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S.B. Karavashkin and O.N. Karavashkina

Chapter 1. Acoustic field produced by a single pulsing sphere

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

e-mail: selftrans@yandex.ru , selflab@mail.ru

Abstract

We analyse conventional conformal and alternative non-conformal mapping and prove that this last describes most exactly an acoustic field produced by a single pulsing sphere. By means of dynamical non-conformal mapping we plot the dynamical pattern of process and show that in case of single pulsing sphere the standing wave in the near acoustic field does not arise, as it was thought till now. The inexact estimation of this process is caused by the incorrect comparison of time- and space-phases of process being the argument of trigonometric periodical function.

Keywords: Wave physics; Acoustics; Theory of complex variable, Non-conformal mapping, Quasi-conformal mapping

Classification by MSC 2000: 30C62; 30C99; 30G30; 32A30; 76-05; 76-99.

Classification by PASC 2001: 43.20.+g; 43.38.+n; 43.58.+z; 43.90.+v; 43.20.Hq; 43.20.Tb; 46.25.Cc; 46.40.Cd

1. Introduction

Basing on experimental data, we showed in [1] that in case of acoustic sources radiating in anti-phase, the transversal oscillations arise in gas medium, and they have a clearly expressed polarisation plane, near and far fields and progressive pattern of wave propagation in both these regions.

Theoretical substantiation presented in [1] and [2] leads to the conclusion that in constructing the mathematical model of this acoustic dipole we have no need to operate with additional regularities. It is sufficient to take into account the wave superposition in the near and far fields, remaining in calculation not only the amplitude of resulting acoustic potential but the unit vector of the power field directed across the wave propagation [2].

In this paper we will show that such investigation can be carried out conveniently and visually in frames of mathematical tool of theory of analytical mapping, in particular, with the use of dynamical non-conformal mapping having been used in [3] to study complex-structure dynamical fields. One of principal advantages of such approach is the possibility to take into account correctly the superposition of force and equipotential lines in frames of unified dynamical power function of the field.

To keep the sequence of investigation, in the first chapter we will consider the features of acoustic oscillation process produced by a single pulsing sphere. This will enable us to form the necessary basis for further studying the fields of more complex structure.

2. Statement of problem

Thinking the conventional theoretical basis for studying the acoustic waves in gas to be sufficient, we will proceed from the standard description of an acoustic field produced by a single pulsing sphere as the basis for further construction of fields superposition.

As we know from [4, p.38], in gas medium the particle velocity produced by a pulsing sphere obeys the following regularity:

where (r,t)  is the velocity potential, A   is some constant determined by the initial and boundary conditions, r is the distance from the source, is the wave process frequency, v_r   is the radial component of the momentary velocity of particles of medium at the studied point, and k  is the wave number.