Continuous Frequency Spectrum Research Articles

VOCALIZATION AND BEHAVIOR OF THE SOOTY-HEADED BULBUL (<i>PYCNONOTUS AURIGASTER</i>), THE STREAK-EARED BULBUL (<i>PYCNONOTUS CONRADI</i>) AND THE STRIPE-THROATED BULBUL (<i>PYCNONOTUS FINLAYSONI</i>) (PYCNONOTIDAE, AVES) IN SOUTHERN VIETNAM

The bulbul family Pycnonotidae is an Old-World group of tropical birds that plays important roles in tree seed dispersal. The behavior and vocalization of bulbuls have been studied only in a few species. The present study’s objective is to describe, for the first time, the acoustic and social behavior of three species of bulbuls of the genus Pycnonotus: the Sooty-headed bulbul (P. aurigaster), the Streak-eared bulbul (P. conradi), and the Stripe-throated bulbul (P. finlaysoni). Field studies were conducted from November 2021 to May 2022 in the Cattien National Park, Dong Nai Province, Vietnam, where all three species are sympatric and tend to inhabit disturbed biotopes. We analyzed “observations”, that is, a single record of a bird or a group of birds of a given species. During each observation (about 800 in total), the behavior of the birds was described and, when possible, vocalizations were recorded. The total duration of phonograms of all three species amounted to ca 227 min. The sooty-headed and streak-eared bulbuls are social birds that kept pairs throughout the year, but often, especially during the non-breeding season, gathered in larger groups. Pair members and individuals from neighboring pairs maintained acoustic contact with each other all the time. To do so, they used calls: broadband calls (noise sounds without clear structure, but with a wide continuous frequency spectrum) emitted by streak-eared bulbuls, vs tonal calls (musical sounds showing one clearly defined frequency) by sooty-headed bulbuls. Both these species singing were only rarely observed, this possibly being due to reduced territorial behavior in these birds. The Stripe-throated bulbul differs from the other two in singing more frequently, along with producing calls, and the song could be used for territorial purposes. We assume that territorial behavior is better expressed in stripe-headed bulbuls than in the other two species. However, even stripe-throated bulbuls did not react to playback simulated territorial intrusion in our experiments. We assume that vocalizations play important roles in the communication of the three species because we noted these or those sounds in almost all observations. Moreover, we observed a general increase in vocal activity at the beginning of the breeding season (February–March). In addition, during this time, bulbuls were most often to sing.

Numerical and experimental investigations on dynamic behaviors of a bolted joint rotor system with pedestal looseness

Pedestal looseness is a common fault in rotating machines. To enhance the economy and convenience of the assembly of the aero-engine rotor system, the bolted joint structures are a useful tool for fastening multiple parts of different materials into an integrated system. To our knowledge, the time-varying bending stiffness and looseness fault at the pedestal significantly influence rotor dynamics. The present work aims to gain insight into the dynamic behaviors and mechanical behaviors of a rotor-bearing system with a bolted joint structure subjected to a pedestal looseness fault. Firstly, the dynamic model of a bolted joint rotor-bearing system supported by pedestals was set up taking into account the time-varying bending stiffness of bolted joint and the pedestal looseness fault. The feature of the pedestal looseness fault that occurs in a bolted joint rotor system was investigated by comparing the variation ranges of bending stiffness and dynamic responses. An experimental study was then performed using a rotor-bearing test rig equipped with a bolted joint and pedestals. Finally, dynamic phenomena such as offset of the equilibrium position of the vibration waveform, appearance of continuous frequency spectra, and irregular shaft orbits can be known as the dynamic features for the pedestal looseness fault.

