Showing papers on "Amplitude published in 1971"
TL;DR: In this paper, the authors show that a large-scale orderly pattern may exist in the noiseproducing region of a round subsonic jet by observing the evolution of orderly flow with advancing Reynolds number.
Abstract: Past evidence suggests that a large-scale orderly pattern may exist in the noiseproducing region of a jet. Using several methods to visualize the flow of round subsonic jets, we watched the evolution of orderly flow with advancing Reynolds number. As the Reynolds number increases from order 102 to 103, the instability of the jet evolves from a sinusoid to a helix, and finally to a train of axisymmetric waves. At a Reynolds number around 104, the boundary layer of the jet is thin, and two kinds of axisymmetric structure can be discerned: surface ripples on the jet column, thoroughly studied by previous workers, and a more tenuous train of large-scale vortex puffs. The surface ripples scale on the boundary-layer thickness and shorten as the Reynolds number increases toward 105. The structure of the puffs, by contrast, remains much the same: they form at an average Strouhal number of about 0·3 based on frequency, exit speed, and diameter.To isolate the large-scale pattern at Reynolds numbers around 105, we destroyed the surface ripples by tripping the boundary layer inside the nozzle. We imposed a periodic surging of controllable frequency and amplitude at the jet exit, and studied the response downstream by hot-wire anemometry and schlieren photography. The forcing generates a fundamental wave, whose phase velocity accords with the linear theory of temporally growing instabilities. The fundamental grows in amplitude downstream until non-linearity generates a harmonic. The harmonic retards the growth of the fundamental, and the two attain saturation intensities roughly independent of forcing amplitude. The saturation amplitude depends on the Strouhal number of the imposed surging and reaches a maximum at a Strouhal number of 0·30. A root-mean-square sinusoidal surging only 2% of the mean exit speed brings the preferred mode to saturation four diameters downstream from the nozzle, at which point the entrained volume flow has increased 32% over the unforced case. When forced at a Strouhal number of 0·60, the jet seems to act as a compound amplifier, forming a violent 0·30 subharmonic and suffering a large increase of spreading angle. We conclude with the conjecture that the preferred mode having a Strouhal number of 0·30 is in some sense the most dispersive wave on a jet column, the wave least capable of generating a harmonic, and therefore the wave most capable of reaching a large amplitude before saturating.
2,108 citations
TL;DR: In this paper, the properties of large amplitude microscale Alfven waves in interplanetary medium were investigated using plasma and magnetic field data from Mariner 5. But the results were limited to a single-dimensional image.
Abstract: Dynamic nonshock properties of large amplitude microscale Alfven waves in interplanetary medium, using plasma and magnetic field data from Mariner 5
1,714 citations
TL;DR: The amplitude and the phase of vibration of the basilar membrane and the bony limbus of the cochlea were measured in living squirrel monkeys using the Mossbauer technique.
Abstract: The amplitude and the phase of vibration of the basilar membrane and the bony limbus of the cochlea were measured in living squirrel monkeys using the Mossbauer technique. In the middle ear, the vibration of the malleus (and occasionally the incus) was measured. The Mossbauer technique makes possible the measurement of very small velocities, e.g., 0.2 mm/sec. This sensitivity permits measurement of the motion of the malleus at sound‐pressure levels (SPLs) of 90 to 110 dB and measurement of the motion of the basilar membrane at 70 to 120 dB SPL, depending on the frequency. The basilar membrane vibrates nonlinearly for frequencies which produce the largest deflections at the spot on the basilar membrane under observation. The ratio of the displacement of the basilar membrane to that of the malleus was observed to have the following characteristics: (1) As the frequency is increased from a low value, its amplitude increases at 6 dB/oct until just below the maximum ratio where the slope increases to about 24 dB/oct; (2) the maximum ratio was about 24 dB for the SPLs used; (3) for frequencies above that producing the maximum ratio, the drop‐off rate was approximately 100 dB/oct; (4) the amplitude ratio did not drop off indefinitely but tended to level off; (5) the motion of the basilar membrane differs from the motion of the malleus by 90° at very low frequencies; (6) for frequencies below that producing the maximum ratio, the phase differences between the motion of the basilar membrane and that of the malleus is a linear function of frequency; (7) near the frequency corresponding to the maximum ratio, the phase difference decreases at a faster rate; and (8) the phase difference approaches a constant value (7π 8π or 9π) for high frequencies. Anatomical constraints allowed only a small portion of the basal turn to be studied (6.5–7.5 kHz produced maximum deflection of the basilar membrane in this region).
942 citations
TL;DR: The amplitude-constrained capacity of a scalar Gaussian channel is shown to be achieved by a unique discrete random variable taking on a finite number of values.
