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Open AccessArticle

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Acoustics 2022, 4(4), 885-893; https://doi.org/10.3390/acoustics4040053 - 10 Oct 2022

Viewed by 1583

The second-order partial differential wave Equation (Cauchy’s first equation of motion), derived from Newton’s force equilibrium, describes a standing wave field consisting of two waves propagating in opposite directions, and is, therefore, a “two-way wave equation”. Due to the second order differentials analytical [...] Read more.

The second-order partial differential wave Equation (Cauchy’s first equation of motion), derived from Newton’s force equilibrium, describes a standing wave field consisting of two waves propagating in opposite directions, and is, therefore, a “two-way wave equation”. Due to the second order differentials analytical solutions only exist in a few cases. The “binomial factorization” of the linear second-order two-way wave operator into two first-order one-way wave operators has been known for decades and used in geophysics. When the binomial factorization approach is applied to the spatial second-order wave operator, this results in complex mathematical terms containing the so-called “Dirac operator” for which only particular solutions exist. In 2014, a hypothetical “impulse flow equilibrium” led to a spatial first-order “one-way wave equation” which, due to its first order differentials, can be more easily solved than the spatial two-way wave equation. To date the conversion of the spatial two-way wave operator into spatial one-way wave operators is unsolved. By considering the one-way wave operator containing a vector wave velocity, a “synthesis” approach leads to a “general vector two-way wave operator” and the “general one-way/two-way equivalence”. For a constant vector wave velocity the equivalence with the d’Alembert operator can be achieved. The findings are transferred to commonly used mechanical and electromagnetic wave types. The one-way wave theory and the spatial one-way wave operators offer new opportunities in science and engineering for advanced wave and wave field calculations. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

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Open AccessArticle

One-Way Vibration Absorber

by

Oskar Bschorr

and

Hans-Joachim Raida

Acoustics 2022, 4(3), 554-563; https://doi.org/10.3390/acoustics4030034 - 13 Jul 2022

Cited by 1 | Viewed by 1544

Abstract

A vibration absorber consisting of a one-dimensional waveguide with a reflectionless termination extracts vibrational energy from a structure that is to be damped. An optimum energy dissipation occurs for the so-called power adjustment, i.e, the same level of resistance and the opposite reactance [...] Read more.

A vibration absorber consisting of a one-dimensional waveguide with a reflectionless termination extracts vibrational energy from a structure that is to be damped. An optimum energy dissipation occurs for the so-called power adjustment, i.e, the same level of resistance and the opposite reactance of structure and absorber. The dimensioning of these impedance parameters on the base of the classic second order “two-way” wave equation provides analytical solutions for a few simple waveguide shapes; solutions for all other waveguides are only accessible via numerical finite-element computation. However, the competing first order “one-way” wave equation allows for an analytical conception of both the known broadband vibration absorber and the “Acoustic Black Hole” absorber. For example, for an exponential waveguide, the two-way calculation shows no resistance (and hence no real wave propagation) below a cut-off frequency, while the one-way wave equation predicts absorption in the whole frequency range. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

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Open AccessArticle

A Method for Modeling Acoustic Waves in Moving Subdomains

by

Milan Brankovic

and

Mark E. Everett

Acoustics 2022, 4(2), 394-405; https://doi.org/10.3390/acoustics4020024 - 13 Apr 2022

Viewed by 2228

Abstract

Forward modeling plays a key role in both the creation of predictive models and the study of the surrounding environment through inversion methods. Due to their competitive computational cost and modest algorithmic complexity, finite difference methods (FDM) are commonly used to model the [...] Read more.

Forward modeling plays a key role in both the creation of predictive models and the study of the surrounding environment through inversion methods. Due to their competitive computational cost and modest algorithmic complexity, finite difference methods (FDM) are commonly used to model the acoustic wave equation. An algorithm has been developed to decrease the computational cost of acoustic-wave forward modeling that can be applied to most finite difference methods. An important feature of the algorithm is the calculation, at each time step, of the pressure in only a moving subdomain which contains the grid points across which waves are passing. The computation is skipped at grid points at which the waves are negligibly small or non-existent. The novelty in this work comes from flexibility of the subdomain and its ability to closely follow the developing wavefield. To demonstrate the efficacy of the algorithm, it is applied to a standard finite difference scheme and validated against 2-D modeling results. The algorithm herein can play an important role in the reduction in computation time of seismic data analysis as the volumes of seismic data increase due to developments in data acquisition technology. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

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Open AccessArticle

Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures

by

Dascha Dobrovolskij

and

Katja Schladitz

Acoustics 2022, 4(1), 139-167; https://doi.org/10.3390/acoustics4010010 - 18 Feb 2022

Viewed by 2243

Abstract

Ultrasonic testing of polycrystalline media relies heavily on simulation of the expected signals in order to detect and correctly interpret deviations due to defects. Many effects disturb ultrasonic waves propagating in polycrystalline media. One of them is scattering due to the granular microstructure [...] Read more.

