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Showing papers on "Amplitude published in 2021"


Journal ArticleDOI
TL;DR: In this article, the authors derived the radial action directly from the amplitude, and determined the corresponding Hamiltonian in isotropic gauge, in agreement with known overlapping terms up to sixth post-Newtonian order, and with the probe limit.
Abstract: Using scattering amplitudes, we obtain the potential contributions to conservative binary dynamics in general relativity at fourth post-Minkowskian order, ${\cal O}(G^4)$. As in previous lower-order calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves polylogarithms with up to transcendental weight two and elliptic integrals. We derive the radial action directly from the amplitude, and determine the corresponding Hamiltonian in isotropic gauge. Our results are in agreement with known overlapping terms up to sixth post-Newtonian order, and with the probe limit. We also determine the post-Minkowskian energy loss from radiation emission at ${\cal O}(G^3)$ via its relation to the tail effect.

128 citations


Journal ArticleDOI
TL;DR: In this article, a detailed analysis of stochastic gravitational wave production from cosmological phase transitions in an expanding universe was performed, where the authors studied both a standard radiation as well as a matter dominated history.
Abstract: We undertake a careful analysis of stochastic gravitational wave production from cosmological phase transitions in an expanding universe, studying both a standard radiation as well as a matter dominated history. We analyze in detail the dynamics of the phase transition, including the false vacuum fraction, bubble lifetime distribution, bubble number density, mean bubble separation, etc., for an expanding universe. We also study the full set of differential equations governing the evolution of plasma and the scalar field during the phase transition and generalize results obtained in Minkowski spacetime. In particular, we generalize the sound shell model to the expanding universe and determine the velocity field power spectrum. This ultimately provides an accurate calculation of the gravitational wave spectrum seen today for the dominant source of sound waves. For the amplitude of the gravitational wave spectrum visible today, we find a suppression factor arising from the finite lifetime of the sound waves and compare with the commonly used result in the literature, which corresponds to the asymptotic value of our suppression factor. We point out that the asymptotic value is only applicable for a very long lifetime of the sound waves, which is highly unlikely due to the onset of shocks, turbulence and other damping processes. We also point out that features of the gravitational wave spectral form may hold out the tantalizing possibility of distinguishing between different expansion histories using phase transitions.

103 citations


Journal ArticleDOI
TL;DR: In this article, the authors compute the scattering amplitude for classical black-hole scattering to third order in the Post-Minkowskian expansion, keeping all terms needed to derive the scattering angle to that order from the eikonal formalism.
Abstract: We compute the scattering amplitude for classical black-hole scattering to third order in the Post-Minkowskian expansion, keeping all terms needed to derive the scattering angle to that order from the eikonal formalism. Our results confirm a conjectured relation between the real and imaginary parts of the amplitude by Di Vecchia, Heissenberg, Russo, and Veneziano, and are in agreement with a recent computation by Damour based on radiation reaction in general relativity.

98 citations


Journal ArticleDOI
TL;DR: The test function method combined with the bilinear form is used to obtain the lump solutions to the generalized Burgers equation with variable coefficients and finds that the shape of kink waves might be parabolic type, and one lump wave can be decomposed into two lump waves.

72 citations


Journal ArticleDOI
TL;DR: In this article, the authors revisited analytical methods for constraining the nonperturbative S-matrix of unitary, relativistic, gapped theories in d ≥ 3 spacetime dimensions.
Abstract: We revisit analytical methods for constraining the nonperturbative S-matrix of unitary, relativistic, gapped theories in d ≥ 3 spacetime dimensions. We assume extended analyticity of the two-to-two scattering amplitude and use it together with elastic unitarity to develop two natural expansions of the amplitude. One is the threshold (non-relativistic) expansion and the other is the large spin expansion. The two are related by the Froissart-Gribov inversion formula. When combined with crossing and a local bound on the discontinuity of the amplitude, this allows us to constrain scattering at finite energy and spin in terms of the low-energy parameters measured in the experiment. Finally, we discuss the modern numerical approach to the S-matrix bootstrap and how it can be improved based on the results of our analysis.

