Showing papers on "Harmonics published in 1984"
TL;DR: A design philosophy, backed by the appropriate mathematics, is presented, which is applicable to the variety of situations encountered in magnet design and drawn to the presence of possibly dominant lower orders of harmonics created by errors in fabrication.
Abstract: A full mathematical framework for the analysis and production of localized magnetic field profiles is presented. Of primary use in the production of highly homogeneous fields for nuclear magnetic resonance studies, the paper details the analysis of fields in terms of spherical harmonics, describes how field plotting in the appropriate manner may be used to obtain a direct measure of which harmonics are present, and shows how to combine basic "building blocks" to produce the various lower-order zonal and tesseral harmonics. "Building blocks" described include coils, arcs, and sinusoids of current as well as rings and arcs of steel. The use of shaped magnets is also briefly mentioned. Attention is drawn to the presence, in high-order designs, of possibly dominant lower orders of harmonics created by errors in fabrication. The goal of the paper is to present a design philosophy, backed by the appropriate mathematics, which is applicable to the variety of situations encountered in magnet design. Practical examples of correcting coils and "shims" are also given.
390 citations
TL;DR: In this article, the distortion and harmonic generation in the near field of a finite amplitude sound beam are considered, assuming time-periodic but otherwise arbitrary on-source conditions, and the amplitude and phase of the fundamental and first few harmonics are calculated along the beam axis, and across the beam at various ranges from the source.
Abstract: Distortion and harmonic generation in the nearfield of a finite amplitude sound beam are considered, assuming time‐periodic but otherwise arbitrary on‐source conditions. The basic equations of motion for a lossy fluid are simplified by utilizing the parabolic approximation, and the solution is derived by seeking a Fourier series expansion for the sound pressure. The harmonics are governed by an infinite set of coupled differential equations in the amplitudes, which are truncated and solved numerically. Amplitude and phase of the fundamental and the first few harmonics are calculated along the beam axis, and across the beam at various ranges from the source. Two cases for the source are considered and compared: one with a uniformly excited circular piston, and one with a Gaussian distribution. Various source levels are used, and the calculations are carried out into the shock region. The on‐axis results for the fundamental amplitude are compared with results derived using the linearized solution modified with various taper functions. Apart from a nonlinear tapering of the amplitude along and near the axis, the results are found to be very close to the linearized solution for the fundamental, and for the second harmonic close to what is obtained from a quasilinear theory. The wave profile is calculated at various ranges. An energy equation for each harmonic is obtained, and shown to be equivalent within our approximation to the three‐dimensional version of Westervelt’s energy equation. Recent works on one‐dimensional propagation are reviewed and compared.
267 citations
TL;DR: In this paper, a six-phase six-step voltage-fed induction motor is presented, where the stator is rewound with two three-phase winding sets displaced from each other by 30 electrical degrees.
Abstract: A six-phase six-step voltage-fed induction motor is presented. The inverter is a transistorized six-step voltage source inverter, while the motor is a modified standard three-phase squirrel-cage motor. The stator is rewound with two three-phase winding sets displaced from each other by 30 electrical degrees. A model for the system is developed to simulate the drive and predict its performance. The simulation results for steady-state conditions and experimental measurements show very good correlation. It is shown that this winding configuration results in the elimination of all air-gap flux time harmonics of the order (6v ±1, v = 1,3,5,...). Consequently, all rotor copper losses produced by these harmonics as well as all torque harmonics of the order (6v, v = 1,3,5,...) are eliminated. A comparison between-the measured instantaneous torque of both three-phase and six-phase six-step voltage-fed induction machines shows the advantage of the six-phase system over the three-phase system in eliminating the sixth harmonic dominant torque ripple.
234 citations
TL;DR: In this paper, phase and group velocities of G_2, G_3, R_2 and R_3 (100-330 s) are measured by the single-station method and are inverted to give a spherical harmonic representation of the velocity lateral variation.
