Constant Phase Lag Research Articles

Dynamics of a magnetic fluid droplet in a rotating field

The response of a magnetic fluid microdrop to a rotating magnetic field is studied numerically in 2D by the boundary element method (BEM). On increasing field frequency, the motion of the droplet goes through a transition from a state where the droplet follows the magnetic field with a constant phase lag to a state where the phase lag increases in a series of kinks when the field frequency passes the critical one. The equations of the droplet motion are derived analytically and good agreement with the BEM is obtained.

The spinal GABA system modulates burst frequency and intersegmental coordination in the lamprey: differential effects of GABAA and GABAB receptors

1. The effect of spinal GABAergic neurons on the segmental neuronal network generating locomotion has been analyzed in the lamprey spinal cord in vitro. It is shown that gamma-aminobutyric acid (GABA)A- and GABAB-mediated effects influence the burst frequency and the intersegmental coordination and that the GABA system is active during normal locomotor activity. 2. Fictive locomotor activity was induced by superfusing the spinal cord with a Ringer solution containing N-methyl-D-aspartate (NMDA, 150 microM). The efferent locomotor activity was recorded by suction electrodes from the ventral roots or intracellularly from interneurons or motoneurons. If a GABA uptake blocker was added to the perfusate, the burst rate decreased. This effect was counteracted by GABAB receptor blockade by phaclofen or 2-(OH)-saclofen. If instead a GABAB receptor agonist (baclofen) was added during fictive locomotion, a depression of the burst rate occurred. It was concluded that a GABAB receptor activation due to an endogenous release of GABA caused a depression of the burst activity with a maintained well-coordinated locomotor activity. 3. If a GABAA receptor antagonist (bicuculline) is applied during fictive locomotion elicited by NMDA, a certain increase of the burst rate occurred. Conversely, if a selective GABAA agonist (muscimol) was administered, the burst rate decreased. Similarly, if the GABAA receptor activity was potentiated by activation of a benzodiazepine site by diazepam, the burst rate was reduced. If, however the GABAergic effect was first enhanced by an uptake blocker (nipecotic acid), an administration of a GABAA antagonist (bicuculline) increased the burst rate, but in addition, the burst pattern became less regular with recurrent shorter periods without clear reciprocal burst activity. The GABAA receptor activity appears important for the rate control and for permitting a regular burst pattern. 4. The intersegmental coordination in the lamprey is characterized by a rostrocaudal constant phase lag of approximately 1% of the cycle duration between the activation of consecutive segments during forward swimming. This rostrocaudal phase lag can be reversed during backward swimming, which can be induced also experimentally in the isolated spinal cord by providing a higher excitability to the caudal segments. In a split-bath configuration, a GABA uptake blocker or a GABAB agonist was administered to the rostral part of the spinal cord, which caused a reversal of the phase lag as during backward swimming. If GABAA receptors were blocked under similar conditions, the intersegmental coordination became irregular. It is concluded that an increased GABA activity in a spinal cord region can modify the intersegmental coordination.(ABSTRACT TRUNCATED AT 400 WORDS)

Physiological and developmental aspects of intersegmental coordination in Xenopus embryos and tadpoles

The rostro-caudal delay in the activation of the segmentally organised trunk muscles is a consistent feature of forward swimming in the tadpoles of Xenopus laevis. Our evidence suggests that such a delay may be achieved because rostral neurons are associated with a higher level of excitability than caudal neurons and so fire first on cycles of swimming. In addition we have found developmentally related changes in the properties of rostro-caudal delays in Xenopus, whereby the duration of motor nerve activity increases, a constant phaselag becomes established and there is increased sensitivity to 5-HT. These changes may be directly related to the incorporation of descending serotonergic neurones into the spinal cord network, so providing a means of determining the role played by identifiable neural elements in intersegmental coordination in the Xenopus preparation.

Local serotonergic modulation of calcium-dependent potassium channels controls intersegmental coordination in the lamprey spinal cord.

