String Oscillations Research Articles

Casimir energy for twisted piecewise uniform bosonic strings

The Casimir energy for the transverse oscillations of piecewise uniform bosonic strings with either untwisted or twisted continuous conditions is discussed. After calculating the analytic values of zeros of the dispersion function under certain conditions, we obtain the Casimir energy for both open and closed bosonic strings composed of two or three segments.

OPEN RIGID STRING WITH GAUSS–BONNET TERM IN THE ACTION

The effect of the Gaussian curvature in the rigid string action on the interquark potential is investigated. The linearized equations of motion and boundary conditions, following from the modified string action, are obtained. The equation which defines the eigen-frequency spectrum of the string oscillations is derived. On this basis the interquark potential generated by the string is calculated in one-loop approximation. A substantial influence of the topological term in the string action on the interquark potential at the distances of order of hadronic size or less is revealed.

FOUR-DIMENSIONAL BLACK HOLES AND STRINGS WITH RESCALED TENSION

We give a microscopic description of extreme and near-extreme Reissner–Nordström black holes in four dimensions in terms of fundamental strings in a background of magnetic five branes and monopoles. The string oscillator numbers and tension are rescaled due to the Rindler space background with the string mass fixed. The entropy of the black holes is reproduced correctly by that of the string by taking into account the tension rescaling.

Studies in bowing point friction in bowed strings

By using force transducers at the terminations of a violin E string, one may reconstruct the force and velocity at the bowing point that produces those termination forces [R. T. Schumacher and S. Garoff, Institute of Acoustics, Proceedings, Vol. 19: Part 5, 43–48 (1997) (ISMA’97)]. That information, combined with examination of the ‘‘friction track’’ [R. T. Schumacher and S. Garoff, CAS J. 3(2), 9–16 (1996)] allows the possibility of determining the frictional properties of the rosin-mediated bow–string interaction. The reconstruction of the force, f(t), and velocity, v(t), allows determination of the force–velocity relation F(v). That reconstruction is a delicate operation requiring careful measurement of the reflection functions of the transverse waves at the terminations. Use of simulations aids in determining the limitations of the reconstructions. Knowledge of F(v), combined with examination of the friction track, variation of the chemical constituents of the rosin, and chemical preparation of the string surface, as well as measurements of the rosin’s physical properties, such as softening temperature and x-ray structural examination, will eventually establish the physical basis for the alternating adhesion–lubrication properties of the rosin that produce the slip-stick string oscillation.

FORCED OSCILLATIONS AND RESONANCE OF INFINITE PERIODIC STRINGS

An infinite string, supported by the equidistantly spaced identical suspensions, is considered. Each suspension consists of a spring and a dashpot with viscous damping, in parallel. The small transverse oscillations of the string are affected by the viscous drag of an external medium. The concentrated harmonic force, moving steadily along the string, causes steady-state oscillations. This means that the displacement along the string across the suspension spacing brings the time delay and the phase lead in the string's transverse deflection. The former is equal to the time of the exciting force motion over the spacing. The latter is equal to the change of the exciting force phase over this time. The steady-state nature of the oscillations and the linear property of the infinite periodic structure allows one to consider just one segment of the string between the neighbouring suspensions.The string deflection is governed within the segment by a partial differential equation and two boundary conditions that include the time delay and the phase lead. Both approach infinity as the speed of exciting force approaches zero and so stationary excitation cannot be directly included in the present consideration. It is supposed that the string segment had been at rest before the force approached and returned to rest after the force had moved away. Fourier transformation is used to solve the boundary problem in the infinite strip. The solution is represented in the form of a single integral that is as good for calculation as for qualitative analysis. These show that the periodic string resonance takes place, if any viscous resistance is absent and the integrand has a real pole of the second order.The string oscillations' dependence on the suspension stiffness is studied. If the stiffness is small enough, then a Doppler effect takes place. Two limit cases which correspond to the stiffness that approaches zero or infinity are considered. In the first case, the string is free. In the second one, the rigid suspensions divide the string into an infinite sequence of isolated segments. Any isolated segment is exposed to the moving exciting force over a limited time. Therefore, string resonance is impossible. The integrand has no second order real pole as well.In order to include stationary excitation into consideration, a suitable limit procedure is used. Resonance in response to such an excitation is studied and resonant frequencies are found. If the excitation point coincides with the suspension location or with the string mid-span, then the exciting force produces symmetric oscillations in the string. In these particular cases of excitation, resonance at some resonant frequencies disappears, but anti-resonance, that seems to be impossible in response to moving excitation, appears instead. In some cases, the exciting force produces a standing wave in the string, each suspension coincides with the wave node and so any suspension is strictly fixed as well as the excitation point. Such anti-resonance is not affected by suspension viscous damping. In other cases, if viscous damping is small, the excitation point and suspensions experience small oscillations.