Evaluation of telluric-associated corrosion on buried pipelines

Telluric currents, often known as geomagnetically induced currents (GIC), are produced by the natural variations of the Earth's magnetic field.The corrosion protection system of a buried pipeline generally comprises the coating and cathodic protection system. Cathodic protection is an electrochemical protection system that maintains the pipe-to-soil potential (PSP) sufficiently negative in order to reduce corrosion to negligible levels. Varying telluric currents alter the PSP, thus interfering with the pipeline corrosion protection system and can create conditions when corrosion might increase above acceptable levels.This paper presents an evaluation of telluric-associated corrosion derived from measured and modelled PSP variations. The corrosion rates (metal loss per year) due to varying telluric currents with continuous frequency spectra (1 Hz–10−5 Hz) are approximated with the use of published experimental results derived for the specific sets of fixed frequencies.Results are presented for PSP observed on Australian and European pipelines during two periods of strong geomagnetic activity (in 2003 and 2004) and for identical hypothetical pipelines located at different latitudes for the entire year 2004.From the analysis of recorded PSP, it was found that the corrosion rates for a near-equatorial pipeline (Australia) could be higher than for mid-latitudes (Europe). This is possible because telluric-associated corrosion rates depend not only on geomagnetic activity, but on the properties of the pipeline coating, the performance of the cathodic protection system and environmental conditions.The study demonstrated that without cathodic protection the estimated corrosion rates exceeded the benchmark values recommended by national and international standards, and in exceptional case can exceed the acceptable values even with cathodic protection.Telluric-associated corrosion estimated for the identical hypothetical pipelines located in zones with different geomagnetic activity, clearly demonstrated the latitudinal dependence. The substantial increase of corrosion rates (about 5 times) has been found with increase of latitude from subauroral to auroral locations.

Stability and receptivity analyses of mixed convection in unstably stratified horizontal boundary layers

A linear stability analysis is employed to investigate thermal effects in shear flows. The cases analysed consist of unstably stratified horizontal boundary layers under a mixed convection regime, where forced and free convection mechanisms compete. Governing equations are given by the incompressible Navier–Stokes equations with the Oberbeck–Boussinesq approximation, where the base flow comes from their boundary layer approximation. Modal and non-modal analyses are used to investigate the behaviour of small-amplitude disturbances superposed to this base flow. An evaluation of the inertial, shearing and buoyancy mechanisms in the mixed convection boundary layer stability is performed through variations in the Reynolds, Prandtl and Richardson numbers. On the one hand, the spectra lead to the parametric conditions for the time-asymptotic onset of instability, which is still caused by Tollmien–Schlichting (TS) waves as in the traditional Blasius case. However, thermal effects have a destabilizing effect on them, more so for liquids than gases. On the other hand, the pseudospectra obtained from a resolvent analysis indicate the existence of transient growth at this same onset. However, contrary to the traditional Blasius case, thermal effects cause it to be dominated by the continuous frequency spectrum instead of the discrete TS modes. In order to elucidate this qualitative change, a componentwise input–output analysis is employed to quantify the receptivity to specific external disturbances. It shows that thermal effects directly impact the conversion of thermal to kinetic linear disturbance energy, causing a strong amplification of the flow response due to the non-normality of the linear operator. Results reveal that heating from below causes the forcing and response modes of the input–output analysis to have a free-stream spatial support due to non-modal excitation of the continuous spectrum. Such a behaviour suggests that the unstably stratified boundary layer is susceptible to free-stream thermal disturbances, which can potentially impact bypass transition.

A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions.

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments.

Employment of Continuous Slope Cumulative Frequency Spectrum in geomorphology quantitative analysis – a case study on Loess Plateau

Slope spectrum is an effective methodology of digital terrain analysis in geomorphology. However, some limitations are still encountered in the construction of the slope spectrum. E.g., the uncertainties caused by slope classification and resolution of digital elevation model (DEM) may restrict the development of slope spectrum and cause a certain spatial information loss. In this paper, a new spectrum method, defined as continuous slope cumulative frequency (CSCF) spectrum, is proposed. Compared with the slope spectrum, the CSCF spectrum does not require slope classification and combines the spatial information of all grids. Hence, it possesses richer terrain feature information and minimal spatial information loss. Meanwhile, it is more convenient to be used in different study areas without requiring heavy work of slope spectrum in determining the suitable slope classification. We also found that the CSCF spectrum can be generally extracted from DEM with different resolutions, whereas the slope spectrum cannot. Experiment results show that the CSCF spectrum is an efficient geomorphology quantitative analysis method that can quantitatively depict the spatial distribution and evolution process of the typical loess landforms from different angles. Besides, the CSCF spectrum can accurately deduce the stable slope spectrum, whereas the slope spectrum cannot deduce the CSCF spectrum, which further suggests that the previous studies in slope spectrum can be conducted on the basis of the CSCF spectrum. Furthermore, the CSCF spectrum shows well application potentials in landform classification with comparative performance to previous methods. Overall, the CSCF spectrum is an effective method in geomorphology quantitative analysis. Considering that the CSCF spectrum solves two key deficiencies in the present model of slope spectrum (slope classification and DEM resolution) and extends the application scope and concept of spectrum method, it is promising to be the research paradigm of the spectrum method in digital terrain analysis.