Ultrasonic testing of polycrystalline media relies heavily on simulation of the expected signals in order to detect and correctly interpret deviations due to defects. Many effects disturb ultrasonic waves propagating in polycrystalline media. One of them is scattering due to the granular microstructure of the polycrystal. The thus arising so-called microstructural noise changes with grain size distribution and testing frequency. Here, a method for simulating this noise is introduced. We geometrically model the granular microstructure to determine its influence on the backscattered ultrasonic signal. To this end, we utilize Laguerre tessellations generated by random sphere packings dividing space into convex polytopes—the cells. The cells represent grains in a real polycrystal. Cells are characterized by their volume and act as single scatterers. We compute scattering coefficients cellwise by the Born approximation. We then combine the Generalized Point Source Superposition technique with the backscattered contributions resulting from the cell structure to compute the backscattered ultrasonic signal. Applying this new methodology, we compute the backscattered signals in a pulse-echo experiment for a coarse grain cubic crystallized Inconel-617 and a fine grain hexagonal crystallized titanium. Fitting random Laguerre tessellations to the observed grain structure allows for simulating within multiple realizations of the proposed model and thus to study the variation of the backscattered signal due to microstructural variation. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

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Open AccessArticle

Sideband Peak Count in a Vibro-Acoustic Modulation Method for Crack Detection

by

,

,

and

Acoustics 2022, 4(1), 74-86; https://doi.org/10.3390/acoustics4010005 - 24 Jan 2022

Cited by 3 | Viewed by 2351

Abstract

This paper presents a new method of signal processing for vibro-acoustic modulation (VAM) methods in order to detect damage accumulation in steel samples. Damage in the tested samples was produced by cycle loading, which, with a small amplitude, was used as a pump [...] Read more.

This paper presents a new method of signal processing for vibro-acoustic modulation (VAM) methods in order to detect damage accumulation in steel samples. Damage in the tested samples was produced by cycle loading, which, with a small amplitude, was used as a pump wave to modulate an ultrasonic probe wave. Multiple sideband peaks were observed, which were used to characterize the modulation effect. We propose the effectiveness sideband peak number (SPN) method as an indicator of any damage accumulation when the load cycle is applied. Moreover, after comparing the SPN with the previously used modulation index (MI), we concluded that, for some of the samples, the SPN provided better damage indication than the MI. The presented results can be explained by a simple model of bilinear crack nonlinearity. This model demonstrates that the amplitude dependences of the sideband components on the pump and the probe wave amplitudes are very different from the quadratic crack model that is usually used for MI test explanation. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

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Open AccessArticle

Factorized One-Way Wave Equations

by

Oskar Bschorr

and

Hans-Joachim Raida

Acoustics 2021, 3(4), 717-722; https://doi.org/10.3390/acoustics3040045 - 09 Dec 2021

Cited by 3 | Viewed by 2566

Abstract

The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- [...] Read more.

The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical finite element (FE) and finite difference (FD) calculations. For seismic applications, a huge number of methods have been developed to reduce these disturbances, but none of these attempts have prevailed to date. However, a priori factorization of the longitudinal wave equation for inhomogeneous media eliminates the above-mentioned ambiguity, and the resulting one-way equations provide the definition of the wave propagation direction by the geometric position of the transmitter and receiver. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

Open AccessCommunication

Phasor Wave-Field Simulation Providing Direct Access to Instantaneous Frequency: A Demonstration for a Damped Elastic Wave Simulation

by

René Hammer

,

Lisa Mitterhuber

and

Roland Brunner

Acoustics 2021, 3(3), 485-492; https://doi.org/10.3390/acoustics3030032 - 06 Jul 2021

Cited by 1 | Viewed by 3578

Abstract

In this work, we describe and simulate a wave field as a phasor field by simultaneously propagating its real and imaginary parts. In this way, the unique phase angle is directly available, and its time derivative determines the instantaneous frequency. We utilize the [...] Read more.

In this work, we describe and simulate a wave field as a phasor field by simultaneously propagating its real and imaginary parts. In this way, the unique phase angle is directly available, and its time derivative determines the instantaneous frequency. We utilize the concept to describe damping in elastic wave propagation, which is of high importance in several engineering and research disciplines, ranging from earth science and medical diagnosis to physics. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

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Open AccessArticle

Preliminary Determination of the Optimal Parameters When Using an Ultrasonic Probe to Measure Cavern Geometry Where a Metal Borehole Pipe Is Present

by

Tomasz Kubacka

and

Chau Nguyen Dinh

Acoustics 2021, 3(2), 425-441; https://doi.org/10.3390/acoustics3020028 - 15 Jun 2021

Viewed by 2872

Abstract

In order to determine the optimal parameters when using an ultrasonic probe to measure cavern geometry when a metal borehole pipe is present, an investigation was firstly carried out on influence of a vertical metal plates with a thickness from 1 mm to [...] Read more.

In order to determine the optimal parameters when using an ultrasonic probe to measure cavern geometry when a metal borehole pipe is present, an investigation was firstly carried out on influence of a vertical metal plates with a thickness from 1 mm to 15 mm immersed in water on transmitted and reflected ultrasonic waves. The results obtained will be used as an indicator for the measurement of underground geometry in which the ultrasonic probe is placed inside a metal pipe lining a borehole. These studies were performed both by experiment and computer simulation. The results show that the wavelength of the incident ultrasonic signals should be equal to half the thickness of the metal plate or an integer times smaller than this thickness. When the thickness of the barrier is unknown, an ultrasonic signal with linear frequency modulation (LFM) should be used. Due to the reverberation of the ultrasonic waves inside the pipe for caverns filled with water, the distance from the transducer to the cavern wall can be measured if it is longer than three times of the pipe diameter. Frequency analysis of both the reflected and the transmitted waves enables an optimal frequency of the incident ultrasonic wave to be selected, which can be used in the measurement of cavern geometry in conditions in which the ultrasonic probe is inside a metal pipe. Full article

(This article belongs to the Special Issue Elastic Wave Scattering in Heterogeneous Media)

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