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At least 577 records · Page 32

Exploring 3D elastic-wave scattering at interfaces using high-resolution phased-array system

The elastic-wave scattering at interfaces, such as cracks, is essential for nondestructive inspections, and hence, understanding the phenomenon is crucial. However, the elastic-wave scattering at cracks is very complex in three dimensions since microscopic asperities of crack faces can be multiple scattering sources. We propose a method for exploring 3D elastic-wave scattering based on our previously developed high-resolution 3D phased-array system, the piezoelectric and laser ultrasonic system (PLUS). We describe the principle of PLUS, which combines a piezoelectric transmitter and a 2D mechanical scan of a laser Doppler vibrometer, enabling us to resolve a crack into a collection of scattring sources. Subsequently, we show how the 3D elastic-wave scattering in the vicinity of each response can be extracted. Here, we experimentally applied PLUS to a fatigue-crack specimen. We found that diverse 3D elastic-wave scattering occurred in a manner depending on the responses within the fatigue crack. This is significant because access to such information will be useful for optimizing inspection conditions, designing ultrasonic measurement systems, and characterizing cracks. More importantly, the described methodology is very general and can be applied to not only metals but also other materials such as composites, concrete, and rocks, leading to progress in many fields.

36 MATERIALS SCIENCE↗

Propeller noise prediction

Analytic propeller noise prediction involves a sequence of computations culminating in the application of acoustic equations. The prediction sequence currently used by NASA in its ANOPP (aircraft noise prediction) program is described. The elements of the sequence are called program modules. The first group of modules analyzes the propeller geometry, the aerodynamics, including both potential and boundary layer flow, the propeller performance, and the surface loading distribution. This group of modules is based entirely on aerodynamic strip theory. The next group of modules deals with the actual noise prediction, based on data from the first group. Deterministic predictions of periodic thickness and loading noise are made using Farassat's time-domain methods. Broadband noise is predicted by the semi-empirical Schlinker-Amiet method. Near-field predictions of fuselage surface pressures include the effects of boundary layer refraction and (for a cylinder) scattering. Far-field predictions include atmospheric and ground effects. Experimental data from subsonic and transonic propellers are compared and NASA's future direction is propeller noise technology development are indicated.

Zorumski, W. E.↗

Linear Interaction Approximation for the Prediction of Noise Generated by Turbulence/Shock Interaction

In this paper we review the basic development of Linear Interaction Approximation (LIA), a linear inviscid approach to study the interaction of weak turbulence with flow discontinuities, and implement a formulation to calculate the acoustic field due to the incidence of vortical and entropic disturbances on a planar shock. The work is motivated by the observance in nozzles with internal mixers and external plugs of a shock formed by local flow accelerations, even at sub-critical pressure ratios. Exhaust systems with internally mixed/externally plugged nozzles are currently under consideration for use in commercial supersonic aircraft and it is important to have an assessment of the impact of this shock on the noise radiated by these nozzles. Numerical results are presented for the fluctuating pressure emitted by (separate) incident single-wave vorticity and entropy modes and for incident isotropic fields of turbulence and entropy. Further applications of the formulation to the flow fields in internally mixed/externally plugged nozzles, and to the modification of mixing noise sources and the scattering of internally generated noise by the shock, are discussed as future work.

: Jet noise, shock waves, aircraft noise↗

Linear Interaction Approximation for the Prediction of Noise Generated by Turbulence/Shock Interactions

In this paper we review the basic development of Linear Interaction Approximation (LIA), a linear inviscid approach to study the interaction of weak turbulence with flow discontinuities, and implement a formulation to calculate the acoustic field due to the incidence of vortical and entropic disturbances on a planar shock. The work is motivated by the observance in nozzles with internal mixers and external plugs of a shock formed by local flow accelerations, even at sub-critical pressure ratios. Exhaust systems with internally mixed/externally plugged nozzles are currently under consideration for use in commercial supersonic aircraft and it is important to have an assessment of the impact of this shock on the noise radiated by these nozzles. Numerical results are presented for the fluctuating pressure emitted by (separate) incident single-wave vorticity and entropy modes and for incident isotropic fields of turbulence and entropy. Further applications of the formulation to the flow fields in internally mixed/externally plugged nozzles, and to the modification of mixing noise sources and the scattering of internally generated noise by the shock, are discussed as future work.

