The multiple scattering of waves in irregular media

1968 ◽  
Vol 262 (1133) ◽  
pp. 609-640 ◽  
Keyword(s):  
Wave Field ◽  
Probability Distributions ◽  
Essential Feature ◽  
Mean Square ◽  
Angular Power Spectrum ◽  
Absorbing Medium ◽  
Phase Fluctuations ◽  
Field Fluctuations ◽  
The Mean ◽  
Wave Normal

When a wave passes through a large thickness of a non-absorbing medium containing weak random irregularities of refractive index, large amplitude and phase fluctuations of the wave field can develop. The probability distributions of these fluctuations are important, since they may be readily observed and from them can be found the mean square amplitudes of the fluctuations. This paper shows how to calculate these distributions and also the ‘ angular power spectrum ’ for an assembly of media which are statistically stationary with respect to variations in time, and in space for directions perpendicular to the wave normal of the incident wave. The scattered field at a given point is resolved into two components in phase and in quadrature with the residual unscattered wave at that point. The assembly averages of the powers in these two components, and of their correlation coefficient are found, and a set of three integro-differential equations is constructed which show how these three quantities vary as the medium is traversed. The probability distributions of amplitude and phase of the wave field at any point in the medium are functions of these three quantities which are found by integrating the equations through the medium. An essential feature of these equations is that they include waves which have been scattered several or m any times (multiple scatter). The equations are solved analytically for some particular cases. Solutions for the general case have been obtained numerically and are presented, together with the corresponding probability distributions of the field fluctuations and their average values.

On the dynamics of unsteady gravity waves of finite amplitude Part 2. Local properties of a random wave field

1961 ◽  
Vol 11 (1) ◽  
pp. 143-155 ◽  
Author(s):  
O. M. Phillips
Keyword(s):  
Kinetic Energy ◽  
Wave Field ◽  
Root Mean Square ◽  
Surface Displacement ◽  
Higher Order ◽  
Random Wave ◽  
Order Effects ◽  
Mean Square ◽  
Local Properties ◽  
The Mean

Expressions in closed form are derived for a number of local properties of a random, irrotational wave field. They are: (i) the mean potential and kinetic energies per unit projected area; (ii) the energy balance among the processes of energy input from the surface pressure fluctuations, rate of growth of potential and kinetic energy and horizontal energy flux; and (iii) the partition between potential and kinetic energy. These expressions are mainly in terms of quantities measured at the free surface, which are therefore functions of only two spatial variables (x, y) and of time t.Approximations for these expressions can be found simply by subsequent expansion methods; the fourth order being the highest for which the assumption of irrotational motion is appropriate in a real fluid. It is shown that the mean product of any three first-order quantities is of fourth or higher order in the root-mean-square wave slope, and this result is applied in estimating the magnitude of some higher order effects. In particular, the skewness of the surface displacement is of the order of the root-mean-square surface slope, which has been confirmed observationally by Kinsman (1960).

The multiple scattering of waves in irregular media. II. Spatial autocorrelation functions

1968 ◽  
Vol 307 (1491) ◽  
pp. 471-492 ◽  
Keyword(s):  
Refractive Index ◽  
Spatial Autocorrelation ◽  
Probability Distributions ◽  
Essential Feature ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Autocorrelation Functions ◽  
Phase Fluctuations ◽  
Multiple Scatter ◽  
Special Cases

When a wave passes through a large thickness of a non-absorbing medium containing weak random irregularities of refractive index, large amplitude and phase fluctuations of the wave field can develop. In a previous paper it was shown how to calculate the probability distributions and average values of these fluctuations. An essential feature of the treatment was that it took into account waves which have been scattered many times (multiple scatter). The present paper considers the spatial autocorrelation functions of the intensity and phase fluctuations in a plane parallel to the wave front of the incident wave for conditions of multiple scatter. These autocorrelation functions are important since they are used in studying the scintillation of radio signals from stellar sources, and yield information about the scattering medium causing the scintillations. The autocorrelation functions of the field quantities depend on the moment matrix of a four-dimensional joint probability distribution. The moments, which are elements of the matrix, are certain average values of the scattered field. A set of integro-differential equations is formulated for the moments and solved analytically for some special cases. General solutions for the moments are obtained by numerical integration, in the case where the irregularities of refractive index have a Gaussian autocorrelation function. Curves for the spatial autocorrelation functions of intensity and phase in this case are given for different distances in the medium.

