SELF

14

S.B. Karavashkin and O.N. Karavashkina

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 deltabig.gif (843 bytes)r  of equipotential lines of the grid. We can determine it, integrating (1) over time for some arbitrary point having the radius-vector equal to r0 in non-disturbed state. With it we yield

(22)
Note that the value deltabig.gif (843 bytes)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 deltabig.gif (843 bytes)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:

(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

(24)
If the disturbing pressure  Pa   has the form (8), then with (5) the expression (24) takes the form

(25)
where .
Knowing that according to the introduces map (21) (x + a) follow.gif (847 bytes) r, we can rewrite (25) as

(26)
and
 
Having the expression for deltabig.gif (843 bytes)r (x,t) , it will be sufficient for us to change a little (21), writing

,

(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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