Beyond the Frenkel-Kac-Segal construction of affine Lie algebras

This contribution reviews recent progress in constructing affine Lie algebras at arbitrary level in terms of vertex operators. The string model describes a completely compactified subcritical chiral bosonic string whose momentum lattice is taken to be the (Lorentzian) affine weight lattice. The main feature of the new realization is the replacement of the ordinary string oscillators by physical DDF operators, whereas the unphysical position operators are substituted by certain linear combinations of the Lorentz generators. As a side result we obtain simple expressions for the affine Weyl translations as Lorentz boosts. Various applications of the consdtruction are discussed.

A periodic problem for the inhomogeneous equation of string oscillations

We study a periodic problem for the equation u tt−uxx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ2 and establish conditions of the existence and uniqueness of the classical solution.

The string spectrum on the horizon of a non-extremal black hole

We investigate the conformal string σ-model corresponding to a general five-dimensional non-extremal black hole solution. In the horizon region the theory reduces to an exactly solvable conformal field theory. We determine the modular invariant spectrum of physical string states, which expresses the Rindler momentum operator in terms of three charges and string oscillators. For black holes with winding and Kaluza-Klein charges, we find that states made with only right-moving excitations have ADM mass equal to the black hole ADM mass, and thus they can be used as sources of the gravitational field. A discussion on statistical entropy is included.

Extremal black hole entropy from conformal string sigma model

We present a string-theory derivation of the semiclassical entropy of extremal dyonic black holes in the approach based on the conformal sigma model (NS-NS embedding of the classical solution). We demonstrate (resolving some puzzles existing in previous related discussions) that the degeneracy responsible for the entropy is due to string oscillations in four transverse dimensions ‘intrinsic’ to the black hole: four non-compact directions of the D = 5 black hole case and three non-compact and one compact (responsible for embedding of magnetic charges) dimensions in the D = 4 black hole case. Oscillations in other compact internal dimensions give subleading contributions to statistical entropy in the limit when all charges are large. The dominant term in the statistical entropy is thus universal (i.e. is the same in type II and heterotic string theory) and agrees with the Bekenstein-Hawking expression.

Musical oscillator simulations: From curiosity‐driven science to commercial product

The first computer simulations of bowed string oscillations were done by McIntyre and Woodhouse in 1977. Generalizations to wind instruments followed within five years. The first commercial product using the principles of these simulations appeared in 1994. This talk will give a brief history of these developments, and will then concentrate on recent work on computer simulation of bowed string motion, much of which is presented in detail by other papers in this and an associated session. the relative success of bowed string simulations compared to simulations of wind instrument oscillations requires some explanation, which will be provided. The talk will conclude with a summary of the recent work by the author with James Woodhouse using fast computation to explore the enormous parameter space of the string instrumentalists’ world.

Out-of-Plane Oscillation of a String under Periodic In-Plane Excitation.

The purpose of this paper is to study the out-of-plane oscillations of a string which is fixed at one end and excited harmonically at the other end. The steady-state oscillation of the string is investigated theoretically and experimentally in the following three cases : the axial, lateral and arbitrary in-plane excitations. The frequency of the excitation is near the natural frequency of the even mode of the string in all cases. As a main result, it is clarified theoretically and experimentally that the first mode of the string oscillation (in-plane oscillation) becomes saturated due to the second mode (out-of-plane oscillation) in the case of arbitrary in-plane excitation.

Forced oscillations of elastic strings with nonlinear damping

Forced oscillations of elastic strings with nonlinear damping

ACOUSTIC CHAOS

The history of chaos pertaining to acoustics is briefly reviewed from the first period-doubling experiments reported by Faraday (vibrated liquid layer) and Melde (string oscillations) to today's investigations on chaos in thermoacoustics, musical instruments, the hearing process, and ultrasonics. A closer analysis is given of the sound produced in a liquid by standing acoustic waves of high intensity.

Particle transmutation from the scattering of strings and superstrings in curved space-time

We give a general formulation of the scattering of strings by a curved space-time geometry, for both open and closed bosonic strings and for supersymmetric ones. We assume that the space-time admits flat regions in order to define ingoing and outgoing states, and we analyze the particle content of the scattering. Due to the interaction with the geometry, we find inelastic processes in which the initial particle of mass m and spin s transmutes into a different final particle of mass m′ and spin s′. We also find elastic processes in which initial and final states describe the same particle (although the momentum and the spin polarization may change). At first order in √ α′, for string oscillations small compared with the energy scales of the metric, outgoing and ingoing oscillator operators are related by a linear, or Bogoliubov transformation. We compute the more relevant transition amplitudes: from the ground state to a dilation, to a graviton and to massive states; and the transmutation of a dilaton into a graviton. For supersymmetric strings, massless particles cannot transmute among themselves (this is true to all orders in √ α′). Several properties of the particle transmutation processes follow directly from the symmetry properties of the geometry. Explicit examples as black holes and gravitational shock-wave space-times are given.