Abstract: Phase and group velocities of G_2, G_3, R_2 and R_3 (100-330_s) are measured by the single-station method and are inverted to give a spherical harmonic representation of the velocity lateral variation. Approximately 200 paths have been studied. The results are presented for degrees and orders up to 6. The even harmonics of the phase velocity representation are consistent with those obtained from great circle phase velocities (Paper I). The odd harmonics are less constrained and generally have larger standard deviations than the even harmonics. To suppress the poorly determined harmonics in the velocity contour maps we construct a filter which is derived from an inverse problem formulation. The filter reduces the amplitudes of regional variations, but does not change the overall pattern. The patterns of the regional variations are generally consistent with those obtained by regionalized inversion of great circle data (Paper I). The velocity maps show significant differences within oceans and continents. An analysis is made of correlations of surface wave velocities with heat flow and the non-hydrostatic geoid. The slownesses correlate well with heat flow for l = 1-6. The correlation peaks at l = 2 and 5. The geoid has an anticorrelation with the slownesses at l = 2 and 3, and a
positive correlation from l = 4 to 6.
121 citations
IBM1
TL;DR: In this article, the authors proposed a method for measuring an optical frequency relative to the primary time standard, the cesium beam standard at ~9.2 GHz, utilizing a complex frequency synthesis chain involving harmonics of laser and klystron sources.
Abstract: The current method for measuring an optical frequency relative to the primary time standard, the cesium beam standard at ~9.2 GHz, utilizes a complex frequency synthesis chain involving harmonics of laser and klystron sources. The method has been extended recently to the visible region,1 to the 633 nm He-Ne laser locked to a molecular iodine line, with an impressive accuracy of 1.6 parts in 1010. With the new definition of the meter, the distance traversed by light in vacuum during the fraction 1/299 792 458 of a second, the speed of light is now fixed and both time and length measurements can be realized with the same accuracy as an optical frequency measurement. In view of the complexity of optical frequency synthesis, these developments set the stage for originating complementary techniques for stabilizing and measuring laser frequencies which are more convenient.
111 citations
TL;DR: In this article, the construction and operation of a cross-correlation phase and modulation fluorometer using the synchrotron radiation facility at the ADONE-Frascati electron storage ring is described.
Abstract: The construction and operation of a cross‐correlation phase and modulation fluorometer using the synchrotron radiation facility at the ADONE–Frascati electron storage ring is described. In the frequency domain the high repetition rate pulsed source gives a large series of equally spaced harmonic frequencies. Use of cross‐correlation techniques in conjunction with such a light source permits one to isolate one harmonic frequency from the adjacent frequencies with high precision. The cross‐correlation frequency required for the analysis of the phase delay and modulation ratio is obtained using two coupled frequency synthesizers, one of which drives the radio‐frequency cavity of the storage ring and the other which modulates the response of the photomultipliers used for the signal detection. The accuracy, reproducibility, and sensitivity of the instrumentation have been determined on a number of systems and are reported.
108 citations
TL;DR: An excitation-pattern model was used to determine whether the frequency DLs of harmonics within complex tones could be explained in terms of place mechanisms, and it is concluded that place mechanisms are not adequate, and that information about the frequencies of individual harmonics is probably carried in the time patterning of neural impulses
Abstract: A two‐interval, two‐alternative forced choice task was used to estimate frequency difference limens (DLs) for individual harmonics within complex tones, and DLs for the periodicity (i.e., number of periods per s) of the whole complexes. For complex tones with equal‐amplitude harmonics, the DLs for the lowest harmonics were small (less than one percent). The DLs increased rather abruptly around the fifth to seventh harmonic. The highest harmonic in each complex was also well discriminated, and the discriminability of a single high harmonic was markedly improved by increasing its level relative to the other components. The DL for a complex tone was generally smaller than the frequency DL of its most discriminable component. The DL for a complex was found to be predictable from the DLs of the harmonics comprising the complex, using a formula derived by Goldstein [J. Acoust. Soc. Am. 54, 1496–1516 (1973)] from his optimum processor theory for the formation of the pitch of complex tones. The DL for a complex i...
