Nonlinear Energy Exchange Research Articles

Analysis of nonlinear water wave interaction solutions and energy exchange between different wave modes

In this study, we consider the ideal fluid model of an inviscid fluid, assuming that the fluid motion is adiabatic; the flow is irrotational, that is, the individual fluid particles do not rotate; vorticity ω̃=0; and the flow is incompressible, in which the density of fluid particles does not vary significantly with fluid motion and can be considered constant throughout the fluid volume and throughout the motion. We start with equations representing continuity, conservation of momentum, conservation of entropy, and streamline equations, respectively. It is then reduced to a standard system of equations describing motion in two dimensions, defined by the Laplace equation with appropriate kinematic and dynamic boundary conditions, in terms of velocity potential and surface elevation. Finally, the one-dimensional nonlinear Korteweg–De Vries (KdV) equation is derived. Then, we further investigate the interaction of multiple periodic waves using the KdV equation and explain the interaction wave energy transfer procedure between the primary and higher order harmonics, and the Phillips [“On the dynamics of unsteady gravity waves of finite amplitude. I. The elementary interactions,” J. Fluid Mech. 9, 193–217 (1960)] wave resonance criterion is employed for capturing the periodic wave interaction whose energy conversion is analyzed via Fourier spectra. It is also found that for solitons, multiple collisions of different solitons eventually regain their original shape and that higher-energy solitons have faster velocities than lower-energy solitons, which, to the best of our knowledge, is overlooked.

Analytical model for extremum analysis of moistened fins involving all nonlinear energy exchange processes

Dehumidification on a fin surface happens when the thermal interface state maintains an undervalue corresponding to saturation condition, and a temperature boundary layer develops during heat transmission. The local thickness of the temperature boundary layer changes, and consequently, sensible and latent heat transfer coefficients become highly non-uniform. In this study, a new mathematical procedure depending on the Adomian decomposition method (ADM) is established to predict the heat transfer duty from a moistened fin where all heat transfer modes involved are functions of temperature. A cubic algebraic equation for connecting the specific humidity of air and the thermal state level of a fin is closer to the actual relation determined based on the regression analysis, which converts the governing equation into a highly nonlinear character. The present model has no restriction using the fractional power factor of temperature-dependent convection coefficients. The current results highlight that the variable convection coefficients decrease both the fin performance index and heat load rate, and this influence becomes significant at the optimum design condition. Therefore, establishing the optimum state of a fin adopting all the variability events in the practical design system is essential for the correct analysis. As the equipment implementation analysis always requires an extremum study, the proposed research will help to predict the optimum information in the practical design with the highest accurate prediction of results.

Nonlinear vibratory energy exchanges in a meta-cell

Meta-materials are utilized for bringing passive attenuation solution especially in the acoustics and vibration domains. Generally speaking, they are composed of an array or a chain of meta-cells. Here the vibratory energy exchanges between particles of a nonlinear meta-cell are studied. The meta-cell is composed of an outer mass which houses an inner mass with a compound nonlinear restoring forcing term. It is globally non-smooth, containing pure cubic and piece-wise linear terms, which constitutes a new type of nonlinearity for such mass-in-mass cells. The complexified form of system equations is treated by the multiple time scale method to find out its fast and slow dynamics, leading to determination of the slow invariant manifold as well as the singular and equilibrium points of the system. The compound nonlinear restoring forcing function of the inner mass makes the global geometry of the slow invariant manifold to be different from those of systems with pure cubic nonlinearities, for example, including four singular lines and two distinct unstable zones. Amplitude dependency of the frequency of such a nonlinear system in the conservative form is represented by the backbone curves. Furthermore, detected equilibrium points for different external forcing amplitudes are represented by three-dimensional frequency response curves. Finally, analytical predictions are confronted with numerical results obtained by direct time integration of the system equations. An application of such system can be the passive control of main systems via embedding another nonlinear oscillator inside it. Moreover, such system can also be extended in the form of an array to create meta-materials.