1. The intersegmental coordination during undulatory locomotion in lamprey is characterized by a constant phase lag between consecutive segments, that is, the ratio between the intersegmental time lag and the cycle duration remains constant. It is shown that the spinal 5-HT (serotonin) system can, in a graded fashion, control the phase lag value from a rostrocaudal to a caudorostral lag corresponding to a reversed direction of swimming. These effects can be explained by a 5-HT-induced depression of Ca(2+)-dependent K+ channels (KCa channels) in network neurons. 2. The actions of the spinal 5-HT system were analyzed in the lamprey spinal cord preparation in vitro. Fictive swimming was induced by bath application of N-methyl-D-aspartate (NMDA). The intersegmental phase lag between ventral root burst activities was measured along the ipsilateral side of the spinal cord. The chamber with the preparation was partitioned into two pools so that the rostral and caudal halves of the preparation could be perfused independently with solutions containing the same level of NMDA (100-150 microM) with or without additional 5-HT or a 5-HT uptake blocker (citalopram). 3. Addition of 5-HT to one of these partitioned pools changed the intersegmental phase lag in this pool, whereas the cycle duration remained unchanged. It was determined by the activity in the "non-5-HT" pool. Addition of 5-HT to the caudal pool resulted in an increased rostrocaudal phase lag. When 5-HT was added to the rostral pool, on the other hand, the phase lag shifted direction to a backward coordination.(ABSTRACT TRUNCATED AT 250 WORDS)

Intersegmental co-ordination of undulatory movements— a “trailing oscillator” hypothesis

Intersegmental co-ordination in undulatory locomotion was investigated in the lamprey spinal cord in vitro. It is characterized by a constant phase lag between the activation of consecutive segments. By increasing the excitability (NMDA level) locally, a "travelling wave" could be induced to start in any segment and to be conducted in both rostral and caudal directions. This leading segment, the one with the highest excitability, determined the frequency of all other segments, and the ascending and descending couplings were functionally symmetric. The intersegmental phase lag can be explained by a simple neural mechanism in which a leading oscillator will entrain the adjacent "trailing" segments, which in turn entrain their neighbours and so forth.

Evolution of the lunar orbit with temperature‐ and frequency‐dependent dissipation

The coupled thermal‐dynamical evolution of the Earth‐Moon system is modeled. The lunar orbit is assumed noncircular and inclined to Earth's equator plane. Solid planet dissipation in the Earth and Moon is assumed to occur by temperature‐ and frequency‐dependent solid friction. The dissipative response of Earth's oceans is represented as an equilibrium ocean with constant phase lag. Subsolidus convection cools the Earth and Moon, removing primordial, radioactive, and (in the Moon) tidal heating. A model without oceanic dissipation results in the Moon moving out to about 50 RE (Earth radii) in less than about 1.5 b.y. Thereafter, dynamic evolution slows considerably because the Earth has cooled and become less dissipative. A model that includes oceanic dissipation shows that about 700 m.y. of large oceanic dissipation (equivalent phase lag equals 1/11) is required to move the Moon to 60 RE. In one interpretation of this result, consistent with variable ocean basin configuration, several episodes of large oceanic dissipation have occurred since the end of the Archean with a cumulative time span of about 700 m.y. In our model, about 90% of the lunar angular momentum received from the Earth is the result of solid friction. Uncertainties in the thermal evolution and dissipation law preclude an unequivocal partitioning of tidal dissipation into solid Earth and oceanic parts. Our model likely underestimates solid Earth dissipation, however, and so emphasizes the proposition that solid Earth tidal friction might have been the main energy sink in the Earth‐Moon system and that oceanic dissipation may have been relatively much less important than solid Earth dissipation over geologic time.

On “radiational” tides and their interpretation as frictional effects

When the observed sea levels are fitted linearly with the tidal potential, using cross-spectral methods, an apparent discontinuity in the response of neighbouring harmonics, like S 2 and K 2, becomes noticeable. The hypothesis of radiational tides would reproduce this discontinuity by introducing an additional input. However, a closer scrutiny of the hypothesis shows it to be untenable: it would require a diffuse and irregular source of energy, the radiation at the water surface, to induce oscillations of a fixed frequency, of constant amplitude and phase lag. The radiational tides calculated, assuming a linear response to gravitational inputs by the ocean, would reach amplitudes of 15–20 cm around the North Sea and of only 1–2 cm in the subtropics; they should even exist in the High Canadian Arctic. A more plausible cause for the discontinuity in response is quadratic friction: friction creates a linkage between the spectral elements of the observed water levels, which is evident for pairs of close frequency. In fact such linkage exists right across the tidal bands. Consequently the spectra of water levels, whenever nonlinear effects are present, will exhibit unusual features such as discontinuities in amplitude and phase, line splitting, besides other irregularities.