Jet noise, shock waves, aircraft noise↗

Linear Interaction Approximation for the Prediction of Noise Generated by Turbulence/Shock Interaction

In this paper we review the basic development of Linear Interaction Approximation (LIA), a linear inviscid approach to study the interaction of weak turbulence with flow discontinuities, and implement a formulation to calculate the acoustic field due to the incidence of vortical and entropic disturbances on a planar shock. The work is motivated by the observance in nozzles with internal mixers and external plugs of a shock formed by local flow accelerations, even at sub-critical pressure ratios. Exhaust systems with internally mixed/externally plugged nozzles are currently under consideration for use in commercial supersonic aircraft and it is important to have an assessment of the impact of this shock on the noise radiated by these nozzles. Numerical results are presented for the fluctuating pressure emitted by (separate) incident single-wave vorticity and entropy modes and for incident isotropic fields of turbulence and entropy. Further applications of the formulation to the flow fields in internally mixed/externally plugged nozzles, and to the modification of mixing noise sources and the scattering of internally generated noise by the shock, are discussed as future work.

Jet noise↗

Temperature-dependent phonon lifetimes and thermal conductivity of silicon by inelastic neutron scattering and ab initio calculations

In this paper, inelastic neutron scattering on a single crystal of silicon was performed at temperatures from 100 to 1500 K. These experimental data were reduced to obtain phonon spectral intensity at all wave vectors $\overset{→}{Q}$ and frequencies ω in the first Brillouin zone. Thermal broadenings of the phonon peaks were obtained by fitting and by calculating with an iterative ab initio method that uses thermal atom displacements on an ensemble of superlattices. Agreement between the calculated and experimental broadenings was good, with possible discrepancies at the highest temperatures. Distributions of phonon widths versus phonon energy had similar shapes for computation and experiment. These distributions grew with temperature but maintained similar shapes. Parameters from the ab initio calculations were used to obtain the thermal conductivity from the Boltzmann transport equation, which was in good agreement with experimental data. Despite the high group velocities of longitudinal acoustic phonons, their shorter lifetimes reduced their contribution to the thermal conductivity, which was dominated by transverse acoustic modes.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Comparing the DES-SN5YR and Pantheon+ SN cosmology analyses: investigation based on ‘evolving dark energy or supernovae systematics’?

Recent cosmological analyses measuring distances of type Ia supernovae (SNe Ia) and baryon acoustic oscillations (BAO) have all given similar hints at time-evolving dark energy. To examine whether underestimated SN Ia systematics might be driving these results, Efstathiou (2025) compared overlapping SN events between Pantheon+ and DES-SN5YR (20 per cent SNe are in common), and reported evidence for an $\sim$0.04 mag offset between the low- and high-redshift distance measurements of this subsample of events. If this offset is arbitrarily subtracted from the entire DES-SN5YR sample, the preference for evolving dark energy is reduced. In this paper, we show that this offset is mostly due to different corrections for Malmquist bias between the two samples; therefore, an object-to-object comparison can be misleading. Malmquist bias corrections differ between the two analyses for several reasons. First, DES-SN5YR used an improved model of SN Ia luminosity scatter compared to Pantheon+ but the associated scatter-model uncertainties are included in the error budget. Secondly, improvements in host mass estimates in DES-SN5YR also affected SN standardized magnitudes and their bias corrections. Thirdly, and most importantly, the selection functions of the two compilations are significantly different, hence the inferred Malmquist bias corrections. Even if the original scatter model and host properties from Pantheon+ are used instead, the evidence for evolving dark energy from CMB, DESI BAO Year 1 and DES-SN5YR is only reduced from 3.9$\sigma$ to 3.3$\sigma$, consistent with the error budget. Finally, in this investigation, we identify an underestimated systematic uncertainty related to host galaxy property uncertainties, which could increase the final DES-SN5YR error budget by 3 per cent. In conclusion, we confirm the validity of the published DES-SN5YR results.