Neimark Sacker bifurcations and non-linear energy exchange in chains of non-linear oscillators

We treat a chain of oscillators with linear stiffness and internal and external cubic non-linearities. The method of harmonic balance is used to determine the non-linear modes of the Hamiltonian system. For each equilibrium point, a stability analysis is performed by means of the associated monodromy matrix. The numerical results shows that branch points, limit points and Neimark–Sacker bifurcations exist in the system. The aim of the paper is to study how they drive the energy in the system: branch points make energy transfer between non-linear modes possible, while Neimark–Sacker bifurcations can lead to chaotic behaviour. As the number of oscillators increases, the energy required to reach the first Neimark–Sacker bifurcation follows a remarkable regularity.

Dispersive characteristics of non-linear waves propagating and breaking over a mildly sloping laboratory beach

The dispersive characteristics of unidirectional irregular waves propagating and breaking over a mildly sloping beach are examined using a highly-resolved laboratory dataset. Cross-spectral analyses are used to determine the cross-shore evolution of (single-valued) dominant wavenumber κ and phase velocity c spectra, and lead to the identification of four different regimes of propagation: I - a linear regime where short waves mostly propagate as free components; II - a shoaling regime where non-linear effects at high harmonics are significant but primary components follow the linear wave dispersion relation; III - a shoaling regime near the mean breaking point location, where amplitude dispersion effects at primary components are important; IV - a surf zone regime, where all components propagate slightly faster than non-dispersive shallow water waves. Bispectral analyses performed onshore of the shoaling region show that the presence of forced energy at high harmonics, which originate from non-linear interactions between triads of frequencies, are responsible for the deviations of wavenumber and phase velocity spectral estimates from predictions by the linear dispersion relation, confirming the findings from previous field-based studies. A Boussinesq approximation of the non-linear energy exchanges between triads is then used to quantify the relative amount of forced energy at high harmonics and explain the differences in dispersion properties observed in the shoaling region between broad and narrow-band spectra. Larger relative amounts of forced energy at high frequencies, which suggest more efficient non-linear energy transfers, are found to be associated with larger deviations of dominant κ and c from predictions by the linear dispersion relation.

On the spectral changes of seismic wave energy by a partially saturated crack due to the hysteresis of liquid bridges phenomenon

Low-frequency shadows are frequently interpreted as attenuation phenomena due to partial saturation with free gas. However, several researchers have argued that shadows are not necessarily a simple attenuation phenomenon because low-frequency energy must have been added or amplified by some physical or numerical process. Attenuation alone should attenuate higher frequencies, not boost lower frequencies. The physical or numerical effects explaining this phenomenon are still debatable in the literature. To better understand the elastic wave energy’s spectral changes in partially saturated rock, we have considered the hysteresis of liquid bridges phenomena inside the crack. We determine that liquid bridges’ hysteresis leads to the nonlinear energy exchange between frequencies, explaining the wave energy boost at lower frequencies. We find that the energy exchange between different frequencies depends on the wave amplitude and the seismic wave spectrum. The low-frequency energy boost is stronger for a continuous spectrum of seismic waves, smaller for the discrete spectrum, and zero for the monochromatic spectrum of seismic waves. In addition, we find that at seismic frequencies, the attenuation 1/ Q-factor due to the friction of the contact line can be much larger than the attenuation due to viscous fluid flow inside the partially saturated crack. Our model depends on the wave amplitude and weakly depends on the wave frequency. The suggested model can help interpret the low-frequency shadows, bright spots, and attenuation anomalies frequently observed around hydrocarbon fields.