104 citations
TL;DR: In this paper, a method to detect slip frequency from rotor slot harmonics of a three-phase squirrel-cage induction motor is proposed, which is obtained by summing the three phase voltages, and after being sampled with a multiple of the stator frequency, it is changed into slip frequency waves, from which a voltage proportional to the slip frequency is obtained.
Abstract: A method to detect slip frequency from rotor slot harmonics of a three-phase squirrel-cage induction motor is proposed. The rotor slot harmonic voltage Is obtained by summing the three phase voltages, and after being sampled with a multiple of the stator frequency, it is changed into slip frequency waves, from which a voltage proportional to the slip frequency is obtained. Sophisticated sampling techniques allow elimination of the third harmonic voltage induced in the sum of the three-phase voltages and also allow reduction of time constant of the slip frequency detector. Expenmental results show that the slip frequency detector has a good linearity in the range of slip frequency of about -50 to +30 percent of the stator frequency.
93 citations
89 citations
TL;DR: In this article, the frequency of the same harmonic is different from one spacecraft to another and each spacecraft observes a decrease in the fundamental frequency as it moves from morning (15 mHz) to afternoon (10 mHz).
Abstract: Harmonically related Pc 3-4 pulsations (7-100 mHz) are observed simultaneously by the three geosynchronous satellites ATS 6, SMS 1, and SMS 2, which are separated 20 deg from one another. At a given instant the frequency of the same harmonic is different from one spacecraft to another and each spacecraft observes a decrease in the fundamental frequency as it moves from morning (15 mHz) to afternoon (10 mHz). This frequency behavior is explained in terms of standing Alfven waves, for which the frequency is determined by the local magnetic field and plasma density. Occurrence of harmonic Pc 3-4 waves only during daytime hours (0400-2000 LT) and their frequency characteristics suggest a broadband energy source located on the dayside. Possible azimuthal wave number m and azimuthal phase velocity V(phi) of the second through fourth harmonics are determined from an unusual interval during which identical harmonic frequencies were observed at SMS 1 and ATS 6. Under the assumption of tailward propagation of constant-phase fronts at the same velocity for all these harmonics, V(phi) of about 1700 km/s is obtained.
86 citations
TL;DR: In this article, the authors used a simple physical model and a gaussian beam analysis to elucidate energy exchange between the focused fundamental beam and the higher harmonics, which results in improved imaging resolution.
Abstract: By operating the reflection mode scanning acoustic microscope at nonlinear power levels, resolution beyond the linear diffraction limit can be achieved. To demonstrate this effect, imaging experiments were performed in liquid nitrogen, liquid argon, and water at frequencies between 2000 and 2800 MHz. The spatial frequency response of the nonlinear imaging system was found to have a cutoff frequency which is at least 1.4 times higher than the cutoff frequency of the linear system. It is proposed that the resolution improvement is the result of interaction between the transmitted fundamental beam and harmonics. The harmonics are generated in the converging portion of the beam and have focal spots which are significantly smaller than the focal spot of the fundamental beam. The harmonic power reflected by the object influences the detected fundamental power due to nonlinear (parametric) interaction and results in the improved imaging resolution. A simple physical model and a gaussian beam analysis are presented to elucidate energy exchange between the focused fundamental beam and the higher harmonics.
TL;DR: Two-body potentials dependent only on inter-particle separation are shown to have a simple expansion in S-state hyperspherical harmonics as mentioned in this paper, and this expansion is helpful when solving the Nparticle Schrodinger equation.
Abstract: Two-body potentials dependent only on inter-particle separation are shown to have a simple expansion in S-state hyperspherical harmonics. This expansion is helpful when solving the N-particle Schrodinger equation in hyperspherical coordinates.