Current source density analysis of linear and non-linear components of the primate electroretinogram.

1. We have used the method of current source density analysis to locate the generators of harmonic electroretinogram (ERG) responses to contrast-modulated pattern and uniform-field stimuli in the primate retina. 2. Sinusoidal steady-state analysis was used, with a stimulus temporal frequency of 8 Hz. Fundamental and second-harmonic response components were measured for the uniform-field response. The second harmonic of the average of contrast-reversal pattern responses obtained at a series of spatial phases was also determined in the same experiments. In addition, retinal tissue resistance was measured. All of these measurements were obtained at a series of equally spaced depths in the retina. 3. Retinal resistivity was not observed to vary systematically with depth. In addition, any plausible undetected inhomogeneities of resistivity with depth were found to slightly affect the relative magnitudes of estimated current sources and sinks, but to have little effect on their localization. 4. In a given penetration, the phase lag of each harmonic component was relatively constant with depth in most cases; however the magnitude of this phase lag sometimes varied in different penetrations. To compare data from different penetrations, the constant phase lag for each harmonic was estimated, and the response data phase-shifted so as to bring all data into a standard (cosine) phase. 5. The resulting current source density analyses were found to be quite consistent in overall form for different penetrations and in different animals. These data were averaged to obtain a final estimate of the depth profiles for generators of different ERG components. 6. The uniform-field fundamental response was found to have a predominant source-sink pair in the distal half of the retina (receptor layer to outer plexiform layer). The pattern (second-harmonic) response generators had a quite different depth profile, consisting mainly of a source-sink pair in the proximal 20% of the retina (encompassing the nerve fibre layer to the middle of the inner plexiform layer). The uniform-field second-harmonic response showed a current source density (CSD) depth profile with multiple sources/sinks, as if it contained contributions from the other two.

Secular effects of oceanic tidal dissipation on the Moon's orbit and the Earth's rotation

The earth and moon are considered as a two‐body system in gravitational isolation from the sun and other planets. The lunar orbit is taken as circular, and the solid earth is assumed to be a rigid sphere (with no tidal deformation) so that there are no precessional torques other than those arising from the tidally deformed ocean. Numerical solutions to Laplace's tidal equations, with dissipation by linear bottom friction, are used to obtain ocean‐wide distributions of tidal amplitude for two idealized continentalities: a single circular continent (spherical cap) centered at the north pole, and a single spherical cap centered on the equator. Calculations are made for two different values of the frictional resistance coefficient, thus giving rise to four sets of solutions. The computed tidal amplitudes are used to calculate the oceanic tidal torque, which in turn is used to integrate the orbital equations backward in time for solution of the two‐body problem. The Coriolis parameter and the tidal frequencies change with time, thus requiring that the tidal equations be solved several times during the course of each orbital integration. In this manner, the earth's rotational and the moon's orbital histories are determined on a geologic time scale for each of the four models. The calculated position of the lunar orbit at 4.5 billion years ago is found to range from 38 to 53 earth radii in the four models and corresponds to a sidereal month of 330 to 550 hours. The sidereal day would have been 12 to 18 hours, with a relative inclination of 3° to 22° between the terrestrial and lunar poles. These results are in sharp contrast to those from previous studies of the earth‐moon system, most of which indicated a Roche limit approach of the two bodies roughly 1 to 2 billion years ago and presented therefore a time scale difficulty in theories of lunar origin. This contrast arises mainly from the fact that previous modelers avoided solution of Laplace's tidal equations by prescribing a constant frictional phase lag angle between the angular position of the moon and the major axis of the second‐degree harmonic of the tidally deformed surface of the earth. The amount of phase lag was established from the present astronomically deduced rate of tidal dissipation, but this precluded dynamic variations in tidal torque over geologic time, which are critical for determination of the orbital time scale. The present rate of oceanic tidal dissipation is evidently anomalously high because of the near resonance of the oceanic response in both frequency and shape to the tidal forcing. For the earth and moon to have evolved to their present separation from a distance of less than about 35 earth radii solely on the basis of oceanic tidal dissipation would have required the M2 response to have been near resonance for a significant portion of geologic history, a rather implausible scenario. The calculations reported here show that, on the contrary, frictional coupling between the earth and moon was much weaker than at present throughout most of the orbital history, and consequently the moon's closest approach could not have been as small as 35 earth radii. Although idealized continentalities were used in these calculations, the general conclusion of a weaker frictional coupling in the past is believed to be relatively insensitive to the continental configuration. This is because at faster paleorotation, the semidiurnal tidal frequencies would be resonant only with smaller‐scale normal modes of the ocean rather than with global scale modes as at present. The time scale difficulty heretofore present in models of orbital evolution is thereby eliminated. It should be noted that the orbital history developed here supports the theory of binary planetary accretion for lunar origin as opposed to fission or tidal capture.