The Influence of the Earth’s Atmosphere Rotation on the Spectrum of Acoustic-Gravity Waves

It was shown in a recent study [11] that taking into account the rotation of the Earth’s atmosphere leads to the appearance of a new region of evanescent waves with a continuous frequency spectrum on the diagnostic diagram of acoustic-gravity waves. The region is located below the lower limit of gravity waves, which is equal to $$2\Omega $$ for all wavelengths, where $$\Omega $$ is the angular frequency of the atmospheric rotation. This result was obtained for high-latitude regions of the atmosphere in which one can be limited to considering only the vertical component of the Earth’s rotation frequency. This paper shows that taking into account both components of the vector $$\vec {\Omega }$$ of the atmospheric rotation frequency $$\vec {\Omega }$$ —horizontal, $$\Omega \cos \varphi ,$$ where $$\varphi $$ is the local latitude, and vertical, $$\Omega \sin \varphi $$ —the dominant role in the acoustic-gravity wave propagation is played by the vertical component. It is shown that the horizontal component leads to a negligible modification of the boundaries of the regions of acoustic and gravity waves on the diagnostic diagram. It is also shown that the vertical component of the frequency affects most strongly the lower limit of gravity waves, which depends on the latitude of the observation site for all wavelengths and is equal to 2 $$\Omega \sin \varphi $$ .

Influence of the Earth’s atmosphere rotation on the spectrum of acoustic-gravity waves

It was shown in a recent paper by Cheremnykh et al. (Advances in Space Research, 2022) that taking into account the rotation of the Earth's atmosphere leads to the appearance of a new region of evanescent waves with a continuous frequency spectrum on the diagnostic diagram of acoustic-gravity waves. The region is located below the lower limit of gravity waves, which is equal $2\Omega$ at all wavelengths, where $\Omega$ is the angular frequency of the atmosphere rotation. This result was obtained for high-latitude regions of the atmosphere, in which we can limit ourselves to considering only the vertical component of the Earth's rotation frequency. In the proposed paper we show that taking into account both components of the vector of the atmosphere rotation frequency $\overrightarrow{\Omega}$ - horizontal, $\Omega\cos\varphi$ , and vertical, $\Omega\sin\varphi$, where $\varphi$ is the Earth`s latitude, the dominant role in the acoustic-gravity waves propagation is played by the vertical component. It is shown that the horizontal component leads to an insignificant modification on the diagnostic diagram of the boundaries of the areas of acoustic and gravity waves, which can be neglected. Also shown that the strongest vertical component of the frequency affects the lower limit of gravity waves, which depends at all wavelengths on the latitude of the observation site and is equal to $2\Omega\sin\varphi$.

MARFE movement and density fluctuations after deuterium pellet injections in H-mode plasmas on EAST tokamak

The multifaceted asymmetric radiation from the edge (MARFE), which is generally considered to be the result of a radiation thermal instability in the edge and usually occurs in high density operation, has been first observed to move up and down along the poloidal cross-section due to edge cooling after cryogenic deuterium pellet injections in EAST tokamak with tungsten divertor. A maximum electron density of 0.84 × n GW has been obtained using continuous cryogenic pellet fueling. In the meantime, MARFEs, initially located near the divertors of EAST, moves to the inner wall on high field side after each pellet injection. This movement should be attributed to the asymmetry of the power flow to the two sides of the MARFE after pellet injections. Accompanied with MARFE movement, two kinds of strong density fluctuations have been observed. The ones with continuous and regular frequency spectrum, which does not cause a reduction of main plasma density, are confirmed to be induced by MARFE. The others, appearing with magnetic fluctuations, have been determined to be induced by the m/n = 2/1 magnetohydrodynamic activities after pellet injections. All the investigations in this paper will be meaningful for the steady high density operation of future fusion reactors, such as ITER.