TL;DR: In this article, the ion cyclotron harmonic damping effects on slow and fast waves in the lower hybrid frequency range for tokamak reactor parameters are studied, and preliminary estimates show that energetic alpha particles tend to absorb both the slow and the fast waves.
Abstract: The ion cyclotron harmonic damping effects on slow and fast waves in the lower hybrid frequency range for tokamak reactor parameters are studied. Inclusion of the higher-order terms in the hot-plasma dielectric tensor introduces ion cyclotron harmonic damping; these terms also contribute to the real part of the dispersion relation and affect the wave trajectories. However, wave absorption by 15 keV deuterium and tritium ions can be avoided by choosing the slow-wave frequency above the lower-hybrid frequency and the fast-wave frequency below the lower hybrid frequency. But preliminary estimates show that energetic alpha particles tend to absorb both the slow and the fast waves. This absorption may become a serious obstacle to fusion-reactor current drive in the lower hybrid frequency range.
TL;DR: A general expression for the Fourier transform of basis functions of exponential class has been derived in this paper, which is used in many particular cases to reveal some important properties (reduction to four-dimensional harmonics, quadratic transformations, etc.) which considerably simplify the mathematical treatment of these functions.
Abstract: A general expression for the Fourier transform of the basis functions of exponential class has been derived. Particular cases of Slater functions, hydrogen-like functions, Shull and Lowdin functions, Shavitt, Filter, and Steinborn functions have been considered. In many particular cases the Fourier transforms have been shown to reveal some important special properties (reduction to four-dimensional harmonics, quadratic transformations, etc.) which considerably simplify the mathematical treatment of these functions and lead to new possibilities in the development of calculation methods for multicenter integrals.
TL;DR: In this article, a semi-empirical induction motor loss model is developed as a function of motor power rating, where losses in rotor, stator and iron (including most important stray losses) are separately estimated and suitably combined.
Abstract: A semiempirical induction motor loss model is developed as a function of motor power rating. The aim is the estimation of losses caused by harmonic-distorted (e.g., PWM) current and voltage supplies. The model may consider harmonics between about 100 and 20 000 Hz and is constructed for power ratings between 1 and 1000 kW. Losses in rotor, stator, and iron (including most important stray losses) are separately estimated and suitably combined. This leads to a simple penalization factor for the time-harmonic currents, which is a function of motor size and which is quasi-linearly depending on harmonic frequency. Total losses are then calculated by the superposition of separate harmonic current or voltage effects. The penalization factor is experimentally verified. The importance of iron (plus stray) losses has been clearly shown. The model predictions are in close agreement with loss measurements on inverter-fed induction motors.
TL;DR: In this paper, the correlation distribution function between normally distributed component harmonics is derived and then verified for low-degree harmonics by using a Monte Carlo technique which takes into account the three-dimensional rotation group.
Abstract: Studies of correlation coefficients between different sets of global geophysical data may lead to useful inferences concerning their relationship or independence. If one data set is allowed to rotate with respect to another, the statistical theory is complicated and extra care is required before one can conclude that there is any statistical significance to a maximized correlation coefficient. If, for some relative rotation, two spherical harmonic fields are significantly correlated, then their individual degree component harmonics of dominant power must also be significantly correlated. Rotations can be found that result in high correlations between the dominant low-degree spherical harmonics of the geomagnetic and tertestrial gravity field potentials, but rotations can also be found that result in equally high, yet meaningless, correlations if the lunar gravity field is substituted for the geomagnetic field. To explain such high correlations, the theoretical correlation distribution function between normally distributed component harmonics is derived and then verified for lowdegree harmonics by using a Monte Carlo technique which takes into account the three-dimensional rotation group. Some curious properties surface: (1)the correlation distribution function for all possible relative orientations is almost the same between identical and uncorrelated fields; and (2)a system for determining the correlation distribution function from randomly selected fields or from randomly rotated fields is almost ergodic.