79 ASTRONOMY AND ASTROPHYSICS↗

Excitation of solar p-modes

We investigate the rates at which energy is supplied to individual p-modes as a function of their frequencies nu and angular degrees l. The observationally determined rates are compared with those calculated on the hypothesis that the modes are stochastically excited by turbulent convection. The observationally determined excitation rate is assumed to be equal to the product of the mode's energy E and its (radian) line width Gamma. We obtain E from the mode's mean square surface velocity with the aid of its velocity eigenfuction. We assume that Gamma measures the mode's energy decay rate, even though quasi-elastic scattering may dominate true absorption. At fixed l, E(Gamma) arises as nu(exp 7) at low nu, reaches a peak at nu approximately equal 3.5 mHz, and then declines as nu(exp 4.4) at higher nu . At fixed nu, E(Gamma) exhibits a slow decline with increasing l. To calculate energy input rates, P(sub alpha), we rely on the mixing-length model of turbulent convection. We find entropy fluctuations to be about an order of magnitude more effective than the Reynolds stress in exciting p-modes . The calculated P(sub alpha) mimic the nu(exp 7) dependence of E(Gamma) at low nu and the nu(exp -4.4) dependence at high nu. The break of 11.4 powers in the nu-dependence of E(Gamma) across its peak is attributed to a combination of (1) the reflection of high-frequency acoustic waves just below the photosphere where the scale height drops precipitously and (2) the absence of energy-bearing eddies with short enough correlation times to excite high-frequency modes. Two parameters associated with the eddy correlation time are required to match the location and shape of the break. The appropriate values of these parameters, while not unnatural, are poorly constrained by theory. The calculated P(sub alpha) can also be made to fit the magnitude of E(Gamma) with a reasonable value for the eddy aspect ratio. Our resutls suggest a possible explanation for the decline of mode energy with increasing l at fixed nu. Entropy fluctuations couple to changes in volume associated with the oscillation mode. These decrease with decreasing n at fixed nu, becoming almost zero for the f-mode.

Goldreich, Peter↗

Pressure transfer function of a JT15D nozzle due to acoustic and convected entropy fluctuations

An acoustic transmission matrix analysis of sound propagation in a variable area duct with and without flow is extended to include convected entropy fluctuations. The boundary conditions used in the analysis are a transfer function relating entropy and pressure at the nozzle inlet and the nozzle exit impedance. The nozzle pressure transfer function calculated is compared with JT15D turbofan engine nozzle data. The one dimensional theory for sound propagation in a variable area nozzle with flow but without convected entropy is good at the low engine speeds where the nozzle exit Mach number is low (M=0.2) and the duct exit impedance model is good. The effect of convected entropy appears to be so negligible that it is obscured by the inaccuracy of the nozzle exit impedance model, the lack of information on the magnitude of the convected entropy and its phase relationship with the pressure, and the scatter in the data. An improved duct exit impedance model is required at the higher engine speeds where the nozzle exit Mach number is high (M=0.56) and at low frequencies (below 120 Hz).

Miles, J. H.↗

DESI 2024 III: baryon acoustic oscillations from galaxies and quasars

We present the DESI 2024 galaxy and quasar baryon acoustic oscillations (BAO) measurements using over 5.7 million unique galaxy and quasar redshifts in the range 0.1 < z < 2.1. Divided by tracer type, we utilize 300,017 galaxies from the magnitude-limited Bright Galaxy Survey with 0.1 < z < 0.4, 2,138,600 Luminous Red Galaxies with 0.4 < z < 1.1, 2,432,022 Emission Line Galaxies with 0.8 < z < 1.6, and 856,652 quasars with 0.8 < z < 2.1, over a ∼ 7,500 square degree footprint. The analysis was blinded at the catalog-level to avoid confirmation bias. All fiducial choices of the BAO fitting and reconstruction methodology, as well as the size of the systematic errors, were determined on the basis of the tests with mock catalogs and the blinded data catalogs. We present several improvements to the BAO analysis pipeline, including enhancing the BAO fitting and reconstruction methods in a more physically-motivated direction, and also present results using combinations of tracers. We employ a unified BAO analysis method across all tracers. We present a re-analysis of SDSS BOSS and eBOSS results applying the improved DESI methodology and find scatter consistent with the level of the quoted SDSS theoretical systematic uncertainties. With the total effective survey volume of ∼ 18 Gpc3, the combined precision of the BAO measurements across the six different redshift bins is ∼0.52%, marking a 1.2-fold improvement over the previous state-of-the-art results using only first-year data. We detect the BAO in all of these six redshift bins. The highest significance of BAO detection is 9.1σ at the effective redshift of 0.93, with a constraint of 0.86% placed on the BAO scale. We find that our observed BAO scales are systematically larger than the prediction of the Planck 2018-ΛCDM at z < 0.8. We translate the results into transverse comoving distance and radial Hubble distance measurements, which are used to constrain cosmological models in our companion paper.