Quantum string scattering in a cosmic-string space-time.

We exactly quantize fundamental strings propagating in a straight cosmic-string space-time (conical space-time with deficit angle $8\ensuremath{\pi}G\ensuremath{\mu}$, $\ensuremath{\mu}$ being the cosmic-string tension). If the fundamental string collides with the cosmic string the scattering is inelastic since the internal modes of the fundamental string become excited. If there is no collision, the fundamental string only suffers a deflection of $\ifmmode\pm\else\textpm\fi{}4\ensuremath{\pi}G\ensuremath{\mu}$. If there is collision we find inelastic particle production from the interaction of the string with the (classical) geometry. The string oscillator modes only suffer a change of polarization (rotation) in the elastic case and a Bogoliubov transformation in the inelastic case. As a consequence, for a given initial state, the final particle may be in any state associated with the string oscillators. All transformations are explicitly calculated in closed form. Finally, the quantum scattering amplitude for the lowest scalar (tachyon) is computed exactly. In this calculation the vertex operator in the conical geometry and the oscillator linear transformations we find here are used thoroughly. The peculiar features of the string propagation in this topologically nontrivial space-time are discussed including the question whether string splitting can occur at the classical level.

Bulk vibrations and twists in global cosmic strings

The Higgs fields surrounding loops of global cosmic strings contain the major component of the loop energy [ E ≈ R ln ( R/ λ)]. These Higgs fields can undergo bulk vibrations and we demonstrate that these vibrations have a frequency which is larger, by a factor of ln(R/λ) , than that of previously considered oscillations of strings. These bulk modes should dominate the processes of radiation emission. We also study configurations where a string threads a loop. Both strings are twisted. Twisted global strings usually have an energy which grows as R CUTOFF 2, but in the above configuration we find the boundary conditions eliminate this extra energy. The final energy is just the sum of the energies of the untwisted loop and the untwisted string, with no interaction energy. The absence of interaction energy suggests that the twists may play an important role in the interaction of global strings, at least when one of the strings is a loop.

Covariant light-cone algebra

By introducing a variable lightlike vector k μ the light-cone gauge algebra of the bosonic string oscillators is rewritten in an apparently covariant form and it is shown that the apparent covariance becomes true covariance if, and only if, the parameters take the usual critical values. In particular it is shown that the k-independence of physical quantities is equivalent to the usual closure of the Lorentz group for fixed k μ and also to the zero-curvature of the light-cone-gauge surface. The connection of the k-formalism with the BRST formalism, and with other aspects of physics, notably spontaneous symmetry breaking mechanisms, is discussed.

Bosonization of the ghost fields and one-loop partition function.

Using an explicit mode expression of a bosonized Virasoro generator for the ghost system, we calculate the one-loop ghost partition function in the Hamiltonian framework, which compensates the unphysical modes of string oscillations.

Properties of an anisotropic powder magnet made of barium ferrite produced using the resonance forming method

Magnetic fluidization of the barium ferrite powder in a highly nonuniform alternating magnetic field makes it possible to disrupt the powder floccules, as indicated by the improvement of the magnetic properties of the nonsintered specimens. Forming the magnetic substructure at the resonance frequency of the oscillations of the magnetic strings increases the specific energy and sharpness of the texture of the compacted specimens. Comparison of the results shows that replacing the aqueous suspension with the dry magnetically fluidized powder with subsequent resonance forming increases the remanent induction of the sintered specimens by 28% and specific energy by 42%.

Splitting of torsional oscillation due to lower mantle lateral heterogeneity.

The splitting of the free oscillations of the laterally heterogeneous Earth is studied to remove the degeneracy due to odd order harmonics of lateral heterogeneity. Calculation is made for normal modes of a circular string and torsional oscillation of the Earth to compare the perturbation theory with the variational method. The results of the calculation show that the second order perturbation is more feasible than the variational method for angular order l≤15. To demonstrate the applicability of the second order perturbation theory, the splitting of torsional oscillation of a realistic, laterally heterogeneous model of the Earth is calculated using the lower mantle heterogeneity model of DZIEWONSKI et al. (1977). The results show that the width of the splitting due to lower mantle heterogeneity is an order of magnitude smaller than that due to the rotation of the Earth.