In ultrasonic polar scans, the reflected or transmitted amplitude is recorded as a function of each possible angle of incidence. A schematic is shown in the following figure. If this amplitude is plotted in a polar diagram (θ being the polar angle, the radius being the angle φ), then we speak about a ‘polar’ scan. It is known that such a scan is sensitive to the stiffness of the composite, to the resin fraction, porosity, …



The influence of stress on polar scans

The figure to your right shows the difference for polar scan on a Ti isotropic laminate in the absense (top) and presence (bottom) of stress in the direction corresponding to θ=0. It is seen that the pattern becomes slightly directional dependent.



Polar scans on composites






radiation mode theory

The radiation mode theory is a major counterpart of finite element methods and finite difference methods. The drawback is its enormous complexity. The advantage is that analytical predictions are possible and that therefore the theory does more than simply predicting phenomena by means of simulations.

Inhomogeneous Wave Theory

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.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, hencenotion 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. The 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 S. 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. Another 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. 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. Nowadays the team always considers the most general solution of the wave equation, therefore incorporating inhomogeneous waves, though specific studies on the behaviour of inhomogeneous waves is not further undertaken.

complex harmonic waves

The frequency of plane waves can be considered complex valued. This results in a harmonic oscillation that is damped in time. It has been examined how the presence of a complex frequency can change the focal length of a focused bounded beam. It is also shown that this phenomenon is not only true in the mathematical case of complex harmonic waves, but also in the realistic case of sound being limited in time and having temporal characteristics that correspond, within a limited time window, to the complex harmonic wave. This is very important, because it shows that adapting the temporal variation of the amplitude (a variation that is much more slowly than the oscillatory  harmonic  variation) makes it possible to change the focal length. Therefore, expensive phased array technology becomes unnecessary and makes the treatment of cancer by means of focused ultrasound better achievable, also for developing countries. The interaction of complex harmonic plane waves with a periodically corrugated surface has also been studied. It is shown that their ability to stimulate leaky Rayleigh waves on a corrugated surface is similar to the ability of harmonic inhomogeneous waves to stimulate such waves on a smooth interface. Because the experimental generation of harmonic inhomogeneous waves is more complicated and less flexible than the generation of complex harmonic waves, the combination with a diffraction grating therefore offers an excellent alternative. The theoretical model developed here is based on the well know and experimentally verified complex harmonic wave theory and the famous Rayleigh decomposition of the diffracted field. A study of the influence of a complex frequency on the generation of Scholte  Stoneley waves is performed as well.

Druckbare Version