TL;DR: In this paper, an intrinsic method of harmonic analysis for application to problems concerning non-linear periodic oscillations is presented, which eliminates the danger of omitting any possible contributions of various harmonics to a particular approximation by engaging all the necessary harmonics and excluding the others automatically.
Journal Article•
TL;DR: In this paper, the authors used ISEE-3 spacecraft observations of 120 relatively simple, isolated bursts in the 30-1980 kHz range are the basis of the present study of Type III bursts in solar wind.
Abstract: ISEE-3 spacecraft observation of 120 relatively simple, isolated bursts in the 30-1980 kHz range are the basis of the present study of Type III bursts in the solar wind. Several characteristics are identified for many of these bursts which imply that the mode of emission changes from predominantly fundamental plasma radiation during the rise phase to predominantly second harmonic during decay. The fundamental emission begins in time coincidence with the start of Langmuir waves, confirming the conventional belief in these waves' causation of Type III bursts. Attention is given to the characteristics of fundamental components, by comparison to harmonics, at km-wavelengths.
01 Mar 1984
TL;DR: In this article, the authors describe a slip-ring induction machine operating as a doubly-fed generator above and below synchronous speed, where a prime mover simulates a wind turbine by producing a torque increasing in proportion to the square of the speed.
Abstract: The paper describes a slip-ring induction machine operating as a doubly-fed generator above and below synchronous speed. A prime mover simulates a wind turbine by producing a torque increasing in proportion to the square of the speed. This torque is balanced in steady-state operation by the doubly-fed generator. The stator, or secondary of the generator is arranged in two electrically separate, magnetically coupled layers connected to a cycloconvertor operating with continuous circulating current. This reduces the harmonics injected into the supply and prevents line-to-line short circuits under shock conditions. The secondary frequency is locked to the difference between actual speed and synchronous speed. A deadband is built into the frequency demand loop to ensure steady-state synchronous rather than asynchronous performance of the generator. Control of the secondary voltage amplitude enables the system to generate power proportional to the cube of the speed over a wide speed range. Three alternative methods of secondary control are considered. These are: (i) secondary current modification, (ii) a voltage-speed function and (iii) constant secondary current feed. The efficiences and peak power capabilities of the methods of systems (ii) and (iii) are contrasted in operation as a quasi-wind-energy transducerss.
TL;DR: In this article, the possibility of increase in frequency of millimetre-wave gyrotron operation by transition to higher EC harmonics at a fixed magnetic field by the use of high-selectivity two-resonator cavities is discussed.
Abstract: In experiments aimed at the development of gyrotrons for electron-cyclotron (EC) plasma heating, output power of 2 MW at a wavelength of 3 mm was attained. The possibility of increase in frequency of millimetre-wave gyrotron operation by transition to higher EC harmonics at a fixed magnetic field by the use of high-selectivity two-resonator cavities is discussed. Outputs of over 100 kW were generated at the second and third harmonics of EC frequency. The results of calculation of pulsed magnetic systems, capable of providing the operation of near millimetre-wave long-pulsed gyrotrons, are given. The principles of increase in efficiency of gyrotrons by means of a depressed two-stage collector of electrons are discussed.
TL;DR: In this article, the angular functions in Chandrasekhar's separation of the variables of Dirac's equation in Kerr geometry are solved by an expansion procedure based on spin-weighted spherical harmonics.
Abstract: The angular functions in Chandrasekhar's separation of the variables of Dirac's equation in Kerr geometry are solved by an expansion procedure based on spin-weighted spherical harmonics. The characteristic values are obtained as a series in a$\sigma$, where a is the Kerr parameter and $\sigma$ is the frequency. Closed expressions are obtained for the successive terms in the expansion. Numerical tables are provided.
TL;DR: In this article, an analysis of the kinetic quantities of the electrons in the steady-state HF neon plasma is performed by a time consuming direct solution of the time dependent electron Boltzmann equation up to the establishment of the periodic state.
