Monopolization Cases Research Articles

Doppler effect of sources in subsonic oscillator motion

The sound fields of a plane-wave source and a monopole source (both at frequency ω) in oscillatory motion (at frequency Ω) are presented. In either case, the source motion is represented in the wave equation by a Dirac delta function. Problems are solved by utilizing Fourier transform and contour integral in the plane-wave case and by utilizing Fourier-Hankel transforms in the monopole case. Solutions are represented by sidebands ω+mΩ, m=…,−2, −1,0,1,2,3,…, which means that the sound waves are modulated due to the oscillatory motion of the sources. In both cases, the amplitude of each sideband ω+mΩ can be represented by a constant factor times the Bessel function of the first kind of order m with an argument in proportion to the amplitude of the source oscillation. These sidebands represent an acoustic signal, the frequency of which changes continuously and periodically at a period of 2π/Ω. [Work supfsported by NSF at Pennsylvania State University.]

Exact solution to the Einstein-Maxwell field equations for a charge and mass distribution with a quadrupole moment

The method introduced by Erez and Rosen to obtain exact multipole-type solutions to the static axially symmetric Einstein field equations is generalized to the Einstein-Maxwell system for the case of a static axially symmetric distribution of mass and charge. This is accomplished by defining an auxiliary function which, when assumed to be related to the electrostatic potential through certain derivative conditions, reduces the Einstein-Maxwell equations for this case to the Weyl form of the pure gravitational field equations. A general expression is then given for the electrostatic potential in terms of the auxiliary function. We interpret the solutions, which contain the Reissner-Nordstr\"om solution as a special case, as fields produced by mass and charge distributions possessing multipole structure, and we give the explicit form of the metric for the monopole plus quadrupole case.

Rotating Magnetosphere: Far-Field Solutions

General principles are established for the solution of the distant magnetic field structure about a rotating axisymmetrically magnetized object surrounded by plasma of negligible inertia. These principles are applied to the interesting case of a dipole magnetic field (as well as the monopole case) and can readily be extended to any other axisymmetric multipole. Analytic solutions are shown, and we thereby compute relevant physical quantities (charge density, magnetic field intensity, torque, etc.). The result, that the magnetic field lines form a perfect universal Archimedean spiral according to the relationship tan = wp/c, is found to be general and applies to all axisymmetric multipoles in the zero inertia limit. In the above equation, is the angular deviation of the field line from radial, w is the rotation rate, p is the distance from the rotation axis, and c is the velocity of light. Also general, the flow velocity is everywhere purely radial and equal to c. Subject headings: hydromagnetics - magnetic stars - rotation

Electromagnetic Fields and Massive Bodies

We consider an example of the effects of massive bodies on static electromagnetic fields in general relativity which yields considerable insight into the fadeaway of multipole moments in nonspherical perturbations of gravitational collapse. We calculate the electromagnetic field of an electrostatic or magnetostatic multipole of fixed strength placed at the center of a massive, nonrotating, spherical shell. If we consider a sequence of static solutions in which the massive shell approaches its own Schwarzschild radius, we find that except in the monopole ($l=0$) case the value of the multipole moment measured by a distant observer goes to zero. Thus, for an arbitrary (but finite) stationary charge and current distribution inside the shell, in the limit as the shell approaches its Schwarzschild radius the only property of the distribution which can be measured by an external observer is the total electric charge.

Size of Firm, Market Structure, and Innovation

Size of Firm, Market Structure, and Innovation