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Article

Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

by
Pilar Ruiz Gordoa
and
Andrew Pickering
*
Área de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, C/Tulipán s/n, Móstoles, 28933 Madrid, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(11), 1309; https://doi.org/10.3390/math9111309
Submission received: 21 May 2021 / Revised: 1 June 2021 / Accepted: 1 June 2021 / Published: 7 June 2021
(This article belongs to the Special Issue Advances in Mathematical Analysis with Applications)

Abstract

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.
Keywords: ultrasonic waves; bubble layer; exact solutions; traveling wave ultrasonic waves; bubble layer; exact solutions; traveling wave

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MDPI and ACS Style

Gordoa, P.R.; Pickering, A. Ultrasonic Waves in Bubbly Liquids: An Analytic Approach. Mathematics 2021, 9, 1309. https://doi.org/10.3390/math9111309

AMA Style

Gordoa PR, Pickering A. Ultrasonic Waves in Bubbly Liquids: An Analytic Approach. Mathematics. 2021; 9(11):1309. https://doi.org/10.3390/math9111309

Chicago/Turabian Style

Gordoa, Pilar Ruiz, and Andrew Pickering. 2021. "Ultrasonic Waves in Bubbly Liquids: An Analytic Approach" Mathematics 9, no. 11: 1309. https://doi.org/10.3390/math9111309

APA Style

Gordoa, P. R., & Pickering, A. (2021). Ultrasonic Waves in Bubbly Liquids: An Analytic Approach. Mathematics, 9(11), 1309. https://doi.org/10.3390/math9111309

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