66 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived the tidal action for a tidally deformed spin-1/2 particle by combining the Hilbert series with an on-shell amplitude basis to construct the wave action.
Abstract: Expanding on the recent derivation of tidal actions for scalar particles, we present here the action for a tidally deformed spin-1/2 particle. Focusing on operators containing two powers of the Weyl tensor, we combine the Hilbert series with an on-shell amplitude basis to construct the tidal action. With the tidal action in hand, we compute the leading-post-Minkowskian tidal contributions to the spin-1/2–spin-1/2 amplitude, arising at $$ \mathcal{O} $$ (G2). Our amplitudes provide evidence that the observed long range spin-universality for the scattering of two point particles extends to the scattering of tidally deformed objects. From the scattering amplitude we find the conservative two-body Hamiltonian, linear and angular impulses, eikonal phase, spin kick, and aligned-spin scattering angle. We present analogous results in the electromagnetic case along the way.

61 citations


Journal ArticleDOI
TL;DR: In this paper, the authors analyzed the leading contributions of higher-dimension operators to the four-graviton amplitude in four spacetime dimensions, including constraints that follow from distinct helicity configurations, and showed that the leading EFT coefficients obtained from both string and one-loop field-theory amplitudes lie in small islands.
Abstract: We analyze constraints from perturbative unitarity and crossing on the leading contributions of higher-dimension operators to the four-graviton amplitude in four spacetime dimensions, including constraints that follow from distinct helicity configurations. We focus on the leading-order effect due to exchange by massive degrees of freedom which makes the amplitudes of interest infrared finite. In particular, we place a bound on the coefficient of the $R^3$ operator that corrects the graviton three-point amplitude in terms of the $R^4$ coefficient. To test the constraints we obtain nontrivial effective field-theory data by computing and taking the large-mass expansion of the one-loop minimally-coupled four-graviton amplitude with massive particles up to spin 2 circulating in the loop. Remarkably, we observe that the leading EFT coefficients obtained from both string and one-loop field-theory amplitudes lie in small islands. The shape and location of the islands can be derived from the dispersive representation for the Wilson coefficients using crossing and assuming that the lowest-spin spectral densities are the largest. Our analysis suggests that the Wilson coefficients of weakly-coupled gravitational physical theories are much more constrained than indicated by bounds arising from dispersive considerations of $2 \to 2$ scattering. The one-loop four-graviton amplitudes used to obtain the EFT data are computed using modern amplitude methods, including generalized unitarity, supersymmetric decompositions and the double copy.

60 citations


Journal ArticleDOI
01 Jan 2021
TL;DR: In this article, the amplitude-frequency relationship of a strongly nonlinear oscillator is determined by a simple and excusive approach, which leads to an approximate solution with relatively high accuracy.
Abstract: This paper recommends a simple and excusive approach to a strongly nonlinear oscillator. Its frequency property can be immediately obtained by the simplest calculation. The results show that the method leads to an approximate solution with relatively high accuracy. Considering the simplest solution process, this paper provides a highly efficient tool for fast determination of the amplitude-frequency relationship of a nonlinear oscillator. The large amplitude vibration of a string is used as an example to illustrate the solution process.

52 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived the conserved stress energy tensor linearly coupled to gravity and the classical probability amplitude of graviton emission at leading and next-to-leading order in the Newton's constant $G$.
Abstract: We study the gravitational radiation emitted during the scattering of two spinless bodies in the post-Minkowskian Effective Field Theory approach. We derive the conserved stress-energy tensor linearly coupled to gravity and the classical probability amplitude of graviton emission at leading and next-to-leading order in the Newton's constant $G$. The amplitude can be expressed in compact form as one-dimensional integrals over a Feynman parameter involving Bessel functions. We use it to recover the leading-order radiated angular momentum expression of Damour [arXiv:2010.01641]. Upon expanding it in the relative velocity between the two bodies $v$, we compute the total four-momentum radiated into gravitational waves at leading-order in $G$ and up to order $v^8$, finding agreement with Herrmann et al. [arXiv:2101.07255]. Our results also allow to investigate the zero frequency limit of the emitted energy spectrum.

