Mathematical Model of the Process of Ultrasonic wave Propagation in a Relax Environment with its Given Profiles at three Time Moments

Abstract

Objective: The process of ultrasound oscillations in a relaxed environment, provided that the profiles of the acoustic wave at three time moments are known, is modeled by a three-point problem for the partial differential equation of the third order in time. This equation as a partial case contains a hyperbolic equation of the third order, which is widely used in ultrasound diagnostics. Methods: The differential-symbol method is applied to study a three-point in-time problem. The advantage of this method is the possibility to obtain a solution of the problem only through operations of differentiation. Results: We propose the formula to construct the analytic solution of the problem, which describes the process of ultrasound oscillations propagation in a relax environment. Due to this, the profile of the ultrasonic wave is known at any time and at an arbitrary point of space. The class of quasi-polynomials is distinguished as a class of uniqueness solvability of a three-point problem. Conclusion: Using the proposed method, it is possible to analyze the influence of the main parameters of ultrasound diagnostics problems on the propagation of acoustic oscillations in a relaxed environment. The research example of a specific three-point problem is given.