SELF |
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S.B. Karavashkin and O.N. Karavashkina |
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SELF |
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S.B. Karavashkin and O.N. Karavashkina |
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To yield the sought result, it remains only to sum (18) and (19), taking into account (20) and (21): |
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(22) |
As we see from (22), the solution also has the form of progressive wave modulated in amplitude, and at the same time it sufficiently differs from the above solution (6) yielded for the case of stationary source and moving boundary, although at the boundary we still will see a node. The difference is, in this case the amplitude modulation, determined by the argument of sine in the right-hand part, will depend on the observer's position within the studied interval. This will lead us to the pattern of oscillation process that will vary with respect to the moving frame speed. At zero speed the solution will be |
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(23) |
It is easy to see, in this case the oscillation process will have the shape of standing wave, as the spatial and time parameters are divided and do not form the delay function. At non-zero speed of dotted reference frame, the pattern becomes well more complicated due to the complicated dependence of both delay phases on the speed of frame. In Fig. 6 we show typical oscillation diagrams at different speeds of frame.
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Fig. 6. Typical patterns of oscillations with synchronous motion of source and boundary relatively the elastic line in which oscillations propagate
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We see in this diagram that as the velocity in the line grows, the process first has the shape of beating of two oscillations propagating towards the boundary. With growing velocity, one of frequencies falls, though not so much. Because of it the process gains a shape of sum of two superimposed oscillation processes. With it, the high-frequency wave will still propagate from the source to the boundary, and low-frequency wave will propagate in opposite direction. The described pattern fully corresponds to the dynamical pattern by A. Kushelev [4] and supplements it (see Fig. 7).
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Fig. 7. Oscillation process in the moving reference frame after A. Kushelev [4]
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We would like to emphasise, shown here oscillation patterns are possible only in 1 D elastic lines having an ideal reflecting boundary. If the boundary is not ideal, as well as if the oscillation process scatters in space, calculations will considerably change; we have to note it when using these features of oscillations in designing experimental schemes and in developing the experimental techniques. |
References: 1. Karavashkin, S.B. and Karavashkina, O.N. Some features of vibrations in homogeneous 1d resistant elastic line with lumped parameters. SELF Transactions, 2 (2002), 1, p. 17 - 34 2. Discussion with A.M. Chepik on our paper "Notes on physical absolute" , SELF Transactions, 3 (2003), 1, p. 17 - 34 3. Marder, L. Time and the space-traveller. George Allen and Unwin Ltd., London, 1971 4. Kushelev, A. Encyclopaedia Nanoworld, CD#12 (reduced), |
Contents: / 17 / 18 / 19 / 20 / 21 / 22 /