Modal energy exchanges in an impulsively loaded beam with a geometrically nonlinear boundary condition: computation and experiment

The capability of a geometrically nonlinear boundary condition, i.e., a strong local nonlinearity, in “redistributing” a broadband input energy (generated by an impulsive load) among the vibration modes of a cantilever Euler–Bernoulli beam is investigated. It is shown that this modal energy redistribution increases the inherent capacity of the cantilever for passive energy dissipation. The nonlinear boundary condition is realized by grounding the free end of the cantilever through an inclined linear spring–damper pair with initial angle of inclination $$ \phi_{0} $$ relative to the neutral axis of the beam while at rest. For $$ \phi_{0} < 90^{^\circ } $$ , the inclined spring–damper pair is geometrically nonlinear, whereas in the limiting case $$ \phi_{0} = 90^{^\circ } $$ the boundary condition becomes linear. To study the nonlinear modal energy redistribution in the cantilever, a multi-step system identification method to identify the unknown parameters of the experimental fixture is employed; this informs a computational reduced-order finite element (FE) model of the fixture. First, the Multi-input Multi-output Frequency Domain Identification (MFDID) technique to analyze the experimental frequency response functions of the “base” linear cantilever without the boundary condition is employed and its modal parameters are identified. Next, the boundary condition for the limiting angle $$ \phi_{0} = 90^{^\circ } $$ is imposed, so that again a linear fixture is obtained. Through reconciliation of computational and experimental measurements, the (linear) stiffness and damping coefficients of the boundary are identified, as well. Finally, by varying the angle of inclination in the range $$ 0^\circ \le \phi_{0} < 90^{^\circ } $$ , the nonlinear transient responses of the identified FE model with the nonlinear boundary condition are computed and projected onto the linearized modal basis of the system in the limit of zero energy. The computational FE results favorably compare with experimental measurements. Following this, the time-averaged modal energies of the system are computed and used to estimate the portion of the total energy of the beam allocated to each mode. Additionally, by employing these modal energies one may study and track the nonlinear energy exchanges between subsets of modes for different angles of initial inclination $$ \phi_{0} $$ of the nonlinear boundary attachment. The computational results are validated by experimental measurements, thus highlighting the predictive capacity of the computational FE model. In the last step, a scalar measure for modal energy exchange is defined by computing the maximum fluctuation in the percentage of each of the instantaneous modal energies that is the maximum percentage of energy being exchanged by the modes. This measure proves to be dependent on both the initial energy and the initial angle of inclination $$ \phi_{0} $$ . Again, experimental measurements favorably compare to computational FE simulations.

Transformation of Infragravity Waves during Hurricane Overwash

Infragravity (IG) waves are expected to contribute significantly to coastal flooding and sediment transport during hurricane overwash, yet the dynamics of these low-frequency waves during hurricane impact remain poorly documented and understood. This paper utilizes hydrodynamic measurements collected during Hurricane Harvey (2017) across a low-lying barrier-island cut (Texas, U.S.A.) during sea-to-bay directed flow (i.e., overwash). IG waves were observed to propagate across the island for a period of five hours, superimposed on and depth modulated by very-low frequency storm-driven variability in water level (5.6 min to 2.8 h periods). These sea-level anomalies are hypothesized to be meteotsunami initiated by tropical cyclone rainbands. Estimates of IG energy flux show that IG energy was largely reduced across the island (79–86%) and the magnitude of energy loss was greatest for the lowest-frequency IG waves (&lt;0.01 Hz). Using multitaper bispectral analysis, it is shown that, during overwash, nonlinear triad interactions on the sea-side of the barrier island result in energy transfer from the low-frequency IG peak to bound harmonics at high IG frequencies (&gt;0.01 Hz). Assuming this pattern of nonlinear energy exchange persists across the wide and downward sloping barrier-island cut, it likely contributes to the observed frequency-dependence of cross-barrier IG energy losses during this relatively low surge event (&lt;1 m).

A radiation hydrodynamics scheme on adaptive meshes using the Variable Eddington Tensor (VET) closure

AbstractWe present a new algorithm to solve the equations of radiation hydrodynamics (RHD) in a frequency-integrated, two-moment formulation. Novel features of the algorithm include i) the adoption of a non-local Variable Eddington Tensor (VET) closure for the radiation moment equations, computed with a ray-tracing method, ii) support for adaptive mesh refinement (AMR), iii) use of a time-implicit Godunov method for the hyperbolic transport of radiation, and iv) a fixed-point Picard iteration scheme to accurately handle the stiff nonlinear gas-radiation energy exchange. Tests demonstrate that our scheme works correctly, yields accurate rates of energy and momentum transfer between gas and radiation, and obtains the correct radiation field distribution even in situations where more commonly used – but less accurate – closure relations like the Flux-limited Diffusion and Moment-1 approximations fail. Our scheme presents an important step towards performing RHD simulations with increasing spatial and directional accuracy, effectively improving their predictive capabilities.

