Interior Modes Research Articles

The study of oscillations localized near a layer of density discontinuity in an example of solar model

If at some radius r = rj there is a jump of the density and composition then a simple asymptotic formula can be derived which gives ω, the oscillation frequency for large degree modes (ℓ ≫ 1). Using relations of Gabriel and Scuflaire (1979) and taking into account the fact that the oscillation amplitude at R < rj or R > rj is equal to rℓ or r−ℓ multiplied by some functions only weakly dependent on ℓ, we obtain where g is the gravitational acceleration, and ρi(ρe) are the densities at the lover (upper) sides of the layer in question. The comparison with the exact ω-values considered below shows that the asymptotic values are larger by 13, 8.8, and 6 percents for ℓ equal to 5, 7, and 10, respectively. Because ω should be less than the maximum value of the Brunt - Väisälä frequency, we have for a finite thickness Δ of the jump layer and for finite ℓ that ℓ ≪ 2rj/Δ. We have performed numerical study of the above modes which may be called interior modes. The equilibrium model was similar to that studied by Vandakurov (1984 a,b) but now we assume that a small iron-like core (as in Rouse's (1983) model) is present. We assume the hydrogen content in the core to be large enough for convection to occur. Above the core, there are the inhomogeneous convectively neutral zone (rj ≤ r ≤ rj1), the homogeneous radiative zone (rj1 ≤ r ≤ ru), and the convective envelope (ru ≤ r ≤ R), whose structure have been taken from Spruit's (1974) paper.

Experimental Validation of Time-Domain Three-Dimensional Finite-Difference Techniques for Predicting Interior Coupling Responses

The finite-difference code G3DXL3 (generalized 3-dimensional expandable Lawrence Livermore National Laboratory) has been applied to the problem of interior coupling. The coupling test object was a 1-m-tall sheet aluminum cylinder, 0.2 m in diameter with an interior cavity of variable height coupled to the exterior via variable-size rectangular apertures. Inside the cavity, wires were run at different locations with a mix of short and open terminations. Cable current responses at the base of the wires were measured for several geometries and compared with the code predictions. Agreement was very good to excellent up to 3 GHz. In this case, a VAX 11/780 was used with a problem space of 32 × 32 × 88 cells (roughly equivalent to 503 cells) and cells 1.25 cm on all sides. In that many interior modes of the test object were predicted with excellent frequency agreement and typically very good amplitude agreement, the time-domain finite-difference technique should be considered an experimentally validated tool for interior response predictions of idealized interior coupling geometries when employed with the proper resources and necessary care in specifying the inputs and outputs.

Imaginary-frequency interior modes of black holes

Perturbations of black holes (Schwarzschild, Reissner-Nordstrøm and Kerr) can be treated by simple radial wave equations. It is shown that the massless scalar radial equation is a form of the spin-weighted spheroidal wave equation. The region in r corresponding to the usual angular argument (cos 0, 0 real) for such functions is the black hole interior, r E (r - r + ) where r - , r + are the inner and outer horizon radii respectively. We restrict ourselves to axisymmetric scalar waves. (Because of the spherical symmetry this is no restriction in the Schwarzschild and Reissner-Nordstrøm backgrounds, but it is a physical restriction in the Kerr background.) In these cases the spin-weighted spheroidal harmonics correspond to imaginary-frequency waves, i.e. to exponentially growing or decaying waves that fall inward across the outer horizon r+ and are converted to waves moving in the opposite direction as they cross the r _-horizon (r is a timelike coordinate when r e( r _ , r + )). These modes are exactly analogous to the external quasi-normal modes of the black hole. There is always one zero-frequency mode, and l non-zero imaginaryfrequency modes. Here l is the angular momentum eigen-number associated with the angular decomposition.

Accelerated convergence of receptance matrices: comments on a paper by Y. T. Leung

Some results are given which amplify a recent paper by Leung, 1 concerned with combining the advantages of matrix condensation and modal acceleration for linear vibrating systems. The receptance matrix for the active degrees of freedom is expressed in terms of the reduced mass and stiffness matrices associated with those degrees of freedom, together with a series based on the interior modes corresponding to the passive degrees of freedom.

Attenuation in corrugated circular waveguides. Part 1: Theory

A study of the propagation behaviour of corrugated circular waveguides is described. The phase and attenuation characteristics are obtained using space harmonics in the interior and higher order modes in the slot region of the waveguide. It is found that only the first pair of harmonics and higher modes have a significant effect on the propagation behaviour. The HE11 mode, which is required in antenna applications, is shown to have an attenuation coefficient which is lower than that of any other mode, lower in fact than that of the TE010 mode in a smooth wall circular waveguide of comparable size. An investigation of the fields and currents in the waveguide aids a physical understanding of the phenomenon, which is shown to be only weakly sensitive to the waveguide tolerances. Experimental results to confirm the theory are given in Part 2 of the paper (see p. 1180).

Electromagnetic Resonances of Free Dielectric Spheres

A systematic study is made of electromagnetic resonances of a spherical, free, and isotropic sample supposed to be without dielectric loss. The characteristic equation which is both complex and transcendant has been resolved with a computer. The results for the first modes (frequency and Q factor for /spl epsi/ varying between 1 and 100) are presented. The Q factor that is calculated represents the comparison between the energy stored by the resonance system and energy radiated per cycle; this is the theoretical maximum Q in the case of nonlossy materials. The different modes are classed in TE/sub nmr/ and TM/sub nmr/ modes which comprise exterior and interior modes. It is shown that for n >or= r the energy is concentrated in all directions near the surface; these are known as surface modes. This systematic study is confirmed by experiments in which numerous modes have been observed and identified.