Noise generation in a bilayer ferromagnet-piezoelectric heterostructure at the converse magnetoelectric effect

The paper describes first observation of parametric generation of noise in a composite multiferroic resonator. Resonator of a disk shape contains two mechanically coupled layers, one of which is the amorphous ferromagnet (FM) FeBSiC and the other is the piezoelectric (PE) lead zirconate titanate. DC magnetic field of 0–100 Oe is applied parallel to the plane of the resonator. Resonator is excited in the frequency range f = 2–10 kHz by a harmonic electric field with amplitude of up to 330 V cm−1 applied to the PE layer. Changes in the magnetization of the resonator caused by the converse magnetoelectric effect were recorded using an electromagnetic coil. With an increase in the excitation field to the threshold value, the parametric generation of harmonics and subharmonics with a discrete spectrum is observed, which then turns into a stochastic mode and a continuous spectrum of frequencies is generated. Noise density is hysterically dependent on the excitation field and non-monotonically depends on the dc magnetic field. Theory of parametric generation of the noise in a multiferroic resonator is developed, taking into account excitation of acoustic resonances and magnetoacoustic nonlinearity of the FM layer of the resonator. Theory, qualitatively describes the main characteristics of noise generation.

Mitigating parametric instabilities in plasmas by sunlight-like lasers

Sunlight-like lasers that have a continuous broad frequency spectrum, random phase spectrum, and random polarization are formulated theoretically. With a sunlight-like laser beam consisting of a sequence of temporal speckles, the resonant three-wave coupling that underlies parametric instabilities in laser–plasma interactions can be greatly degraded owing to the limited duration of each speckle and the frequency shift between two adjacent speckles. The wave coupling can be further weakened by the random polarization of such beams. Numerical simulations demonstrate that the intensity threshold of stimulated Raman scattering in homogeneous plasmas can be doubled by using a sunlight-like laser beam with a relative bandwidth of ∼1% as compared with a monochromatic laser beam. Consequently, the hot-electron generation harmful to inertial confinement fusion can be effectively controlled by using sunlight-like laser drivers. Such drivers may be realized in the next generation of broadband lasers by combining two or more broadband beams with independent phase spectra or by applying polarization smoothing to a single broadband beam.

Frequency Considerations in GIC Applications

AbstractGeomagnetically induced currents (GIC) are a phenomenon well known for its negative effects on the operations of power systems. To efficiently mitigate them requires different types of power system modeling, from GIC to alternating current harmonic generation, to three‐dimensional finite element models of transformers. GIC are initiated by variations of the geomagnetic field in the presence of the conductive Earth, that is, the geophysical variables characterized by continuous frequency spectra, making GIC also exhibit continuous spectra. In order to adequately estimate their variations and peak values for mitigation purposes, an analysis is required of how sampling rate and spectral frequency content impact the measured characteristics of GIC and harmonics. The study is based on the geomagnetic measurements and the power network data (i.e., GIC and harmonics) with high sampling rates recorded during two geomagnetic storms, March 31, 2001 and July 26–27, 2004. Availability of data covering both the source and the result of geomagnetic storm impacts on power grid allows (a) analysis of the influence of spectral content on adequate representation of both geomagnetic and geoelectric variations during the intervals with significant increases in GIC and harmonics and (b) identifying the sampling rate sufficient to usefully represent the network response presented as GIC and harmonics variations. In summary, the adequate sampling rate is suggested and the deficiencies associated with undersampling of the geoelectric and GIC variations are identified and discussed.

Space-time nonseparable pulses: Constructing isodiffracting donut pulses from plane waves and single-cycle pulses

Maxwell's equations can be satisfied not only by plane electromagnetic waves, but also by more exotic space-time nonseparable electromagnetic pulses which cannot be represented as a product of time- and space-dependent functions. A family of such pulses with finite energy was identified by Ziolkowski [Phys. Rev. A 39, 2005 (1989)]. Later, Hellwarth and Nouchi [Phys. Rev. E 54, 889 (1996)] highlighted a subset of Ziolkowski's pulses, now known as flying donuts, a formation of polarization singularities of toroidal topology traveling at the speed of light. Spurred by recent advances in ultrafast and topological optics, space-time nonseparable electromagnetic excitations are now becoming the focus of growing experimental efforts as they hold promise for topological information transfer, probing and inducing transient excitations in matter such as anapole and toroidal modes. Here we demonstrate that the flying donut can be constructed from an ensemble of monochromatic plane waves with continuous spatial and frequency spectrum and hence can be generated by converting broadband conventional transverse electromagnetic pulses.