Effects of Nonlinear Stores on the Dynamics of a Computational Model Airplane

It has been shown experimentally that local nonlinear stores attached to the wings of a model airplane enhance overall energy dissipation through global nonlinear modal energy exchanges. Here, a computational model is developed to investigate this effect. The dynamics of the airplane–store system is studied under impulse loading with both stores locked (L-L configuration), left-wing store unlocked (U-L), or both stores unlocked (U-U). Each store acts as a rigid mass when locked, or as a strongly nonlinear element otherwise. The results are in agreement with the experimental ones at different impulse levels. The simulated responses for the U-L and U-U configurations are then projected onto the normal modes of the underlying L-L system to study modal energy redistributions caused by the nonlinear stores. The U-L and U-U configurations exhibit irreversible nonlinear energy transfers from the airplane modes to the stores, with faster decay of total energy compared with the L-L configuration, corroborating the experimental results. The stores also drastically affect the distribution of energy among airplane modes, also contributing to enhanced energy dissipation. Finally, harmonic excitation is considered, and the nonlinear effects are confined to a specific frequency band. This work confirms the drastic dynamic effects that local nonlinear attachments can have on the dynamics of the (linear) structures to which they are attached. Whether such local strong nonlinearities are to be purposely designed to enhance energy dissipation or need to be accounted for to mitigate their possibly detrimental effects is of great practical significance in different applications.

Nonlinear normal modes, resonances and energy exchange in single-walled carbon nanotubes

The nonlinear resonance interaction and energy exchange between bending and circumferential flexure modes in single-walled carbon nanotubes is studied. First, the results of an analytical model of the resonance interaction between the considered nonlinear normal modes previously developed are reported. This approach was based on a reduced form of the Sanders–Koiter thin shell theory obtained by using simplifying hypotheses on the shell deformations. The analytical model predicted that the nonlinear resonance interaction leads to energy localization in a certain coherence domain over the carbon nanotube surface within a specific range of the initial oscillation amplitude. Then, a numerical model of the resonance interaction between the analysed nonlinear normal modes in the framework of the complete Sanders–Koiter thin shell theory is reported. Numerical simulations are performed to verify the energy localization phenomenon over the carbon nanotube surface and to compute the threshold values of the initial oscillation amplitude giving rise to energy localization. Finally, from the comparison between the two different approaches, it is obtained that the results of the numerical model for the threshold values of the nonlinear energy localization confirm with very good accuracy the predictions of the analytical model.

Estimation of Growth Rate of Electromagnetic Plasma Wave through Vlasov-Maxwell Mathematical Frame in Ionospheric Plasma

Unique forms of nonlinear wave energy exchange phenomena are observed in the Earth’sionosphere region. Energy upconversion of nonresonant plasma waves in the top ionospheric and auroral zone are noticed.Origin of these phenomena are tried to explained by linear andnonlinear theoretical approach.Wave-wave and wave-particle-wave interaction processes may be possible role takes place here.In this theoretical investigation we wish to derive probable growthrate expression of high frequency electromagnetic O-mode wave in the presence of low frequency electrostatic ion sound wave through wave-particle interaction process known as plasma maserinstability and estimate its value by using observational data.