The dispersion relation for a nonlinear random gravity wave field

1976 ◽  
Vol 75 (2) ◽  
pp. 337-345 ◽  
Author(s):  
Norden E. Huang ◽  
Chi-Chao Tung
Keyword(s):  
Dispersion Relation ◽  
Phase Velocity ◽  
Wave Field ◽  
Gravity Wave ◽  
Root Mean Square ◽  
Surface Velocity ◽  
Mean Deviation ◽  
Mean Square ◽  
Local Phase ◽  
The Mean

The dispersion relation for a random gravity wave field is derived using the complete system of nonlinear equations. It is found that the generally accepted dispersion relation is only a first-order approximation to the mean value. The correction to this approximation is expressed in terms of the energy spectral function of the wave field. The non-zero mean deviation is proportional to the ratio of the mean Eulerian velocity at the surface and the local phase velocity. In addition to the mean deviation, there is a random scatter. The root-mean-square value of this scatter is proportional to the ratio of the root-mean-square surface velocity and the local phase velocity. As for the phase velocity, the nonzero mean deviation is equal to the mean Eulerian velocity while the root-mean-square scatter is equal to the root-mean-square surface velocity. Special cases are considered and a comparison with experimental data is also discussed.

Asteroid and Comet Impact Speeds upon Mars: Significance for Panspermia and the Supply of Organics

1997 ◽  
Vol 161 ◽  
pp. 197-201 ◽  
Author(s):  
Duncan Steel
Keyword(s):  
Probability Distributions ◽  
Regions Of Interest ◽  
Significant Fraction ◽  
Organic Chemicals ◽  
Impact Speed ◽  
The Mean ◽  
The Impact

AbstractWhilst lithopanspermia depends upon massive impacts occurring at a speed above some limit, the intact delivery of organic chemicals or other volatiles to a planet requires the impact speed to be below some other limit such that a significant fraction of that material escapes destruction. Thus the two opposite ends of the impact speed distributions are the regions of interest in the bioastronomical context, whereas much modelling work on impacts delivers, or makes use of, only the mean speed. Here the probability distributions of impact speeds upon Mars are calculated for (i) the orbital distribution of known asteroids; and (ii) the expected distribution of near-parabolic cometary orbits. It is found that cometary impacts are far more likely to eject rocks from Mars (over 99 percent of the cometary impacts are at speeds above 20 km/sec, but at most 5 percent of the asteroidal impacts); paradoxically, the objects impacting at speeds low enough to make organic/volatile survival possible (the asteroids) are those which are depleted in such species.

The Bordeaux Automatic Meridian Circle

1978 ◽  
Vol 48 ◽  
pp. 227-228
Author(s):  
Y. Requième
Keyword(s):  
Mean Square Error ◽  
Mean Square ◽  
Meridian Circle ◽  
The Mean ◽  
Full Automation

In spite of important delays in the initial planning, the full automation of the Bordeaux meridian circle is progressing well and will be ready for regular observations by the middle of the next year. It is expected that the mean square error for one observation will be about ±0.”10 in the two coordinates for declinations up to 87°.

