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Astron. Astrophys. 331, 392-404 (1998)

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2. Impedance matching: selecting eigenmodes and optimal driving modes

Eigenfrequencies and optimal driving frequencies are selected by means of the boundary conditions of the flux tube with its infinite and wave-carrying surroundings. Across the boundary of the flux tube, both the total pressure perturbation [FORMULA] and the velocity component normal to the flux tube boundary, [FORMULA], must be continuous ([FORMULA] denotes the Lagrangian perturbation). Their ratio defines the normal acoustic impedance [FORMULA] (e.g. Morse & Feshbach 1953). As in Keppens (1996) we can split the external wave field into the exciting and the scattered part (part entirely due to the presence of the flux tube). Alternatively we can decompose it into radially incoming and outgoing cylindrical waves, when focussing on the geometry of the scatterer. For each of these wave fields we can define the corresponding impedance.

The continuity of [FORMULA] and [FORMULA] across the boundary can be rewritten to give the transmitted or internal normal acoustic impedance:

[EQUATION]

These equalities can then be manipulated to yield:

[EQUATION]

[EQUATION]

Two important conclusions can be drawn from these equations:

  • Eigenfrequencies can be found by assuming that there is no incoming wave. Thus the impedance criterion to select eigenfrequencies is [FORMULA] (or [FORMULA]).
  • Optimal driving frequencies can be found by assuming that there is no outgoing wave. Thus the impedance criterion to select optimal driving frequencies is [FORMULA].

These conclusions give us a means to select the eigenfrequencies and the optimal driving frequencies. Therefore we define the complex impedance difference functions:

[EQUATION]

[EQUATION]

The eigenfrequencies are these frequencies for which F equals 0. The optimal driving frequencies correspond with the zeroes of G. Note that in general the solutions of [FORMULA] and [FORMULA] are complex.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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