Guy Vandegrift

Curvature‐induced interchange mode in an axisymmetric plasma

A theoretical and experimental description of the curvature‐induced electrostatic interchange mode in a simple mirror confined low‐β plasma is given. The frequency, growth rate , and nonflute‐like effects have been measured and compared with theory. Effects due to radial electric field, finite ion Larmor radius, line‐tying, wall radius, and parallel electron response are discussed.

The maze of quantum mechanics

Students can explore quantum mechanics with a concept map that uses only the most elementary solutions to Schrodingers equation. The content has been slightly modified from the traditional introduction to the subject because the issue of interpretation is postponed until Parsevals theorem is reached and used to postulate two fundamental equations of probability, simultaneously. A set of canonical but approximate equations can describe wavepacket motion for most linear waves. If restricted to certain special cases, these equations of wavepacket motion are easily derived and can serve as a temporary substitute for Ehrenfests theorem. A number of exercises can be incorporated into this maze of quantum mechanics.

The diffraction and spreading of a wavepacket

The spreading of a one-dimensional wavepacket solution of Schrodinger’s equation is related to the diffraction of light, as can be verified by considering the three-dimensional spreading of a wavepacket for an arbitrary dispersion relation. This investigation uncovers a special property of Schrodinger’s equation for a free particle: A wavepacket with initial spherical symmetry will preserve this symmetry in all Galilean reference frames. This property leads to a derivation of de Broglie’s postulate that wave number is proportional to momentum (or velocity). The application to non-Gaussian wavepackets and to Fraunhoffer diffraction also is discussed.

The Mössbauer effect explained

Direct integration of Schrodinger’s equation yields the transition probability for the Mossbauer effect, assuming that the bound nucleus receives a sudden impulse of momentum from a gamma particle. Generalization from two coupled oscillators to a linear chain introduces the discrete Fourier transform, in real variables. This chain of coupled oscillators can be used to suggest how very low order collective modes can remain unexcited by the impact.

EXPERIMENTAL STUDY OF THE HELMHOLTZ RESONANCE OF A VIOLIN

The violin supports a Helmholtz resonance, acting like a driven, damped harmonic oscillator. The amount of damping can be measured from the width of the response curve, and also from the decay of an undriven oscillator. The phase shift between driver and resonator is also consistent with theory. Shifts in resonant frequency due to changing the aspect ratio of the f hole are measured and found to be in qualitative agreement with a convenient formula for the resonant frequency associated with a long thin aperture.

Partial line‐tying of the flute mode in a magnetic mirror

The amount of electron emission required to partially line‐tie the m=1 flute mode is measured on an axisymmetric mirror‐confined plasma. The results are consistent with an ideal magnetohydrodynamic (MHD) calculation with a sheath dissipation term added. The minimum electron emission required is approximately the ion saturation current. The sheath impedance is also calculated for the case when the electron emission exceeds the threshold required for the formation of a virtual sheath.

Transverse bending waves and the breaking broomstick demonstration

When a broomstick is supported at both ends by two wine glasses, a strong downward blow to the center will break the stick, leaving the wine glasses undisturbed, provided care is taken to cushion the wine glasses against an initial and brief downward motion of the ends of the broomstick. This downward motion is analyzed and estimated to be about 1 mm in magnitude. Qualitative experimental evidence of this motion is easily obtained using a force probe to monitor a light and nondestructive tap to a 2-m measuring stick. The method of analysis developed here leads to a simple derivation of the dispersion relation for transverse bending waves on a long rod.

The spatial inhomogeneity of pressure inside a violin at main air resonance

The fluctuating pressure inside a violin is investigated at the A0 (main air) resonance, under the assumption that the walls are kept rigid. Three effects are shown to contribute to a reduction in the fluctuating pressure near the violin’s f-holes. The strongest effect arises from a tendency to form a standing wave within the relatively long, thin shape of the upper bout. A second and smaller effect arises from the violin’s nonuniform shape, and is modeled by treating the upper bout as an acoustical waveguide with nonuniform cross section. A third effect is associated with the Green’s function of an acoustical radiator, and will significantly reduce the fluctuating pressure in the immediate vicinity of the violin’s f-hole. Experimental verification of the theory is obtained by measuring the fluctuating pressure inside a rectangular box, with the resonance being driven by a small loudspeaker located outside the aperture.

Line tying of interchange modes in a nearly collisionless mirror‐trapped plasma

If the confining potential of electrostatically trapped electrons fluctuates, then the number of trapped electrons also fluctuates. The linear relationship between potential fluctuations and the number of trapped electrons is investigated, considering two loss mechanisms for a nearly collisionless plasma: (1) small‐angle collisions (diffusion), and (2) large‐angle collisions. This nearly collisionless model predicts the line tying of interchange modes in a mirror‐trapped plasma is orders of magnitude larger than previously thought for a fusion plasma, yet still not strong enough to completely line tie an axisymmetric mirror. Also, a nonlinearity in the response can occur at low amplitude.

Temperature-Driven Convection

Warm air aloft is stable. This explains the lack of strong winds in a warm front and how nighttime radiative cooling can lead to motionless air that can trap smog. The stability of stratospheric air can be attributed to the fact that it is heated from above as ultraviolet radiation strikes the ozone layer. On the other hand, fluid heated from below is unstable and can lead to Bernard convection cells. This explains the generally turbulent nature of the troposphere, which receives a significant fraction of its heat directly from the Earths warmer surface. The instability of cold fluid aloft explains the violent nature of a cold front, as well as the motion of Earths magma, which is driven by radioactive heating deep within the Earths mantle.1 This paper describes how both effects can be demonstrated using four standard beakers, ice, and a bit of food coloring.