47 citations


Journal ArticleDOI
TL;DR: In this paper, the scattering between a wobbling kink and an antikink in the standard φ-4 model is numerically investigated and the dependence of the final velocities, wobbling amplitudes and frequencies of the scattered kinks on the collision velocity and on the initial wobbling amplitude is discussed.
Abstract: In this paper the scattering between a wobbling kink and a wobbling antikink in the standard ${\ensuremath{\phi}}^{4}$ model is numerically investigated. The dependence of the final velocities, wobbling amplitudes and frequencies of the scattered kinks on the collision velocity and on the initial wobbling amplitude is discussed. The fractal structure becomes more intricate due to the emergence of new resonance windows and the splitting of those arising in the nonexcited kink scattering. Outside this phase the final wobbling amplitude exhibits a linear dependence of the collision velocity, which is almost independent of the initial wobbling amplitude.

47 citations


Journal ArticleDOI
TL;DR: This paper considers a model that accounts for the intertwinement between the amplitude and phase response, and derive closed-form expressions for the outage probability and ergodic capacity of an RIS-assisted single-input single-output system over Rayleigh fading channels.
Abstract: Reconfigurable intelligent surfaces (RISs) have drawn significant attention due to their capability of controlling the radio environment and improving the system performance. In this paper, we study the performance of an RIS-assisted single-input single-output system over Rayleigh fading channels. Differently from previous works that assume a constant reflection amplitude, we consider a model that accounts for the intertwinement between the amplitude and phase response, and derive closed-form expressions for the outage probability and ergodic capacity. Moreover, we obtain simplified expressions under the assumption of a large number of reflecting elements and provide tight upper and lower bounds for the ergodic capacity. Finally, the analytical results are verified by using Monte Carlo simulations.


Journal ArticleDOI
Yoshiharu Omura1
TL;DR: In this paper, the nonlinear dynamics of energetic electrons interacting with a coherent whistler-mode wave and the formation of electromagnetic electron holes or hills in the velocity phase space are described.
Abstract: Nonlinear processes associated with the generation process of whistler-mode chorus emissions are summarized. The nonlinear dynamics of energetic electrons interacting with a coherent whistler-mode wave and the formation of electromagnetic electron holes or hills in the velocity phase space are described. The condition for resonant electrons to be free from the anomalous trapping at low pitch angles is obtained. In the presence of the inhomogeneity due to the frequency variation and the gradient of the magnetic field, the electron holes or hills result in resonant currents generating rising-tone emissions or falling-tone emissions, respectively. After formation of a coherent wave at a frequency of the maximum linear growth rate, triggering of the nonlinear wave growth takes place when the wave amplitude is above the threshold amplitude. The wave grows to a level close to the optimum wave amplitude as an absolute instability near the magnetic equator. The nonlinear growth rate at a position away from the equator is derived for a subtracted Maxwellian momentum distribution function with correction to the formulas in the past publications. The triggering process is repeated sequentially at progressively higher frequencies in the case of a rising-tone emission, generating subpackets forming a chorus element. With a higher plasma density as in the plasmasphere, the triggering of subpackets takes place concurrently over a wide range of frequency forming discrete hiss elements with varying frequencies. The mechanism of nonlinear wave damping due to quasi-parallel propagation from the equator is presented, which results in the formation of a gap at half the electron cyclotron frequency, separating a long rising-tone chorus emission into the upper-band and lower-band chorus emissions. The theoretical formulation of an oblique whistler mode wave and its interaction with energetic electrons at the n-th resonance is also presented along with derivation of the inhomogeneity factor.

Journal ArticleDOI
TL;DR: It is found that the 7.5° and 30° configurations have great significance for reducing the amplitude of radial force harmonics and the results calculated by the multiphysics vibration prediction model, the 30° configuration has lower vibration acceleration in the entire frequency band.
Abstract: This article investigates the effect of phase shift angle on radial force and vibration behavior in the dual three-phase permanent magnet (PM) synchronous motor (DTP-PMSM). First, the elimination principles of stator magneto-motive force and radial force harmonics with different phase shift angles are derived. Afterward, a 48-slot/22-pole DTP-PMSM with appropriate windings configurations are analyzed from the perspectives of the electromagnetic performance and vibration behavior in detail. It is found that the 7.5° and 30° configurations have great significance for reducing the amplitude of radial force harmonics. The 7.5° configuration has the lowest second-order radial force amplitude, while the 30° configuration has the smallest amplitude of fourth-, sixth-, and eighth-order radial forces. According to the results calculated by the multiphysics vibration prediction model, the 30° configuration has lower vibration acceleration in the entire frequency band. Finally, the prototypes of the 48-slot/22-pole DTP-PMSMs with 30° and 60° phase shift angle are fabricated. The tests are conducted to validate the theoretical analysis.