The mean-square generalized delayed decision feedback sequence detector

2003 ◽  
Vol 14 (3) ◽  
pp. 265-268 ◽  
Author(s):  
Maurizio Magarini ◽  
Arnaldo Spalvieri ◽  
Guido Tartara
Keyword(s):  
Decision Feedback ◽  
Mean Square ◽  
The Mean

Limit tolerances for sums of measurement errors

2018 ◽  
Vol 934 (4) ◽  
pp. 59-62
Author(s):  
V.I. Salnikov
Keyword(s):  
Normal Distribution ◽  
Mean Square Error ◽  
Gaussian Distribution ◽  
Measurement Errors ◽  
Theory And Practice ◽  
Mean Square ◽  
The Mean ◽  
Limiting Values ◽  
Limiting Value

The question of calculating the limiting values of residuals in geodesic constructions is considered in the case when the limiting value for measurement errors is assumed equal to 3m, ie ∆рred = 3m, where m is the mean square error of the measurement. Larger errors are rejected. At present, the limiting value for the residual is calculated by the formula 3m√n, where n is the number of measurements. The article draws attention to two contradictions between theory and practice arising from the use of this formula. First, the formula is derived from the classical law of the normal Gaussian distribution, and it is applied to the truncated law of the normal distribution. And, secondly, as shown in [1], when ∆рred = 2m, the sums of errors naturally take the value equal to ?pred, after which the number of errors in the sum starts anew. This article establishes its validity for ∆рred = 3m. A table of comparative values of the tolerances valid and recommended for more stringent ones is given. The article gives a graph of applied and recommended tolerances for ∆рred = 3m.

scholarly journals Electrical stimulator with mechanomyography-based real-time monitoring, muscle fatigue detection, and safety shut-off: a pilot study

2020 ◽  
Vol 65 (4) ◽  
pp. 461-468
Author(s):  
Jannatul Naeem ◽  
Nur Azah Hamzaid ◽  
Amelia Wong Azman ◽  
Manfred Bijak
Keyword(s):  
Muscle Fatigue ◽  
Real Time ◽  
Isometric Force ◽  
Mean Square ◽  
Clear Indication ◽  
Initial Value ◽  
Cord Injury ◽  
The Mean ◽  
Electrical Stimulator ◽  
Fatigue Detection

AbstractFunctional electrical stimulation (FES) has been used to produce force-related activities on the paralyzed muscle among spinal cord injury (SCI) individuals. Early muscle fatigue is an issue in all FES applications. If not properly monitored, overstimulation can occur, which can lead to muscle damage. A real-time mechanomyography (MMG)-based FES system was implemented on the quadriceps muscles of three individuals with SCI to generate an isometric force on both legs. Three threshold drop levels of MMG-root mean square (MMG-RMS) feature (thr50, thr60, and thr70; representing 50%, 60%, and 70% drop from initial MMG-RMS values, respectively) were used to terminate the stimulation session. The mean stimulation time increased when the MMG-RMS drop threshold increased (thr50: 22.7 s, thr60: 25.7 s, and thr70: 27.3 s), indicating longer sessions when lower performance drop was allowed. Moreover, at thr70, the torque dropped below 50% from the initial value in 14 trials, more than at thr50 and thr60. This is a clear indication of muscle fatigue detection using the MMG-RMS value. The stimulation time at thr70 was significantly longer (p = 0.013) than that at thr50. The results demonstrated that a real-time MMG-based FES monitoring system has the potential to prevent the onset of critical muscle fatigue in individuals with SCI in prolonged FES sessions.

Die Molekülstruktur des N-Chlorsdiwefeloxiddifluoridimids ClNSOF2 / The Molecular Structure of ClNSOF2

1974 ◽  
Vol 29 (6) ◽  
pp. 901-904 ◽  
Author(s):  
O. Oberhammer ◽  
O. Glemser ◽  
H. Klüver
Keyword(s):  
Molecular Structure ◽  
Electron Diffraction ◽  
Geometric Parameters ◽  
Mean Square ◽  
The Mean

The molecular structure of ClNSOF2 was determined by electron diffraction of gases. The following geometric parameters were obtained:Cl-N=1.715(5), S=N=1.484(7), S=O=1.394(3), S-F=1.548(3) Å, ∢ ClNS=114.7 (8), ∢ FSF=92.6(.8), ∢ NSF=111.8(.9) ∢ NSO=117.4 (3.1) and ∢ OSF=108.6 (.8)°. The results for the mean square amplitudes of vibration are given in the paper and an attempt is made to explain differences in corresponding parameters of some related molecules.

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