Nonlinear Resonance Interaction between Conjugate Circumferential Flexural Modes in Single‐Walled Carbon Nanotubes

This paper presents an investigation on the dynamical properties of single‐walled carbon nanotubes (SWCNTs), and nonlinear modal interaction and energy exchange are analysed in detail. Resonance interactions between two conjugate circumferential flexural modes (CFMs) are investigated. The nanotubes are analysed through a continuous shell model, and a thin shell theory is used to model the dynamics of the system; free‐free boundary conditions are considered. The Rayleigh–Ritz method is applied to approximate linear eigenfunctions of the partial differential equations that govern the shell dynamics. An energy approach, based on Lagrange equations and series expansion of the displacements, is considered to reduce the initial partial differential equations to a set of nonlinear ordinary differential equations of motion. The model is validated in linear field (natural frequencies) by means of comparisons with literature. A convergence analysis is carried out in order to obtain the smallest modal expansion able to simulate the nonlinear regimes. The time evolution of the nonlinear energy distribution over the SWCNT surface is studied. The nonlinear dynamics of the system is analysed by means of phase portraits. The resonance interaction and energy transfer between the conjugate CFMs are investigated. A travelling wave moving along the circumferential direction of the SWCNT is observed.

On unravelling mechanism of interplay between cloud and large scale circulation: a grey area in climate science

The interaction between cloud and large scale circulation is much less explored area in climate science. Unfolding the mechanism of coupling between these two parameters is imperative for improved simulation of Indian summer monsoon (ISM) and to reduce imprecision in climate sensitivity of global climate model. This work has made an effort to explore this mechanism with CFSv2 climate model experiments whose cloud has been modified by changing the critical relative humidity (CRH) profile of model during ISM. Study reveals that the variable CRH in CFSv2 has improved the nonlinear interactions between high and low frequency oscillations in wind field (revealed as internal dynamics of monsoon) and modulates realistically the spatial distribution of interactions over Indian landmass during the contrasting monsoon season compared to the existing CRH profile of CFSv2. The lower tropospheric wind error energy in the variable CRH simulation of CFSv2 appears to be minimum due to the reduced nonlinear convergence of error to the planetary scale range from long and synoptic scales (another facet of internal dynamics) compared to as observed from other CRH experiments in normal and deficient monsoons. Hence, the interplay between cloud and large scale circulation through CRH may be manifested as a change in internal dynamics of ISM revealed from scale interactive quasi-linear and nonlinear kinetic energy exchanges in frequency as well as in wavenumber domain during the monsoon period that eventually modify the internal variance of CFSv2 model. Conversely, the reduced wind bias and proper modulation of spatial distribution of scale interaction between the synoptic and low frequency oscillations improve the eastward and northward extent of water vapour flux over Indian landmass that in turn give feedback to the realistic simulation of cloud condensates attributing improved ISM rainfall in CFSv2.

Nonlinear vibrations and energy exchange of single-walled carbon nanotubes. Radial breathing modes

In this paper, the nonlinear vibrations and energy exchange of single-walled carbon nanotubes (SWNTs) are analysed. The Sanders-Koiter shell theory is used to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary conditions are applied. The resonant interaction between radial breathing (axisymmetric) modes (RBMs) is analysed. An energy method, based on the Lagrange equations, is considered in order to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is then solved applying the implicit Runge-Kutta numerical method. The present model is validated in linear field comparing the RBM natural frequencies numerically predicted with data reported in the literature from experiments and molecular dynamics simulations. The nonlinear energy exchange between the two halves along the SWNT axis in the time is studied for different amplitudes of initial excitation applied to the two lowest frequency resonant RBMs. The influence of the SWNT aspect ratio on the numerical value of the nonlinear energy beating period under different boundary conditions is analysed.

Extreme nonlinear energy exchanges in a geometrically nonlinear lattice oscillating in the plane

