| SELF | 14 |
S.B. Karavashkin and O.N. Karavashkina |
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| 4. Construction of the
dynamical model of a single pulsing sphere To construct the dynamical non-conformal map, we have to introduce correctly the time parameter into (21). As we agreed before, we will form the field by way of
longitudinal deformation of the initial stationary grid. For it we need to know the
momentary radial shift |
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(22) |
Note that the value  r in (22) is the difference between
the shift of particles of medium at the moment t relatively to their shift at the
moment t0. But we in our problem consider the shift relatively to the
non-disturbed metric to which for any point of the field r = 0. So in (22) we have
to use only the first term in square brackets. In this way we change a little the standard
Riemann integral. Actually, generalising the situation with the expression (22), we could
write the result as follows: |
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(23) |
where in the right part we use the semi-determined integrals taken only for the upper limit. It follows from (23) that when the under-integral functions are equal, (23) reduces to (22). But if we determine the shift relatively to the metric which at the moment t0 obeyed another regularity, then (22) and (23) do not coincide. It is especially important to note it in the problems, studying the field structures, where the space-time phase delay is important. With it, far from always the initial shift is zero in the entire space, then the second semi-integral in (23) will not vanish, as in our case. Given the above, we can write the time regularity of momentary shift of particles of some point of field as |
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(24) |
| If the disturbing pressure Pa has the form (8), then with (5) the expression (24) takes the form | |
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(25) |
where  . |
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Knowing that according to the introduces map (21) (x
+ a)  r,
we can rewrite (25) as |
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(26) |
| and | |
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| Having the expression for r (x,t) ,
it will be sufficient for us to change a little (21), writing |
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(27) |
| to determine completely the dynamical non-conformal map
describing the radiation of a single pulsing sphere. The same as in dynamical maps
presented in [3], the prototype of map proceeded by (25) is time-independent.
Consequently, the equations of equipotential lines still correspond to the condition (19),
and force lines equations - to the condition (20). Now when we finished the construction of the dynamical non-conformal map, we can understand the reason, why the complex form of parameters P and vr did not satisfy us. In the complex form P and vr described themselves in the complex plane. But in constructing the non-conformal dynamical mapping, the field description proceeds by the transformation of equipotential lines of field. With it the parameters P and vr become the characteristics determining the value of the indicated space-time transformations. In this connection the complex form of writing the parameters becomes excessive. |
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