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S.B. Karavashkin and O.N. Karavashkina

To the point, in Fig. 4 we see that for a single pulsing sphere the field pattern in the near field is also progressive and distinguishes from the far field only by the velocity of its amplitude variation with the distance. Thus, on the basis of conventional mathematical formalism there exists no region of the standing field. And all attempts to substantiate this field, e.g. in [5], are based on an incorrect approximation of conventional equations for the case  r << . The incorrectness is first of all in ignoring the spatial phase delay of the process, in comparison with the time characteristic. As the indicated terms are arguments of a trigonometric function, the value equal to the integer number of periods can be deleted from the momentary time phase of process t without any consequence. The rest value will vary in limits from 0 to 2. Thus, however near distance from the source we take, we can always find the time intervals in which the time phase of process is less than that spatial. It is clear that this will violate the rigour of inequality   r << , the result of this we see in Fig. 4. Though visually the dynamics of this process can be delusively seen as a standing wave near the source, because of moving surface of the pulsing sphere. In case of few sources superposition the near field will form and the interference will be seen in it. Now we can study it on the basis worked out in this investigation.

5. Conclusions

In this investigation we have ascertained that the non-conformal mapping of a semi-belt of the plane Z  into the uniform radial metric of the plane W  models in the best way the processes in an acoustic field produced by a single pulsing sphere.

On the basis of carried out construction of dynamical non-conformal mapping we have ascertained that with the correct application of conventional mathematical formalism the standing wave in the near field does not form, and all attempts to describe it in the region r << are the consequence of incorrectly taken into account relationship of time and spatial phases of process in the argument of trigonometric function.

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