Journal ArticleDOI
TL;DR: The first nonperturbative determination of an energy-dependent three-hadron scattering amplitude from first-principles QCD is presented, which shows rich analytic structure and a complicated dependence on the two-pion invariant masses.
Abstract: Focusing on three-pion states with maximal isospin (π^{+}π^{+}π^{+}), we present the first nonperturbative determination of an energy-dependent three-hadron scattering amplitude from first-principles QCD. The calculation combines finite-volume three-hadron energies, extracted using numerical lattice QCD, with a relativistic finite-volume formalism, required to interpret the results. To fully implement the latter, we also solve integral equations that relate an intermediate three-body K matrix to the physical three-hadron scattering amplitude. The resulting amplitude shows rich analytic structure and a complicated dependence on the two-pion invariant masses, represented here via Dalitz-like plots of the scattering rate.

Journal ArticleDOI
TL;DR: In this paper, the fractal velocity pattern in symmetric kink-antikink collisions in ''ensuremath{\phi}}^{4}$ theory was shown to emerge from a dynamical model with two effective moduli: the kinkantikinks separation and the internal shape mode amplitude.
Abstract: The fractal velocity pattern in symmetric kink-antikink collisions in ${\ensuremath{\phi}}^{4}$ theory is shown to emerge from a dynamical model with two effective moduli: the kink-antikink separation and the internal shape mode amplitude. The shape mode usefully approximates Lorentz contractions of the kink and antikink, and the previously problematic null vector in the shape mode amplitude at zero separation is regularized.


Journal ArticleDOI
TL;DR: In this paper, the effects of the amplitude and frequency of the sinusoidal current on the firing mode of the neuron were studied by using bifurcation analysis and it was found that there is a peak of firing interval of neurons with the increasing of stimulation current intensity.
Abstract: The physical electric variable is included into the known Hindmarsh-Rose (HR) model for estimating the depolarization field effect and then external current forcing is applied to detect the firing responses. Based on the proposed model, the effects of the amplitude and frequency of the sinusoidal current on the firing mode of the neuron are studied by using bifurcation analysis. It is found that there is a peak of firing interval of neurons with the increasing of stimulation current intensity. In the presence of electric field, the firing pattern of neuron is transformed from single busting to intermittent multimodal busting with the increasing of frequency and amplitude of electric field. The largest Lyapunov exponent is drawn for verification. In the absence of electric field, the two neurons coupled in the first variable achieve synchronization in busting mode. If there is an external electric field, the two neurons can achieve intermittent multimodal busting firing synchronization even without direct variable couple since the energy is injected into the coupled system by the external electric field. Our results show that the periodic external electric field and external current stimulation play an important role in the neuronal firing pattern.

Journal ArticleDOI
TL;DR: In this paper, a variable scale-convex-peak method is constructed to identify the frequency of weak harmonic signal, where the key of this method is to find a set of optimal identification coefficients to make the transition of dynamic behavior topologically persistent.
Abstract: A variable scale-convex-peak method is constructed to identify the frequency of weak harmonic signal. The key of this method is to find a set of optimal identification coefficients to make the transition of dynamic behavior topologically persistent. By the stochastic Melnikov method, the lower bound of the chaotic threshold continuous function is obtained in the mean-square sense. The intermediate value theorem is applied to detect the optimal identification coefficients. For the designated identification system, there is a valuable co-frequency-convex-peak in bifurcation diagram, which indicates the state transition of chaos-period-chaos. With the change of the weak signal amplitude and external noise intensity in a certain range, the convex peak phenomenon is still maintained, which leads to the identification of frequency. Furthermore, the proposition of the existence of reversible scaling transformation is introduced to detect the frequency of the harmonic signal in engineering. The feasibility of constructing the hardware and software platforms of the variable scale-convex-peak method is verified by the experimental results of circuit design and the results of early fault diagnosis of actual bearings, respectively.

