A Fascinating Expedition of Complex Analysis

The rapid development of mathematics is largely due to the creation of complex numbers. Although complex numbers are just strange and seemingly meaningless notations for most people, complex numbers play important roles in engineering fields. A better understanding of complex numbers' geometric and algebraic structure enables us to study complex analysis better. Complex numbers convince scientists that our world is magical, full of wonderful insights, and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group under the operation of the multiplication and under the operation of addition. Then I visualize complex numbers by building a bijective connection between complex numbers and points on the complex plane. I also give several alternative forms of complex numbers, namely the trigonometric and general forms. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.

Acoustic wave propagation in liquid media containing many parallel air-filled cylinders is considered. A self-consistent method is used to compute rigorously the propagation, incorporating all orders of multiple scattering. It is shown that, under proper conditions, multiple scattering leads to a peculiar phase transition in the acoustic propagation. When the phase transition occurs, a collective behavior of the cylinders appears and the acoustic waves are confined in a region of space in the neighborhood of the transmission source. A novel phase diagram is used to illustrate the distinct phase states in the acoustic propagation.

There is a one academic year theoretical acoustics course for postgraduate students in the Department of Acoustic Science and Engineering of Nanjing University. The course introduces the physical principles and mathematical methods for acoustics in fluids, and the main objective is to deepen understanding of acoustical principle for postgraduate students in the field of acoustics following their undergraduate fundamental acoustics course. The contents include acoustic waves in ideal fluids, acoustic radiation in infinite space, acoustic scattering and diffractions, propagation and radiation of acoustic waves in dust, acoustic fields in enclosed space, acoustic waves in dissipative fluids, acoustic waves in layered fluid, acoustic waves in moving fluids, propagation of acoustic waves with finite amplitude, and effects generated by acoustic waves with finite amplitude. More details can be found in the book title “Principle of Acoustics in Fluids” published by Science Press in China in 2012.

The work presents a ray acoustic based analysis of the effect of ocean internal waves and its fluctuations on acoustic wave propagation. In the present work, it is shown that consideration of only perfect internal waves simplifies the real problem, disregarding the fluctuations of internal waves. The focus of the paper is to study numerically the influence of internal wave fluctuations and the role of an initial ray angle onto underwater acoustic propagation. In the present formulation the eikonal equation is considered in the form of a second order, nonlinear ordinary differential equation with harmonic excitation due to internal wave. The harmonic excitation is taken imperfect, i.e., with a random phase modulation due to Gaussian white noise. The amplitude and wavelength of the acoustic waves are used as the principle signal characteristics in bifurcation analysis. The regions of instability are identified using the bifurcation and phase diagrams. The effect of internal waves phase modulation is demonstrated by means of comparison of the numerical data obtained in the present work with data, obtained perfect internal wave. The preliminary analysis shows very strong dependence of acoustic propagation on intensity of fluctuations.

We investigate the excitation and propagation of acoustic waves in polycrystalline aluminum nitride films along the directions parallel and normal to the c-axis. Longitudinal and transverse propagations are assessed through the frequency response of surface acoustic wave and bulk acoustic wave devices fabricated on films of different crystal qualities. The crystalline properties significantly affect the electromechanical coupling factors and acoustic properties of the piezoelectric layers. The presence of misoriented grains produces an overall decrease of the piezoelectric activity, degrading more severely the excitation and propagation of waves traveling transversally to the c-axis. It is suggested that the presence of such crystalline defects in c-axis-oriented films reduces the mechanical coherence between grains and hinders the transverse deformation of the film when the electric field is applied parallel to the surface.

The generation of acoustic-gravity waves by meteorological heat sources in the troposphere and the propagation of these waves to the heights of the upper atmosphere are studied theoretically. Equations that describe separately the generation and propagation of acoustic and gravity waves by a local heat source are derived. The heat source of the waves is divided into partial sources of gravity and acoustic waves (AWs). The power of these partial sources is estimated and it is shown that the power of the sources differs by about a factor of two, regardless of the shape, size, and frequency of the heat source. It is shown that in the case of heat sources of the waves, the generation of gravity waves cannot occur without the corresponding generation of AWs: these waves are generated only in pairs. The splitting of the problem of waves from a heat source into problems of waves from gravity and acoustic sources is illustrated by the direct modeling of these waves. The application of the obtained results to the problem of parametrization of acoustic-gravity waves in general circulation and climate models is discussed.

