| SELF | 12 |
S.B. Karavashkin and O.N. Karavashkina |
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| 3. Choice of mapping function
The main distinction of studied model of pulsing sphere from the considered in [3] model of harmonic sink is that in present case we need to form the basis for the next construction of the pattern given by two dynamical sources, when each of them has its own dynamical regularity. This is why we cannot use such standard expression as |
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(13) |
that maps the region between the concentric circumferences into the space between the circumferences spaced apart in the complex plane [7, p.216]. This is because the force and equipotential lines in (13) are formed in disregard of the complex dynamical pattern of process. If we correct the map (13) by introducing the time multiplier, it will lead us to the improper results, because (13) forms the force lines described by this map simultaneously in the entire space. We need to take into account the amplitude and phase characteristics of two wave processes at once; this necessity predetermines the direction of construction of the field pattern by superposition of fields of single pulsing spheres. To construct the force and equipotential lines of a single pulsing sphere, analyse briefly the possible alternatives. Consider the map [7, p.217] |
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(14) |
| It reflects a semi-plane with the unit-radius semi-circumference thrown to the semi-plane. Since the reverse map from w into z is of our interest, transform (14) as follows: | |
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(15) |
| which corresponds to the following algebraic form: | |
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(16) |
where  |
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To correlate the boundaries of regions of mapping, we have to take (+) for positive u and (-) for negative u in the right part of (16). |
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| The force lines and equipotential lines plotted in Fig. 1 on the basis of (16) show that the map (14) flows the boundary with the semi-circumference cut out. Of course, it is irrelevant to the modelled process. | |
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