Analytic solutions of the Teukolsky equation for massless perturbations of any spin in de Sitter background

We present analytic solutions to the Teukolsky equation for massless perturbations of any spin in the four-dimensional de Sitter background. The angular part of the equation fixes the separation constant to a discrete set, and its solution is given by hypergeometric polynomials. For the radial part, we derive an analytic power series solution that is regular at the poles and determine a transcendental function whose zeros give the characteristic values of the wave frequency. We study the existence of explicit polynomial solutions to the radial equation and obtain two classes of singular closed-form solutions, one with discrete wave frequencies and the other with continuous frequency spectra.

New Concepts of the Origin of Chemical Elements on the Earth and in the Universe. Part 1

The formation of space began with the formation of the first star, and then the formation of our Solar System ensued. This is how the evolution of stars and planets, in which chemical elements are synthesized, is described. The dimensions of the system are determined by the relationship of atomic matter with physical vacuum and gravitational forces of attraction. Both micro- and macrosystems have their own oscillation frequencies, and thus a continuous spectrum of different frequencies emerges and fills the entire space of the Universe; resonance is the most plausible mechanism underlying the interactions of the oscillations of these micro- and macrosystems. The development of the material world proceeds from the simple to the complex. The frequency of electromagnetic radiation of atomic hydrogen precedes the frequency of radiation of all subsequent chemical elements of the periodic table, and the rhythms of each subsequent element are harmonized with the vibrations of the hydrogen atom according to the resonance relations. This study touches upon global issues regarding the origin of chemical elements with the whole set of their stable and radioactive isotopes and their prevalence in the Universe. Some aspects of the constancy of the speed of light and mode oscillations on the Sun, as well as Schumann resonances, are addressed. Vibration is the single connection and the single source of motion in all this diversity.

Development of a mathematical model of a vibrating polyfrequency screen as a dynamic system with distributed parameters

A mathematical model of a vibrating polyfrequency screen as a dynamic system with distributed parameters has been developed. The dynamic system of solids of finite sizes was chosen as the design scheme for the screen: framework, sieves with bulk material and impactors, the contact interaction of which occurs through two-side bonds and collisions on surface areas that have elastic-damping coverings. It is shown that a change in the amplitude of the exciting force has a significant effect on the dynamics of vibration impactors of a polyfrequency vibrating. There is an amplitude value at which the impactor passes from the mode without interacting with elastic bonds to the vibro-impact mode. The impactor movements begin to change disproportionately altered by the exciting force amplitude. It is shown that the start of the impactors in the screen substantially depends on the exciting force. Changes in the amplitude of the exciting force make it possible to achieve chaotic oscillations of impactors, which in turn leads to oscillations of screen surfaces with a continuous frequency spectrum, i.e. to the operation mode of the screen, which is most appropriate for dehydration and separation of fine mineral fractions.

Peak-to-Average Power Ratio Analysis for OFDM-Based Mixed-Numerology Transmissions

In this paper, the probability distribution of the peak to average power ratio (PAPR) is analyzed for the mixed numerologies transmission based on orthogonal frequency division multiplexing (OFDM). State of the art theoretical analysis implicitly assumes continuous and symmetric frequency spectrum of OFDM signals. Thus, it is difficult to be applied to the mixed-numerology system due to its complication. By comprehensively considering system parameters, including numerology, bandwidth and power level of each subband, we propose a generic analytical distribution function of PAPR for continuous-time signals based on level-crossing theory. The proposed approach can be applied to both conventional single numerology and mixed-numerology systems. In addition, it also ensures the validity for the noncontinuous-OFDM (NC-OFDM). Given the derived distribution expression, we further investigate the effect of power allocation between different numerologies on PAPR. Simulations are presented and show the good match of the proposed theoretical results.