Abstract: In this paper an analysis of the kinetic quantities of the electrons in the steady-state HF neon plasma is performed by a time consuming direct solution of the time dependent electron Boltzmann equation up to the establishment of the periodic state. In particular, the reduction of the modulation degree and the phase shift with respect to the HF field of the isotropic distribution function and of some macroscopic electron quantities are systematically determined with increasing field frequency. Furthermore, using the Fourier expansion technique, an equation system for the Fourier coefficients of the isotropic distribution function was derived from the time dependent Boltzmann equation which immediately gives a description of the electron kinetics in the steadystate HF plasma. On the basis of this system the coupling of the harmonics of the isotropic distribution with its dc part was studied for different HF fields. In addition, appropriate conditions could be found which determine a field frequency limit above which the coupling impact of the harmonics on the determination of the dc part of the isotropic distribution may be neglected, making a simplified treatment of the electron kinetics possible. The exact determination of its validity limits and the systematic investigations on the basis of this simplified kinetics will be reported in a second paper.
01 Sep 1984
TL;DR: In this article, a probabilistic model is proposed to represent harmonic currents generated by thyristor-controlled AC locomotives, which is used in a Monte Carlo simulation to predict traction feeder-station currents.
Abstract: A probabilistic model is proposed to represent harmonic currents generated by thyristor-controlled AC locomotives. The model is used in a Monte Carlo simulation to predict traction feeder-station currents. The results of the simulation are compared with results obtained from tests on a 25 kV AC railway.
Patent•
23 Jan 1984
TL;DR: In this paper, a system for converting three-phase AC to DC voltage employs three separate converter circuits having their outputs connected in parallel to reduce current harmonics, each converter circuit includes a full wave rectifier feeding a pulse width current modulator which modulates the DC current flow to the load in a manner to maintain the magnitude of the current proportional to the instantaneous magnitude of DC voltage.
Abstract: A system for converting three-phase AC to DC voltage employs three separate converter circuits having their outputs connected in parallel to reduce current harmonics. Each converter circuit includes a full wave rectifier feeding a pulse width current modulator which modulates the DC current flow to the load in a manner to maintain the magnitude of the DC current proportional to the instantaneous magnitude of the DC voltage. The current modulators act as substantially resistive loads to reduce source current harmonics and correct the power factor. The modulators include a pair of transistors connected in a push-pull circuit which are controlled by an oscillator circuit to modulate the DC current at a frequency which is many orders of magnitude greater than the line frequency of the AC source. The DC current outputs of the modulators are coupled through DC/DC isolation transformers to the load.
TL;DR: In this paper, the authors analyzed the effect of the Brillouin scattering in finite systems which have a boundary reflective to light and showed that in the nonlinear regime the scattered light intensity may behave chaotically in time and exhibit rich frequency spectra including shifts corresponding to fractional harmonics of the fundamental acoustic wave.
Abstract: Stimulated Brillouin scattering is analyzed in finite systems which have a boundary reflective to light. The linear instability is considerably altered from that for the usual transmitting boundaries. In the nonlinear regime the scattered light intensity may behave chaotically in time and exhibit rich frequency spectra including shifts corresponding to fractional harmonics of the fundamental acoustic wave. Simulation results compare favorably with data from laser-plasma-interaction experiments.
TL;DR: In this paper, a novel design for millimeter and sub-millimeter wavelength varactor frequency triplers and quadruplers is described, where the varactor diode is coupled to the pump source via waveguide and stripline impedance matching and filtering structures.