79 ASTRONOMY AND ASTROPHYSICS↗

The Local Seismoacoustic Wavefield of a Research Nuclear Reactor and Its Response to Reactor Power Level

In this article, we describe the seismoacoustic wavefield recorded outdoors but inside the facility fence of the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory (Tennessee). HFIR is a research nuclear reactor that generates neutrons for scattering, irradiation research, and isotope production. This reactor operates at a nominal power of 85 MW, with a full-power period between 24 and 26 days. This study uses data from a single seismoacoustic station that operated for 60 days and sampled a full operating reactor cycle, that is, full-power operation and end-of-cycle outage. The analysis presented here is based on identifying signals that characterize the steady, that is, full-power operation and end-of-cycle outage, and transitional, that is, start-up and shutdown, states of the reactor. We found that the overall seismoacoustic energy closely follows the main power cycle of the reactor and identified spectral regions excited by specific reactor operational conditions. In particular, we identified a tonal noise sequence with a fundamental frequency around 21.4 Hz and multiple harmonics that emerge as the reactor reaches 90% of nominal power in both seismic and acoustic channels. We also utilized temperature measurements from the monitoring system of the reactor to suggest links between the operation of reactor’s subsystems and seismoacoustic signals. We demonstrate that seismoacoustic monitoring of an industrial facility can identify and track some industrial processes and detect events related to operations that involve energy transport.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Random and systematic measurement errors in acoustic impedance as determined by the transmission line method

The effect of random and systematic errors associated with the measurement of normal incidence acoustic impedance in a zero-mean-flow environment was investigated by the transmission line method. The influence of random measurement errors in the reflection coefficients and pressure minima positions was investigated by computing fractional standard deviations of the normalized impedance. Both the standard techniques of random process theory and a simplified technique were used. Over a wavelength range of 68 to 10 cm random measurement errors in the reflection coefficients and pressure minima positions could be described adequately by normal probability distributions with standard deviations of 0.001 and 0.0098 cm, respectively. An error propagation technique based on the observed concentration of the probability density functions was found to give essentially the same results but with a computation time of about 1 percent of that required for the standard technique. The results suggest that careful experimental design reduces the effect of random measurement errors to insignificant levels for moderate ranges of test specimen impedance component magnitudes. Most of the observed random scatter can be attributed to lack of control by the mounting arrangement over mechanical boundary conditions of the test sample.

Parrott, T. L.↗

Direct Measurement of the Return Current Instability in a Laser-Produced Plasma

Measurements were made of the return current instability growth rate, demonstrating its concurrence with nonlocal transport. Thomson scattering was used to measure a maximum growth rate of 5.1 × 10 9 Hz, which was 3 times less than classical Spitzer-Härm theory predicts. Here, the measured plasma conditions indicate the heat flux was nonlocal, and Vlasov–Fokker-Planck simulations that account for nonlocality reproduce the measured growth rates. Furthermore, the threshold for the return current instability was measured (δ T = 0.017 ± 0.002) to be in good agreement with previous theoretical models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Noise Radiation from a Continuous Mold-Line Link Flap Configuration