Journal ArticleDOI
TL;DR: In this paper, the authors presented a method to determine the leading-order (LO) contact term contributing to the neutrino amplitude through the exchange of light Majorana neutrinos, based on the representation of the amplitude as the momentum integral of a known kernel.
Abstract: We present a method to determine the leading-order (LO) contact term contributing to the $nn \to pp e^-e^-$ amplitude through the exchange of light Majorana neutrinos Our approach is based on the representation of the amplitude as the momentum integral of a known kernel (proportional to the neutrino propagator) times the generalized forward Compton scattering amplitude $n(p_1) n(p_2) W^+ (k) \to p(p_1^\prime) p(p_2^\prime) W^- (k)$, in analogy to the Cottingham formula for the electromagnetic contribution to hadron masses We construct model-independent representations of the integrand in the low- and high-momentum regions, through chiral EFT and the operator product expansion, respectively We then construct a model for the full amplitude by interpolating between these two regions, using appropriate nucleon factors for the weak currents and information on nucleon-nucleon ($N\! N$) scattering in the $^1S_0$ channel away from threshold By matching the amplitude obtained in this way to the LO chiral EFT amplitude we obtain the relevant LO contact term and discuss various sources of uncertainty We validate the approach by computing the analog $I = 2$ $N\! N$ contact term and by reproducing, within uncertainties, the charge-independence-breaking contribution to the $^1S_0$ $N\! N$ scattering lengths While our analysis is performed in the $\overline{\rm MS}$ scheme, we express our final result in terms of the scheme-independent renormalized amplitude ${\cal A}_ u(|{\bf p}|,|{\bf p}^\prime|)$ at a set of kinematic points near threshold We illustrate for two cutoff schemes how, using our synthetic data for ${\cal A}_ u$, one can determine the contact-term contribution in any regularization scheme, in particular the ones employed in nuclear-structure calculations for isotopes of experimental interest

Journal ArticleDOI
TL;DR: In this article, the authors present the results of a search in LIGO O2 public data for continuous gravitational waves from the neutron star in the low-mass X-ray binary Scorpius X-1.
Abstract: We present the results of a search in LIGO O2 public data for continuous gravitational waves from the neutron star in the low-mass X-ray binary Scorpius X-1. We search for signals with $\approx$ constant frequency in the range 40-180 Hz. Thanks to the efficiency of our search pipeline we can use a long coherence time and achieve unprecedented sensitivity, significantly improving on existing results. This is the first search that has been able to probe gravitational wave amplitudes that could balance the accretion torque at the neutron star radius. Our search excludes emission at this level between 67.5 Hz and 131.5 Hz, for an inclination angle $44^\circ \pm 6^\circ$ derived from radio observations (Fomalont et al. 2001), and assuming that the spin axis is perpendicular to the orbital plane. If the torque arm is $\approx $ 26 km -- a conservative estimate of the \alfven\ radius -- our results are more constraining than the indirect limit across the band. This allows us to exclude certain mass-radius combinations and to place upper limits on the strength of the star's magnetic field. We also correct a mistake that appears in the literature in the equation that gives the gravitational wave amplitude at the torque balance (Abbott et al. 2017b, 2019a) and we re-interpret the associated latest LIGO/Virgo results in light of this.

Journal ArticleDOI
TL;DR: The amplitude (Higgs) mode associated with longitudinal fluctuations of the order parameter at the continuous spontaneous symmetry breaking phase transition is investigated in a system of weakly coupled spin chains with the help of quantum Monte Carlo simulations, stochastic analytic continuation, and a chain-mean field approach combined with a mapping to the field-theoretic sine-Gordon model.
Abstract: We investigate the amplitude (Higgs) mode associated with longitudinal fluctuations of the order parameter at the continuous spontaneous symmetry breaking phase transition. In quantum magnets, due to the fast decay of the amplitude mode into low-energy Goldstone excitations, direct observation of this mode represents a challenging task. By focusing on a quasi-one-dimensional geometry, we circumvent the difficulty and investigate the amplitude mode in a system of weakly coupled spin chains with the help of quantum Monte Carlo simulations, stochastic analytic continuation, and a chain-mean field approach combined with a mapping to the field-theoretic sine-Gordon model. The amplitude mode is observed to emerge in the longitudinal spin susceptibility in the presence of a weak symmetry-breaking staggered field. A conventional measure of the amplitude mode in higher dimensions, the singlet bond mode, is found to appear at a lower than the amplitude mode frequency. We identify these two excitations with the second (first) breather of the sine-Gordon theory, correspondingly. In contrast to higher-dimensional systems, the amplitude and bond order fluctuations are found to carry significant spectral weight in the quasi-1D limit.

