eISSN 2658–5782

DOI 10.21662/mfs

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2026. Vol. 21. Issue 1, Pp. 32–41
URL: https://multiphasesystems.online/mfs2026.1.006,en
DOI: https://doi.org/10.21662/mfs2026.1.006
Numerical analysis of the total cross-section scattering of acoustic wave on a two sound-permeable spheres system
E.Sh. Nasibullaeva 🖂
Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russia

Abstract

In this paper, an analysis of the total cross section of scattering of a spherical wave from a monopole radiation source on a system of two sound-permeable spheres (water droplets in air or air bubbles in water) is performed using the orthogonal central compositional design method. The numerical method is implemented using a three-factor computational experiment, where one physical (wave radius) and two geometric (the ratio of the radii of the spheres and the distance between the centers of the spheres) parameters of the system are selected as factors. The main objectives of the study are to determine the main parameters, changes to which significantly affect the entire system, as well as to find the values of the variable parameters of the system for which the objective function (the total scattering cross section) takes the smallest and largest values, and to study the system for these parameter values. It is shown that for a small wave radius, all coefficients are significant for both droplets and bubbles; with an increase in the wave radius, the number of insignificant coefficients increases; the objective function takes the smallest and largest values only on the boundary of the parametric region; the average value of the objective function with an increase in the wave radius increases for a system with droplets and decreases for a system with bubbles; the objective function is more sensitive for systems with droplets than for systems with bubbles, while for the two types of systems, with an increase in the wave radius, the sensitivity of the objective function decreases. Pressure distribution diagrams were constructed, which made it possible to determine the zones of high or low pressure for the found optimal values of the factors.

Received: 25.02.2026
Accepted: 23.03.2026
Published: 31.03.2026
Citation

Nasibullaeva ESh. Numerical analysis of the total cross-section scattering of acoustic wave on a two sound-permeable spheres system. Multiphase Systems. 2026;21(1):32–41 (in Russian).

Keywords

acoustic scattering;
sound-permeable sphere;
computational experiment;
orthogonal central composite design method;
monopole radiation source;
total scattering cross-section

Funding

The work was carried out with the support of the state budget under the state assignment 124030400064-2 (FMRS-2024-0001).

Article outline

When studying the mechanism of an acoustic wave scattering on a set of spherical particles, the problem of wave scattering on a system consisting of two spheres is of particular interest. In such systems, there is an interaction of fields scattered from spherical particles; however, this interaction is quite simple, which allows it to be examined in detail. This paper presents the results of studying the total scattering cross-section (the main characteristic of the scattering field) on a system consisting of two bubbles or drops, when a spherical wave is incident from a monopole radiation source using the orthogonal central compositional design method. In the context of the study, the orthogonal central compositional design method was applied to the case of two sound-permeable spheres (air bubbles in water or water droplets in air) based on a three-factor computational experiment, where one physical (wave radius) and two geometric (the ratio of the radii of the spheres and the distance between the centers of the spheres) parameters of the system were also selected as factors.

The main purposes of the work are to determine the main parameters, the change of which significantly affects the entire system, as well as to find the values of the variable parameters of the system at which the objective function (the total scattering cross-section) takes the minimal and maximal values, and to study the system for these parameter values.

Methodology. To take into account the quadratic contributions of factors, the orthogonal central compositional design method based on regression analysis methods was used. For the obtained regression equation, the significance of the coefficients was checked using Student's t-test and the adequacy of the model using Fisher's F-test. The minimum and maximum values of the loss functions were determined using analytical methods of the theory of several variable functions.

Findings. Analysis of the obtained results showed that:

  • for a small wave radius, all coefficients are significant for both droplet and bubble systems, and with increasing wave radius, the amount of insignificant coefficients increases;
  • for a large wave radius in the case of drops, all nonlinear coefficients are insignificant, and for bubbles, all nonlinear coefficients are insignificant for the factor corresponding to the distance between the centers of the spheres;
  • the objective function takes its smallest and largest values at the parametric domain boundary. In the case of a bubble system, the largest value is achieved for all wave radii with one set of coded system parameters;
  • the average value of the objective function with increasing wave radius increases for a system with drops and decreases for a system with bubbles, which corresponds to the nature of the change in the total scattering cross-section for single soundproof (hard and soft) spheres;
  • the objective function is more sensitive for droplet system than for bubble system, and for the two types of systems, the sensitivity of the objective function decreases with increasing wave radius.

The phase diagrams constructed for the found optimal parameters of the system showed zones of decrease and increase of pressure, which allow one to make an analogy with both the case of a two impermeable spheres (with a wave radius value equal to 1) and the cases of single spheres (Poisson spot and spherical lens), and the higher the value of the wave radius, the more pronounced these zones are.

Value. A rigorous consideration of multiple scattering on simple model systems is the first step in understanding the phenomena associated with multiple scattering of waves. The application of a numerical technique based on the orthogonal central compositional design method to a system of two sound-permeable spheres and the analysis of the results obtained on such systems will allow one to move to the next stage of studying the acoustic wave scattering mechanism on multiple gas bubbles in a liquid and liquid droplets in a gas over a wider range of parameter variations. This will allow one to consider problems with real systems containing a finite number of scatterers.

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