Abstract: This paper describes a novel design for millimeter and sub-millimeter wavelength varactor frequency triplers and quadruplers. The varactor diode is coupled to the pump source via waveguide and stripline impedance matching and filtering structures. Output power at the various harmonics of the pump frequency is fed to quasi-optical filtering and tuning elements. The low-loss quasi-optical structures enable near-optimum control of the impedances seen by the varactor diode at the idler and output frequencies, resulting in efficient high-order harmonic conversion. A minimum efficiency of 4 percent with 30-mW input power has been obtained for a tripler operating between 200 and 280 GHz, with a peak efficiency of 8 percent between 250 and 280 GHz. Another tripler, designed for the 260-350-GHz band, gave a minimum conversion efficiency of 3 percent with 30-mW input power, with a peak efficiency of 5 percent at 340 GHz.
TL;DR: In this article, a theory model for the reduction of stimulated Brillouin scattering by the generation of higher ion wave harmonics is derived, and a simplified model based on the truncation to second harmonic ion waves reveals several interesting features.
Abstract: Reduction of stimulated Brillouin scattering by the generation of higher ion wave harmonics is investigated. A theory model is derived which includes all harmonics. A simplified model based on the truncation to second harmonic ion waves reveals several interesting features. When damping and detuning are neglected the reflectivity is drastically reduced in a low‐density plasma (ne<0.25ncr), but the solution turns phase unstable at low reflectivity levels. Finite damping of ion waves extends the range of validity of the model. Still nonlinear damping of the primary decay may considerably exceed the linear damping, rendering the generation of the second harmonic an important mechanism. A large detuning of the second harmonic is required before it can be considered decoupled. The strong interaction between the ion waves introduces a power shift which is able to dynamically reduce the Brillouin reflectivity. Undersone parameter conditions the reflectivity approaches an oscillatory steady state. The accuracy of...
Patent•
21 May 1984
TL;DR: In this article, a low power transmitter for garage door operator systems and the like wherein harmonics of the fundamental frequency are suppressed by class A or AB oscillator operation and fundamental radiation is enhanced by way of a combined inductor/radiator having conductive printed circuit extrusions on the ends thereof to effectively form a center-fed dipole antenna.
Abstract: A low-power transmitter for garage door operator systems and the like wherein harmonics of the fundamental frequency are suppressed by class A or AB oscillator operation and fundamental radiation is enhanced by way of a combined inductor/radiator having conductive printed circuit extrusions on the ends thereof to effectively form a center-fed dipole antenna.
TL;DR: In this article, the propagation of surface acoustic waves in a nonlinear isotropic elastic material is investigated and the effects of heat conduction and viscous internal damping, both assumed small, are included.
Abstract: The propagation of surface acoustic waves in a nonlinear isotropic elastic material is investigated. The effects of heat conduction and viscous internal damping, both assumed small, are included. The method of multiple scales is used to investigate the slow modulation of a wave of arbitrary initial profile and a coupled system of integrodifferential equations is obtained for the variation of the different harmonic constituents. Explicit numerical solutions are obtained for the generation of higher harmonics by an initially sinusoidal wave and for the parametric amplification of a weak signal by a pump wave of twice the frequency.
TL;DR: In this article, the phase difference of the H components between the conjugate points as a function of local time was calculated for the Pc 4/5 magnetic pulsations observed on the ground.
Abstract: Strong evidence for the harmonic structure of Pc 3–5 magnetic pulsations observed on the ground has been provided by extracting coherent oscillations from magnetic pulsation data recorded at the Syowa-Husafell conjugate pair near L = 6 and then calculating the phase difference of the H components between the conjugate points as a function of local time. The model calculation suggests that the Pc 4/5 band observed in the frequency range ∼3–10 mHz is the fundamental mode of standing shear Alfven waves near L = 6, while the Pc 3 band with multiple harmonic structure observed in the frequency range ∼20–100 mHz is the higher harmonics (usually third to sixth harmonics). The second harmonic is not notable in comparison with other harmonics. It is found further that the fundamental mode and the higher harmonics are not always excited simultaneously. Therefore it is likely that when the frequency of the external driving source is in the Pc 3 range, it excites higher harmonics of standing shear Alfven waves near L = 6, while when the frequency of the external driving source is in the Pc 4/5 range, it excites the fundamental mode.