Journal ArticleDOI
TL;DR: In this article, the generalized partial wave expansion for N→M scattering amplitude in terms of spinor helicity variables is derived and the basis amplitudes of the expansion with definite angular momentum j consist of the Poincare Clebsch-Gordan coefficients.
Abstract: We derive the generalized partial wave expansion for N→M scattering amplitude in terms of spinor helicity variables. The basis amplitudes of the expansion with definite angular momentum j consist of the Poincare Clebsch-Gordan coefficients. Moreover, we obtain a series of selection rules that restrict the anomalous dimension matrix of effective operators and how effective operators contribute to some 2→N amplitudes at the loop level.


Journal ArticleDOI
TL;DR: In this paper, the Trap-Release-Amplify (TaRA) model was proposed to explain the frequency chirping of chorus in a uniform background magnetic field and of electromagnetic ion cyclotron waves in the magnetosphere.
Abstract: Whistler mode chorus waves are quasi-coherent electromagnetic emissions with frequency chirping. Various models have been proposed to understand the chirping mechanism, which is a long-standing problem in space plasmas. Based on analysis of effective wave growth rate and electron phase space dynamics in a self-consistent particle simulation, we propose here a phenomenological model called the "Trap-Release-Amplify" (TaRA) model for chorus. In this model, phase space structures of correlated electrons are formed by nonlinear wave particle interactions, which mainly occur in the downstream. When released from the wave packet in the upstream, these electrons selectively amplify new emissions which satisfy the phase-locking condition to maximize wave power transfer, leading to frequency chirping. The phase-locking condition at the release point gives a frequency chirping rate that is fully consistent with the one by Helliwell in case of a nonuniform background magnetic field. The nonlinear wave particle interaction part of the TaRA model results in a chirping rate that is proportional to wave amplitude, a conclusion originally reached by Vomvoridis et al. Therefore, the TaRA model unifies two different results from seemingly unrelated studies. Furthermore, the TaRA model naturally explains fine structures of chorus waves, including subpackets and bandwidth, and their evolution through dynamics of phase-trapped electrons. Finally, we suggest that this model could be applied to explain other related phenomena, including frequency chirping of chorus in a uniform background magnetic field and of electromagnetic ion cyclotron waves in the magnetosphere.

Journal ArticleDOI
TL;DR: In this paper, a damped variable-coefficient fifth-order modified Kortewegde Vries equation for the small-amplitude surface waves in a strait or large channel of slowly-varying depth and width and non-vanishing vorticity is investigated.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the problem of a temporal discontinuity in the permittivity of an unbounded medium with Lorentzian dispersion and showed that the second-order nature of the dielectric function now gives rise to two shifted frequencies.
Abstract: We study the problem of a temporal discontinuity in the permittivity of an unbounded medium with Lorentzian dispersion. More specifically, we tackle the situation in which a monochromatic plane wave forward-traveling in a (generally lossy) Lorentzian-like medium scatters from the temporal interface that results from an instantaneous and homogeneous abrupt temporal change in its plasma frequency (while keeping its resonance frequency constant). In order to achieve momentum preservation across the temporal discontinuity, we show how, unlike in the well-known problem of a nondispersive discontinuity, the second-order nature of the dielectric function now gives rise to two shifted frequencies. As a consequence, whereas in the nondispersive scenario the continuity of the electric displacement D and the magnetic induction B suffices to find the amplitude of the new forward and backward wave, we now need two extra temporal boundary conditions. That is, two forward and two backward plane waves are now instantaneously generated in response to a forward-only plane wave. We also include a transmission-line equivalent with lumped circuit elements that describes the dispersive time-discontinuous scenario under consideration.


Journal ArticleDOI
TL;DR: In this paper, the authors examined the presence and maturation of low-frequency (<12 Hz) cortical speech tracking in infants by recording EEG longitudinally from 60 participants when aged 4-, 7- and 11- months as they listened to nursery rhymes.

Journal ArticleDOI
TL;DR: In this paper, the attenuation rate of the longitudinal wave through the granite after different high-temperature treatments was obtained, and the effects of temperature, amplitude and wavelength of longitudinal waves on the propagation coefficient and attenuation rates were discussed.