The use of continuous spectral analysis for the assessment of postural stability changes after sports-related concussion

Impaired postural stability is associated with a variety of pathologies including sports-related concussion (SRC). Quantification of centre of pressure (COP) movement is the most common focus of instrumented assessment. Frequency-domain COP analyses have focused primarily on summary measures or pre-defined frequency bands but continuous analysis may provide novel and complementary insight into pathological control mechanisms. Our aims were (i) to compare post-SRC COP trajectory changes identified using clinician scores (Modified Balance Error Scoring System (M-BESS)), time-domain COP variables and continuous frequency spectral comparison; and (ii) to characterise frequency spectra changes. Male rugby players aged 15–19 years (n = 135) completed a pre-season baseline assessment comprising vision-obscured double-leg, single-leg and tandem stances on a force platform. Participants diagnosed with SRC during the season (n = 15) underwent repeat testing (median 4 days post-SRC; IQR 2.5–6.5). Baseline and post-SRC COP trajectories were compared using common time-domain COP variables, M-BESS scores and continuous frequency spectra. Post-SRC changes were identified using all three approaches. Spectral analysis revealed the largest effect size (Cliff’s delta 0.39) and was the only method to identify differences in all three stances and in double-leg stance. All post-SRC increases in spectral content were in the anteroposterior direction; all decreases were in the mediolateral direction. Changes were localised to higher frequencies (1.7–8 Hz) except for double-leg stance anteroposterior direction, for which increases were observed throughout the analysed range. Our findings suggest that this method of spectral comparison may provide a more responsive and meaningful measure of postural stability changes after SRC than other commonly-used variables.

Shear Alfvén and acoustic continuum in general axisymmetric toroidal geometry

The equations describing the continuous spectrum of shear Alfvén and ion sound waves propagating along magnetic field lines are introduced and solved in the ballooning space for general geometry in the ideal MHD limit. This approach is equivalent to earlier analyses by Chu et al. 1992 [Phys. Fluids B 4, 3713 (1992)], but the present formulation in the ballooning space allows us to readily extend it to include gyrokinetic and three-dimensional equilibrium effects. In particular, following Chen and Zonca 2017 [Phys. Plasmas 24, 072511 (2017)], the MHD limit is adopted to illustrate the general methodology in a simple case, and the equations are solved within the framework of Floquet and Hill's equation theory. The connection of shear Alfvén and ion sound wave continuum structures to the generalized plasma inertia in the general fishbonelike dispersion relation is also illustrated and discussed. As an application, the continuous frequency spectrum is calculated for a reference equilibrium of the Divertor Tokamak Test facility. The results are compared with those obtained by the MARS code adopting the standard methodology, demonstrating excellent agreement.

Equatorial Magnetosonic Waves Observed by Cluster Satellites: The Chirikov Resonance Overlap Criterion

AbstractMagnetosonic waves play an important role on the overall dynamics of relativistic radiation belt electrons. Numerical codes modeling the evolution of the radiation belts often account for wave‐particle interaction with magnetosonic waves. The diffusion coefficients incorporated in these codes are generally estimated based on the results of statistical surveys of the occurrence and amplitude of these waves. These statistical models assume that the spectrum of the magnetosonic waves can be considered as continuous in frequency space. This assumption can only be valid if the discrete nature of the waves satisfy the Chirikov overlap criterion. Otherwise, the assumption of a continuous frequency spectrum could produce erroneous results in wave models and hence estimates of the electron diffusion coefficients used in numerical models of the inner magnetosphere. Recently, it was demonstrated, through a case study conducted on a single short (10 s) period snapshot within a longer wave event, that the discrete nature of the equatorial magnetosonic waves do satisfy the Chirikov overlap criterion and so the assumption of a continuous frequency spectrum is valid for the calculation of diffusion coefficients. This paper expands this study to a broader range of time with many magnetosonic wave events to determine whether the discrete nature of the waves always satisfy the Chirikov overlap criterion. The results show that most, but not all, discrete magnetosonic emissions satisfy the Chirikov overlap criterion. Therefore, the use of the continuous spectrum, employed in quasi‐linear theory, may not always be justified.