We simulate the generation of acoustic and tsunami waves generated by submarine landslides using the linear model developed in [1]. The model is able to reproduce both acoustic and surface gravity waves generated by a moving source (e.g. earthquake, landslide) in a vertically stratified ocean.There are only a few studies that focus on the generation of acoustic waves by submarine landslides. In [2], the combined analysis of field data and simulations underline the presence of an interference pattern in the acoustic waves' spectrogram. The interference pattern has a time-varying bandwidth, which is a signature of the submarine landslide dynamics. In a previous work [3], we used the model developed in [1] to reproduce the interference pattern for a static source. Here we use the same model to simulate a submarine landslide in the 2D case. The simulations reproduce the time-varying bandwidth. They are then used to study the influence of two parameters on the acoustic spectrograms, namely landslide velocity and topography. Different velocity profiles available in the literature [4] are tested. For the topography, we use as reference the 2D case simulated in [1]. We also provide illustrations for the tsunami generation by the landslide. [1] Dubois J, Imperiale S, Mangeney A, Bouchut F, Sainte-Marie J. Acoustic and gravity waves in the ocean: a new derivation of a linear model from the compressible Euler equation. Journal of Fluid Mechanics. 2023;970:A28. doi:10.1017/jfm.2023.595[2] Caplan-Auerbach, J., Dziak, R. P., Bohnenstiehl, D. R., Chadwick, W. W., and Lau, T.-K. (2014), Hydroacoustic investigation of submarine landslides at West Mata volcano, Lau Basin, Geophys. Res. Lett.,  41,  5927– 5934, doi:10.1002/2014GL060964.[3] Dubois, J., Imperiale, S., Mangeney, A., and Sainte-Marie, J.: Simulation of the hydro-acoustic and gravity waves generated by a landslide, EGU General Assembly 2023, Vienna, Austria, 24–28 Apr 2023, EGU23-15937, https://doi.org/10.5194/egusphere-egu23-15937, 2023. [4] Farin, M., Mangeney, A., de Rosny, J., Toussaint, R., & Trinh, P.-T. (2019). Relations between the characteristics of granular column collapses and resultant high-frequency seismic signals. Journal of Geophysical Research: Earth Surface, 124, 2987–3021. https://doi.org/10.1029/2019JF005258

Acoustic wave propagation in free standing CVD diamond: Influence of film quality and temperature

The shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was <i>80</i>% and <i>117</i>%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible. The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows. There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range. One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper <i>Summa de arithmetica, geometria, proportioni et proportionalita</i> (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]). We can see, that already then, they were definitely important (complex numbers).

Tbe relationsbip between porosity and the speed of propagation of acoustic waves in fluid-saturated porous rocks as measured by the Sonic log and by ultrasonic techniques is analyzed. Biot's continuum theory is used to explain the difference in acoustic wave propagation between a dry and a liquid-saturated porous material. The porosity is a variable in this theory. However, the acoustic wave propagation in the dry rock depends too on porosity, and this dependence is not predicted by the theory. Frequently in dry sandstones, a nearly linear relationsbip between reciprocal acoustic wave velocity and porosity is observed in the low-porosity range. The physics behind this behavior is outlined. An empirical relationship of the form, 1/V ~ A + B ø, applies accordingly for many porous dry rocks, provided the porosity is the only variable. The presence of a liquid in the pores changes the value of B, and this change is found to be in agreement with the Biot theory. The time-average relation introduced some years ago results in an equation of the same type 1/V = ø/Vf + (1 - ø)/Vr - but is not based on a sound physical picture. Still, this relation sometimes predicts approximately correct A and B values. Carbonate rocks with their complicated pore structures sometimes show a different relationship between wave velocity and porosity, unfavorable for log interpretation. Examples are presented. The simultaneous presence of calcite, dolomite and anhydrite, with their different grain densities and matrix compressibilities, complicates acoustic-log interpretation in carbonate rocks still further. Other complicating effects of acoustic-log interpretation are discussed. They are related to the influence of shale streaks and natural fractures on the average wave velocity observed by the logging tool and to the effect of adsorption phenomena on wave propagation in unstressed rocks particularly in sandstones.