The results of an experimental study of the noise from a Continuous Mold-Line Link (CML) flap are presented. Acoustic and unsteady surface pressure measurements were performed on a main element wing section with a half-span CML flap in NASA Langley s Quiet Flow Facility. The acoustic data were acquired with a medium aperture directional array (MADA) of microphones. The Deconvolution Approach for the Mapping of Acoustic Sources (DAMAS) method is applied to determine the spatial distribution and strength of the noise sources over the surface of the test model. A Coherent Output Power (COP) method which relates the output from unsteady surface pressure sensors to the output of the MADA is also used to obtain more detailed characteristics of the noise source distribution in the trailing edge region of the CML. These results are compared to those obtained for a blunt flap to quantify the level of noise benefit that is achieved with the CML flap. The results indicate that the noise from the CML region of the flap is 5 to 17 dB lower (depending on flap deflection and Mach number) than the noise from the side edge region of the blunt flap. Lower noise levels are obtained for all frequencies. Spectral analysis of the noise from the cove region of the CML and blunt flap models also reveal a spectral peak in the high frequency range that is related to noise scattering at the trailing edge of the main element. The peaks in the CML and blunt flap cove noise spectra are close in level and often exceed blunt side edge noise. Applying a strip of serrated tape to the trailing edge of the CML flap model main airfoil, reduced the peak but increased other noise somewhat. Directivity measurements show that the CML flap can be more directional than the blunt flap.

Hutcheson, Florence V.↗

Noise Radiation from a Continuous Mold-Line Link Flap Configuration

The results of an experimental study of the noise from a Continuous Mold-Line Link (CML) flap are presented. Acoustic and unsteady surface pressure measurements were performed on a main element wing section with a half-span CML flap in NASA Langley s Quiet Flow Facility. The acoustic data were acquired with a medium aperture directional array (MADA) of microphones. The Deconvolution Approach for the Mapping of Acoustic Sources (DAMAS) method is applied to determine the spatial distribution and strength of the noise sources over the surface of the test model. A Coherent Output Power (COP) method which relates the output from unsteady surface pressure sensors to the output of the MADA is also used to obtain more detailed characteristics of the noise source distribution in the trailing edge region of the CML. These results are compared to those obtained for a blunt flap to quantify the level of noise benefit that is achieved with the CML flap. The results indicate that the noise from the CML region of the flap is 5 to 17 dB lower (depending on flap deflection and Mach number) than the noise from the side edge region of the blunt flap. Lower noise levels are obtained for all frequencies. Spectral analysis of the noise from the cove region of the CML and blunt flap models also reveal a spectral peak in the high frequency range that is related to noise scattering at the trailing edge of the main element. The peaks in the CML and blunt flap cove noise spectra are close in level and often exceed blunt side edge noise. Applying a strip of serrated tape to the trailing edge of the CML flap model main airfoil reduced the peak but increased other noise somewhat. Directivity measurements show that the CML flap can be more directional than the blunt flap.

Hutcheson, Florence V.↗

[?]Nonlinear Issues in the Aerothermochemistry of Gases and Materials and the Associated Physics and Dynamics of Interfaces

Our research and technology are focused on nonlinear issues in the aerothermochemistry of gases and materials and the associated physics and dynamics of interfaces. Our program is now organized to aggressively support the NASA Aeronautics Enterprise so as to: (a) develop a new generation of environmentally compatible, economic subsonic aircraft; (b) develop the technology base for an economically viable and environmentally compatible high-speed civil transport; (c) develop the technology options for new capabilities in high-performance aircraft; (d) develop hypersonic technologies for air-breathing flight; and (e) develop advanced concepts, understanding of physical phenomena, and theoretical, experimental, and computational tools for advanced aerospace systems. The implications from our research for aeronautical and aerospace technology have been both broad and deep. For example, using advanced computational techniques, we have determined exact solutions for the Schrodinger equation in electron-molecule scattering allowing us to evaluate atmospheric models important to reentry physics. We have also found a new class of exact solutions for the Navier Stokes equations. In experimental fluid dynamics, we have found explicit evidence of turbulence modification of droplet sizes in shock tube flow with condensation. We have developed a new diagnostic tool for the direct estimation of flow velocities at MHz sampling rates in quasi-one dimensional turbulent flow. This procedure suggests an unexpected confirmation of the possibility of 'natural' closure in Reynolds stresses with deep implications for the development of turbulent models. A transient increase is observed in both the spectral energy decay rate and the degree of chaotic complexity at the interface of a shock wave and a turbulent ionized gas. Even though the gas is apparently brought to rest by the shock wave, no evidence is found either of the expected relaminarization. A unique diamond-shaped nozzle has been designed for a detailed investigation of the effect of significant streamwise vorticity on the acoustic and IR characteristics of supersonic jets. Our results provide convincing evidence of the significant effect of vorticity on the far-field noise for the diamond jet as compared to the conventional round jets. We have found that the countercurrent shear layer mixes much more efficiently than conventional coflowing shear layers. We also developed the fluid thrust vectoring procedures which use counter flow to vector a jet. Our materials research has shown that the steep stress gradients at the fiber-matrix interface could be the primary cause of interface cracks after the processing of metallic and intermetallic matrix composites. New techniques have been evolved for: the microcharacterization of materials including microplastic strain and, point by point, the misorientation and plasticity for matrix composites; thermally induced stress measurements and load relaxation; the growth and characterization of metallic matrix composite interfaces; and for the growth of ferrite materials by pulsed laser deposition. The FAMU commitment to the HBCU Research Center also continues to be broad and deep.