On the controversy over the effect of tidal friction upon the history of the earth-moon system

The reasons for the different results obtained by various authors are discussed, especially pertaining to the value of the minimum distance a min and the corresponding inclination ϵ. The main sources of difficulty appear to be (1) neglect of Fourier analysis required by the assumption of constant phase lag, and (2) improper treatment of the torque equation in a precessing coordinate system. This latter error becomes significant for small distances between Moon and Earth, for which case the nature of the precession of the relevant angular momentum vectors is completely different from the case for large distances. This error results in too-small calculated values of ϵ. Goldreich avoids the cited errors, but in the interval 25 R > a > 10 R he commits a slight calculation error which results in ϵ being approximately 8° too small in this region.

A DETERMINATION OF THE EARTH TIDE FROM TILT OBSERVATIONS AT TWO PLACES

Summary This paper, which was almost completed* by Dr Corkan just before his death in 1952, gives a method of analysing observations of the tilting of the Earth's surface. The method combines observations from two places and assumes only that the body tilt due to the direct yielding to the attractive forces is simply related to the equilibrium form, but with a constant phase lag, and that the semidiurnal constituents in the load tilt have the same ratios as in the loading tide. It is shown that these ratios are very stable over large parts of the oceans, and a useful table is given for the main seas and oceans. This method avoids the uncertainties of computation of the loading tide which have caused many difficulties in previous investigations, and it automatically eliminates the greater part of the secondary effects of the more distant oceanic tides.

A problem in underwater acoustics

IN A recent communication under the above title 1 Prof. A. S. Eve has given an account of a problem which arose in the production of directional hydrophones during the war. As the writer had the privilege of serving as a member of Professor Eve’s staff engaged on the problem, the following remarks may not be inappropriate: Attention is drawn to a paper by Wood and Young 2 dealing with this problem, in which the disturbances produced by small bodies in plane waves in water are discussed and a satisfactory approximate theory of the single plate direction finder is formulated. Briefly, the directional hydrophone depends for its action on phase difference. In the symmetrical ‘I edge-on ” position of the unbaffled instrument, the sound waves reach the opposite faces of the hydrophone in the same phase and the diaphragm is therefore not excited. If, however, the hydrophone is “ broadside-on,” the waves at the opposite faces are not in phase, a lag occurring at the back face due to the longer path travelled by the waves reaching it, and the diaphragm is excited. The function of the baffle is to introduce a constant phase lag near the adjacent face of the diaphragm equal to the lag normally occurring at the back face in the “ broadside-on ” position of the unbaffled instrument. Thus when the baffled side is facing the source of sound the lags at the two faces are equal, the pulses are once more in phase, and the diaphragm is not excited. In the opposite position the two lags add, and a reinforced maximum is produced. This elementary theory is admittedly an approximate attempt to account for a complex phenomenon, and though no simple theory covers all of the facts, some further light may be thrown on the problem from practical experience. 1 JOUR. FRANK. ISST., 202, 627, 1926. ’ PI-oc. Roy. Sot., A IOO, 1921.