By virtue of the diffraction-free propagation of surface acoustic waves (SAWs) on a sphere, we observed multiple round-trips of as many as 100 turns. After experimental verification of the beam shapes, we proposed ball acoustooptic devices utilizing the focussing SAWs.

The purpose of this paper is the generalization of structure factor for rods by polygon section in two dimensional phononic crystals. If we use the plane wave expansion method (PWE) for the propagation of acoustic waves in 2D phononic crystals, structure factor will be an important quantity. In order to confirm the obtained relations, we have calculated the band structure for XY and Z vibration modes in 2D phononic crystals and the propagation of bulk acoustic waves (BAW) are considered. In addition, the effect of sides’ number on the band structure and the complete band gaps width are investigated. Phononic crystals studied in this paper are composites medium of a square lattice consisting of parallel nickel rods embedded in epoxy. The frequency is calculated by PWE in the condition of elastic rigidity to the solid inclusions. The results showed that, when the section of rods have 2n+2 (n is even) and 2n+1ሺn א Nሻ by increasing sides number of the rod sections, the bands of XY mode shift to lowerfrequency, the bands are smoother and the width of the band gap increases, but the band of Z mode has not changed by n variations. Moreover, when the section of rods have 2n+2 (n is odd) by increasing sides number of rod sections, the band structure of XY mode changes slightly and the width of the complete band gap is decreased. This confirms the effect of lattice symmetry on the complete band gap width. But, the band structure of Z mode has not changed.

Based on the modification of the radial pulsation equation of an individual bubble, an effective medium method (EMM) is presented for studying propagation of linear and nonlinear longitudinal acoustic waves in viscoelastic medium permeated with air bubbles. A classical theory developed previously by Gaunaurd (Gaunaurd GC and Uberall H, J. Acoust. Soc. Am., 1978; 63: 1699–1711) is employed to verify the EMM under linear approximation by comparing the dynamic (i.e. frequency-dependent) effective parameters and an excellent agreement is obtained. The propagation of longitudinal waves is hereby studied in detail. The results illustrate that the nonlinear pulsation of bubbles serves as the source of second harmonic wave and the sound energy has the tendency to be transferred to second harmonic wave. Therefore the sound attenuation and acoustic nonlinearity of the viscoelastic matrix are remarkably enhanced due to the system's resonance induced by the existence of bubbles.

A branch of mathematical analysis called complex analysis studies the functions of complex numbers. It is also sometimes referred to as the theory of functions of a complex variable. Complex analysis has been used extensively in mathematics, physics, and engineering over the years, particularly in the areas of algebraic geometry, fluid dynamics, quantum mechanics, and other related fields. It dates back to the 16th century, when Italian mathematicians Girolamo Cardano and Raphael Bombelli first noticed complex numbers while attempting to solve a particular algebra, and was later developed by Cauchy and Riemann in the 19th century. The development of complex numbers has a lengthy history. Mathematicians have advanced the discipline of mathematics significantly after thousands of years of development. During this time, mathematicians also found a great deal of previously unknown mathematical information and proved formulae and phenomena that had previously been impossible to verify. And it covers complex numbers as well as some of the mathematics related to them. In this paper, the complete discovery of complex numbers from cubic equation to topology of complex numbers is detailedly revealed.

What is a complex number? The question may be well solved today, but about 400 years ago it would have baffled many mathematicians. For example, how was it possible to define the square root of a negative number, which was an "impossible" quantity at the time, but they were all real quantities in mathematics. Complex analysis is now recognized as a fundamental part of mathematics because of its many applications in physics and engineering as well as its connections to other branches of mathematics. By solving some equations of more than one degree, complex numbers were produced. Most mathematicians focused on real analysis and physics applications until the nineteenth century. Cauchy was the first figure to make a considerable effort in a challenging examination. He calculated a number of complex integrals using his integral formula, establishing the foundation for defining the operations and characteristics of complex analysis. Gauss furthered the development of the complex number geometrical theory. The first part of this essay focuses on the history discovery of mathematical and geometrical aspects of complex numbers. The topological features of the complex plane are then carefully understood after that. The beautiful characteristics of complex functions and series are finally compiled based on the topology of the complex field.