We study the in-plane damped oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Strong nonlinearity in this system is generated by geometric effects due to the in-plane stretching of the coupling spring elements. The lattice has a finite number of nonlinear transverse standing waves (termed nonlinear normal modes – NNMs), and an equal number of axial linear modes which are nonlinearly coupled to the transverse ones. Nonlinear interactions between the transverse and axial modes under harmonic excitation give rise to unexpected and extreme nonlinear energy exchanges in the lattice. In particular, we directly excite a transverse NNM by harmonic forcing (causing simulataneous indirect excitation of a corresponding axial linear mode due to nonlinear coupling), and identify three energy transfer mechanisms in the lattice. First, we detect the stable response of the directly excited transverse NNM (despite its instability in the absence of forcing), with simultaneous stability of the indirectly excited axial linear mode. Second, by changing the system and forcing parameters we report extreme nonlinear “energy explosions,” whereby, after an initial regime of stability, the directly excited transverse NNM loses stability, leading to abrupt excitation of all transverse and axial modes of the lattice, at all possible wave numbers. This strong instability is triggered by the parametric instability of an indirectly excited axial mode which builds energy until the explosion. This is proved through theoretical analysis. Finally, in other parameter ranges we report intermittent, intense energy transfers from the directly excited transverse NNM to a small set of transverse NNMs with smaller wavelengths, and from the indirectly excited axial mode to a small set of axial modes, but with larger wavelengths. These intermittent energy transfers resemble energy cascades occurring in turbulent flows. Our results show that nonlinear mechanical systems can support extreme energy transfer phenomena, with features resembling “mechanical turbulence”.

A Broadband Internally Resonant Vibratory Energy Harvester

The objective of this paper is twofold: first to illustrate that nonlinear modal interactions, namely, a two-to-one internal resonance energy pump, can be exploited to improve the steady-state bandwidth of vibratory energy harvesters; and, second, to investigate the influence of key system’s parameters on the steady-state bandwidth in the presence of the internal resonance. To achieve this objective, an L-shaped piezoelectric cantilevered harvester augmented with frequency tuning magnets is considered. The distance between the magnets is adjusted such that the second modal frequency of the structure is nearly twice its first modal frequency. This facilitates a nonlinear energy exchange between these two commensurate modes resulting in large-amplitude responses over a wider range of frequencies. The harvester is then subjected to a harmonic excitation with a frequency close to the first modal frequency, and the voltage–frequency response curves are generated. Results clearly illustrate an improved bandwidth and output voltage over a case which does not involve an internal resonance. A nonlinear model of the harvester is developed and validated against experimental findings. An approximate analytical solution of the model is obtained using perturbation methods and utilized to draw several conclusions regarding the influence of key design parameters on the harvester’s bandwidth.

Nonlinear vibrations and energy exchange of single-walled carbon nanotubes. Circumferential flexural modes

In this paper, the nonlinear vibrations and energy exchange of single-walled carbon nanotubes (SWNTs) are studied. The Sanders–Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary conditions are considered. The circumferential flexural modes (CFMs) are investigated. Two different approaches based on numerical and analytical models are compared. In the numerical model, an energy method based on the Lagrange equations is used to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is solved by using the implicit Runge–Kutta numerical method. In the analytical model, a reduced form of the Sanders–Koiter theory assuming small circumferential and tangential shear deformations is used to get the nonlinear ordinary differential equations of motion, which are solved by using the multiple scales analytical method. The transition from energy beating to energy localization in the nonlinear field is studied. The effect of the aspect ratio on the analytical and numerical values of the nonlinear energy localization threshold for different boundary conditions is investigated.

Observations of runup and energy flux on a low‐slope beach with high‐energy, long‐period ocean swell

AbstractThe transformation of surface gravity waves from 11 m depth to runup was observed on the low‐sloped (1/80) Agate Beach, Oregon, with a cross‐shore transect of current meters, pressure sensors, and a scanning lidar. Offshore wave heights H0 ranged from calm (0.5 m) to energetic (&gt;7 m). Runup, measured with pressure sensors and a scanning lidar, increases linearly with (H0L0)1/2, with L0 the deep‐water wavelength of the spectral peak. Runup saturation, in which runup oscillations plateau despite further increases in (H0L0)1/2, is not observed. Infragravity wave shoaling and nonlinear energy exchanges with short waves are included in an infragravity wave energy balance. This balance closes for high‐infragravity frequencies (0.025–0.04 Hz) but not lower frequencies (0.003–0.025 Hz), possibly owing to unmodeled infragravity energy losses of wave breaking and/or bottom friction. Dissipative processes limit, but do not entirely damp, increases in runup excursions in response to increased incident wave forcing.

Identification of mode couplings in nonlinear vibrations of the steelpan

The vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying 1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics.