No abstract

The time-domain description of musical and other nonlinear oscillators complements the more commonly used frequency-domain description, and is advantageous for some purposes. It is especially advantageous when studying large-amplitude oscillations, for which nonlinearity may be severe. It gives direct insight into the physical reasons for the variation of waveform as playing conditions vary, and into certain phenomena which may seem counter-intuitive from the frequency-domain viewpoint, such as the musically undesirable flattening in the pitch of a bowed string when the bow is pressed too hard onto the string. It is easy to set up efficient time-domain simulations on a small computer, a fact that has been surprisingly little exploited in musical acoustics. The simplest relevant model is described here. It demonstrates some of the basic nonlinear behavior of the clarinet, violin, and flute families with very little programming effort. Remarkably, a single set of model equations has relevance to all three cases, at a certain level of idealization, with appropriate choices of parameter values and of linear and nonlinear characteristics. For the flute family, this simplest model gives waveforms and phase relations closely resembling those observed at resonance in the organ-pipe experiments of Coltman [J. Acoust. Soc. Am. 60, 725-733 (1976)], including the triangular pressure and velocity waveforms. It can be shown (again using a time-domain approach) that the triangular waveform is a universal limiting form, independent of detailed acoustic 'loss mechanisms provided losses are small.

The internal oscillating functions volume velocity, pressure, and reed displacement have been calculated for a clarinet. The data used, all in the literature, are the mechanical properties of the reed, the geometrical measurements of the instrument, the properties of air, properties of tone holes [A. H. Benade, J. Acoust. Soc. Am. 41, 1609 (1967)], and data and calculations on the flow of air through a slit. From these, the input impedance Z(ω) is calculated, and is used to find an effective reflection function r(t). The latter is used in an algorithm [first used in calculating bowed string oscillations: M. E. McIntyre and J. Woodhouse, Acoustica 41 (in press)] that permits rapid numerical integration of the integral equations of motion of the system. The normal behavior of clarinets is observed in both chalumeau and clarion registers. Multiphonics are also calculated. Steady state behavior agrees qualitatively with the previous calculations [R. T. Schumacher, Acustica 40, 298 (1978)] to the extent that the latter are comparable with the present more elaborate model.

We report the results of a detailed investigation of the electron spin resonance of a hydrogen atom in an interstitial position in CaF 2 . The following aspects of the problem are treated in detail: (1) apparatus and sample preparation; (2) a spin Hamiltonian with four parameters describing the g value and hfs of the hydrogen atom with the eight equivalent fluorine nuclei surrounding it; (3) the electron-nuclear double resonance (ENDOR) spectrum and the resonance linewidth; and (4) an attempt to calculate the parameters of the spin Hamiltonian starting from atomic and ionic wave functions. Our sample preparation technique has allowed us to deuterate the specimens, and we have obtained spin Hamiltonian parameters for the deuterium center as well. Most of the data were obtained at room temperature, but some data are available on the hydrogen center at 77°K. We have essayed an explanation of the small proton-deuteron and temperature-dependent differences of the spin Hamiltonian parameters.

The maximum bow force for Helmholtz oscillations, and the frequency shift as a function of bow force, have been measured for unstopped notes on a variety of violin strings. A bowing machine is briefly described on which data using bowing velocities at 5 and 10 cm/s were obtained. Computer simulations using the standard friction function model of the bowed string were also done. The combination of the measurements and the simulations allow an assessment of the probable validity of the friction curve model of the force between bow hair and string in bowed string motion.