Johnson, Joseph A., III↗

Spray measurements of aerothermodynamic effect on disintegrating liquid jets

An experimental investigation was made to determine the effect of atomizing gas mass flux and temperature on liquid jet breakup in sonic velocity gas flow. Characteristic drop size data were obtained by using the following atomizing gases: nitrogen, argon, and helium to breakup water jets in high velocity gas flow. A scattered light scanning instrument developed at Lewis Research Center was used to measure Sauter mean diameter (SMD). The three gases gave a molecular weight range of 4 to 40 and atomizing gas mass flux and temperature were varied from 6 to 50 g/sq cm and 275-400 K, respectively. The ratio of liquid jet diameter to SMD, D(sub 0)/D(sub 32), was correlated with aerodynamic and liquid-surface force ratios, i.e., the product of the Weber and Reynolds number, We Re, the gas to liquid density ratio, rho(sub g)/rho(sub 1) g and also the molecular scale dimensionless group, rho(sub 1)(Vm exp 3)/ mu(sub 1) g, to give the following expression: D(sub 0)/D(sub 32) = 0.90 x 10(exp -8) x (We Re rho sub g/rho sub 1)exp 0.44 x (rho sub 1 Vm exp 3/mu sub 1 g)exp 0.67 where We Re = ((rho sub g)exp 2(D sub 0)exp 2(V sub C)exp3))/ mu sub 1 sigma, mu sub 1 is liquid viscosity, sigma is surface tension, V sub C is the acoustic gas velocity, V sub m is the RMS velocity of gas molecules, and g is the acceleration of gas molecules due to gravity. Good agreement was obtained with atomization theory for liquid-jet breakup in the regime of aerodynamic stripping. Also, due to its low molecular weight and high acoustic velocity, helium was considerably more effective than nitrogen or argon in producing small-droplet sprays with values of D(sub 32) on the order of 5 microns.

Ingebo, Robert D.↗

Cross-beam energy transfer in direct-drive ICF. II. Theory and simulation of mitigation through increased laser bandwidth

The response of the cross-beam energy transfer instability (CBET) to laser bandwidth is investigated through a combination of theory and simulation. Existing linear theory is generalized to treat broadband lasers, demonstrating that CBET is most effectively suppressed when the bandwidth exceeds the ion-acoustic wave (IAW) frequency. It is shown that for such bandwidths, reverse (seed to pump) transfer becomes possible, which reduces the net energy transfer rapidly as bandwidth is increased. The CBET gain exponent in this regime scales with bandwidth (Δω) as Δω−3 for Gaussian or Lorentzian laser spectra with different scalings possible for other spectra. Comparison of our theory with linearized fluid and particle-in-cell simulations, performed with the laser-plasma simulation environment (LPSE) and vector particle in cell (VPIC) codes, respectively, finds that the model is accurate in the absence of nonlinear processes. However, linear analysis also finds that the IAW energy density scales as Δω−1, implying that nonlinear effects may be more difficult to control than the CBET scaling would suggest. Indeed, nonlinear effects are found to be present in VPIC simulations with high-intensity lasers, despite minimal apparent CBET. Nonlinear processes in the VPIC cases include particle trapping, the two-ion wave decay, and ion wave self-focusing. In some high intensity VPIC cases, these effects lead to net energy transfer from seed to pump and increases to backscatter stimulated Brillouin scattering reflectivities. Finally, for a given bandwidth, we show that improved control of nonlinear processes can be achieved via smoothing by spectral dispersion.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