Abstract: The amplitude-constrained capacity of a scalar Gaussian channel is shown to be achieved by a unique discrete random variable taking on a finite number of values. Necessary and sufficient conditions for the distribution of this random variable are obtained. These conditions permit determination of the random variable and capacity as a function of the constraint value. The capacity of the same Gaussian channel subject, additionally, to a nontrivial variance constraint is also shown to be achieved by a unique discrete random variable taking on a finite number of values. Likewise, capacity is determined as a function of both amplitude- and variance-constraint values.
542 citations
TL;DR: In this paper, the relationship between the spectrum of the transmitted pulse and the reflection coefficient series is explored for various synthetic and real sequences of layers, and it is shown that the spectrum can be used as an indication of cyclic sedimentation and of the dominant durations of the sedimentary cycles.
Abstract: Modern seismic recording instruments allow precise measurements of the amplitude of reflected signals. Intuitively we would expect that this amplitude information could be used to increase our knowledge of the physical properties of the reflecting earth. The relevant factors defining the amplitude of a reflection signal are: spherical divergence, absorption, the reflection coefficient of the reflecting interface, the cumulative transmission loss at all interfaces above this, and the effect of multiple reflections. Of these factors, three—spherical divergence, the reflection coefficient and the transmission loss—are reasonably clear concepts (though the estimation of transmission loss from acoustic logs caused some difficulties in the hey-day of synthetic seismograms). Absorption still presents considerable problems of detail, but our understanding has increased significantly in recent years. The factor least well understood is undoubtedly the effect of multiple reflections. Multiple paths having an even number of bounces can have the effect of delaying, shaping and magnifying the pulse transmitted through a layered sequence. Simple demonstations of this phenomenon can be made using elementary thin plates, and these can be presented for various synthetic and real sequences of layers. Such demonstrations lead one to explore the relation between the spectrum of the transmitted pulse and the spectrum of the reflection coefficient series. If it were possible to isolate the amplitude and shape variations imposed by absorption within a layer, there would be a chance that this measure of absorption would be useful as a correlatable or diagnostic indication of rock properties. If it were possible to isolate the amplitude and shape variations imposed by multiple reflections, there would be a chance that this measure would be useful as an indication of cyclic sedimentation and of the dominant durations of the sedimentary cycles. However, the separation of these two effects constitutes a formidable challenge. The very difficulty of this separation suggests that it may be opportune to review the quantitative estimates of absorption made by field experiments.
459 citations
TL;DR: In this paper, the initial value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given for values of the Reynolds number slightly greater than the critical value, above which perturbation may grow.
Abstract: The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are et and e½(x+a1rt), where t is the time, x the distance in the direction of flow, e the growth rate of linearized theory and (−a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.
440 citations
TL;DR: In this paper, a simple model for the nonlinear interaction of a low-density monoenergetic electron beam and a relatively cold infinite homogeneous one-dimensional plasma was proposed.
Abstract: Recently, a simple model was proposed for the nonlinear interaction of a low‐density monoenergetic electron beam and a relatively cold infinite homogeneous one‐dimensional plasma. The essential feature of this model is the observation that after several e‐foldings the bandwidth of the growing waves is so narrow that the electrons interact with a very nearly pure sinusoidal field. In terms of this single wave model, a properly scaled solution of the nonlinear beam‐plasma problem which depends analytically on all the basic parameters of the problem (i.e., plasma density, beam density, plasma thermal velocity, and beam drift velocity) is presented. This solution shows that the single wave grows exponentially at the linear growth rate until the beam electrons are trapped. At that time the wave amplitude stops growing and begins to oscillate about a mean value. During the trapping process the beam electrons are bunched in space and a power spectrum of the higher harmonics of the electric field is produced. Both the oscillation in wave amplitude and the power spectrum are given a simple physical interpretation.
384 citations
TL;DR: In this article, a simple device is proposed which effectively suppresses these anomalies by introducing a small random delay in the timing of the radiofrequency pulses, which may be masked by short-term instabilities in the field/frequency ratio or by incoherence introduced by heteronuclear noise decoupling.
318 citations
TL;DR: Large amplitude periodic X ray pulsations from Centaurus X-3, observing abrupt source intensity and pulse rate changes as mentioned in this paper, were used to detect the presence of a nuclear power station in the X-ray channel.
Abstract: Large amplitude periodic X ray pulsations from Centaurus X-3, observing abrupt source intensity and pulse rate changes
249 citations
TL;DR: In this paper, the deformation of a solitary wave due to a slow variation of the bottom topography was examined and the variation of amplitude with depth was determined and compared with some recent experimental results.
Abstract: This paper examines the deformation of a solitary wave due to a slow variation of the bottom topography. Differential equations which determine the slow variation of the parameters of a solitary wave are derived by a certain averaging process applied to the exact in viscid equations. The equations for the parameters are solved when the bottom topography varies only in one direction, and when the wave evolves from a region of uniform depth. The variation of amplitude with depth is determined and compared with some recent experimental results.
235 citations
TL;DR: In this article, the existence and structure of large amplitude, stationary, longitudinal plasma oscillations are studied for the case of a simple waterbag distribution of electrons and an immovable background of ions.
Abstract: The existence and structure of large amplitude, stationary, longitudinal plasma oscillations are studied for the case of a simple waterbag distribution of electrons and an immovable background of ions. The analysis employs the one‐dimensional Vlasov equation for a plasma of infinite spatial extent. An expression for the maximum amplitude of the oscillations is derived. This maximum amplitude decreases monotonically as the ratio of the electron thermal velocity to the wave phase velocity increases. The structure of the oscillations is expressed analytically in terms of hyperelliptic integrals.
TL;DR: In this article, a non-linear growth and decay analysis of a combustion chamber is presented, based on the idea that the gas dynamics throughout most of the volume can be handled in a linear fashion.
Abstract: Due to non-linear loss or gain of energy, unstable oscillations in combustion chambers cannot grow
indefinitely. Very often the limiting amplitudes are sufficiently low that the wave motions appear to be sinusoidal without discontinuities. The analysis presented here is based on the idea that the gasdynamics throughout most of the volume can be handled in a linear fashion. Non-linear behavior is associated with localized energy losses,
such as wall losses and particle attenuation, or with the interaction between the oscillations and the combustion
processes which sustain the motions. The formal procedure describes the non-linear growth and decay of an
acoustic mode whose spatial structure does not change with time. Integration of the conservation equations over
the volume of the chamber produces a single non-linear ordinary differential equation for the time-dependent
amplitude of the mode. The equation can be solved easily by standard techniques, producing very simple results
for the non-linear growth rate, decay rate, and limiting amplitude. Most of the treatment is developed for unstable
motions in solid propellant rocket chambers. Other combustion chambers can be represented as special cases
of the general description.
TL;DR: In this paper, a nonlinear analysis of collisional drift waves is presented in which a systematic expansion is made in powers of the wave amplitude, including the effects of resistivity, viscosity, and thermal transport.
Abstract: A nonlinear analysis of collisional drift waves is presented in which a systematic expansion is made in powers of the wave amplitude. The two‐fluid equations are used, including the effects of resistivity, viscosity, and thermal transport. The result, for the wave amplitude as a function of magnetic field in the linearly unstable region close to marginal stability, agrees reasonably well with experiment.
TL;DR: In this article, a three-dimensional Lippmann-Schwinger-type equation for the elastic scattering amplitude and the corresponding homogeneous Schrodinger equation for two-particle bound states are studied.
Abstract: A three-dimensional Lippmann-Schwinger-type equation for the elastic scattering amplitude and the corresponding homogeneous Schr\"odinger equation for the two-particle bound states are studied. The potential is defined as an infinite power series in the coupling constant which fits the perturbative expansion of the on-energy-shell scattering amplitude. The approximate equation obtained by keeping only the lowest-order term in the potential is local and has the following properties: (i) The scattering amplitude yields the relativistic eikonal approximation for large energies or small exchanged mass and momentum transfer; (ii) for the Coulomb problem the approximate equation is exactly soluble and leads to a relativistic Balmer formula including the fine-structure splitting.
TL;DR: In this article, the frequency spectra for the amplitude, phase, and phase-difference fluctuations of an infinite plane wave propagating through turbulence were calculated for spherical point sources.
Abstract: Tatarski has found the frequency spectra for the amplitude, phase, and phase-difference fluctuations of an infinite plane wave propagating through turbulence. Many practical optical beams, used in atmospheric studies, closely resemble point sources, for which the spherical-wave theory is more applicable. The same spectra, calculated for spherical waves, reveal contributions at higher frequencies for amplitude scintillations, nearly identical phase results, and a phase-difference spectrum with no nulls, in contrast with the plane-wave results. Comparison with recent data is shown.
TL;DR: In this article, a two-time method is used to obtain an expansion, valid for small and large sizes of the vector solution of an abstract ordinary differential equation involving a random function of position, and the results are shown to apply to the solutions of stochastic equations.
Abstract: The two-time method is used to obtain an expansion, valid for $\varepsilon $ small and t large, of the vector solution $u( {t,\varepsilon } )$ of an abstract ordinary differential equation involving $\varepsilon $. The same method is used to get expansions of functions of u. The results are shown to apply to the solutions of stochastic equations. They are used to find the first two moments and the transition probability of the displacement of a harmonic oscillator with spring constant a random function of t. The result contains the condition for mean square stability due to Stratonovich. The results are also applied to one-dimensional wave propagation through a layer with refractive index a random function of position. They are used to find the mean square amplitude reflection and transmission coefficients, which are just the mean power reflection and transmission coefficients. A graph of the mean square transmission coefficient as a function of layer thickness is presented. The results are also compared ...
TL;DR: In this article, a single acoustic resonance, excited by vortex shedding from the trailing edge of a flat plate, has been studied in detail, where the mode was traversed at resonance and the modal shape and natural frequency were found to agree well with computations.
TL;DR: In this paper, a conformal transformation is used to investigate the steady streaming generated by an oscillatory viscous flow over a wavy wall, and the equations are linearized and solved for large and small values of the parameter kR. This parameter is the ratio of the amplitude of oscillation of a fluid particle to the wavelength of the wall.
Abstract: The method of conformal transformation is used to investigate the steady streaming generated by an oscillatory viscous flow over a wavy wall. By assuming that the amplitude of the wall is much smaller than the Stokes layer thickness, the equations are linearized and solved for large and small values of the parameter kR. This parameter is the ratio of the amplitude of oscillation of a fluid particle to the wavelength of the wall. When kR [Lt ] 1, the results due to Schlichting (1932) are recovered, and when kR [Gt ] 1 the equations resemble closely those derived in the theory of stability of plane parallel flows. With the aid of this theory the first-order steady streaming is found.
IBM1
TL;DR: In this article, a model involving head motion for self-consistently computing magnetic recording medium magnetization patterns is given, and the reduction in demagnetizing field due to the presence of the high-permeability head structure is included, as is record head removal, read head replacement, and computation of the readback voltage.
Abstract: A model involving head motion is given for self-consistently computing magnetic recording medium magnetization patterns. The reduction in demagnetizing field due to the presence of the high-permeability head structure is included, as is record head removal, read head replacement, and computation of the readback voltage. The model is capable of handling an arbitrary record current waveform. Optimum record-current amplitude for nonreturn to zero (NRZ) digital recording is first determined, and then single-, double-, and quadruple-transition computations are performed using two different values of hysteresis loop squareness M_{r}/M_{s} and both linear and exponential current reversals. Results are primarily for the Karlquist fringe field, but the recording properties of a head exhibiting regions in which the fringe field reverses sign are also briefly investigated.
TL;DR: In the case of a periodic object, the wave amplitude in every point of a Fresnel diffraction pattern can be obtained from the values of the wave wave amplitude at a finite number of points in the object.
Abstract: In the case of a periodic object, the wave amplitude in every point of a Fresnel diffraction pattern may be obtained from the values of the wave amplitude at a finite number of points in the object...
TL;DR: In this article, the reciprocal theorem was used to recast the problem of finding the field near the obstacle, induced by an incident plane wave, and the prototype problem of a wedge of exterior angle (p/q)π was shown to yield an intensity law I ∝ U4+2q/p for both hard and soft surfaces.
Abstract: According to the Lighthill acoustic analogy, the sound induced by a region of turbulence is the same as that due to an equivalent distribution of quadrupole sources within the fluid. It is known that the presence of scattering bodies situated near such multipoles can convert some of their intense near field energy into the form of sound waves whose amplitude is far greater than that of the incident field. Calculations are here presented to determine the extent of this conversion, for hard and soft bodies of various shapes, making use of the reciprocal theorem to recast the problem into one of finding the field, near the obstacle, induced by an incident plane wave. If the obstacle is small compared with a wavelength, then its presence is equivalent to an additional dipole (or source) whose greater efficiency as a sound radiator implies that the familiar intensity law I ∝ U8, for far field intensity I against typical turbulence velocity U for an unbounded flow, is replaced by I ∝ U6 (or I ∝ U4) for a hard (or soft) body. For the situation where the scatterer is large compared with wavelength, the prototype problem of a wedge of exterior angle (p/q)π is shown to yield an intensity law I ∝ U4+2q/p for both hard and soft surfaces. This result is shown to hold for the more general ‘wedge-like’ surfaces, whose dimensions are large scale and whose edges may be smoothed out on a small scale, compared with wavelength. The method used involves the matching of an incompressible flow, on the fine scales typical of the edge geometry, to an outer flow determined by the large scale features of the surface. Favourable comparisons are made with previous results pertaining to the two-dimensional semi-infinite duct and to the half-plate of finite thickness.
CERN1
TL;DR: In this paper, the moduli and relative phases of the π N scattering amplitudes are extracted directly from 6 GeV/c data and the resulting amplitude structure at the crossover and at t = −0.6 GeV 2 is analyzed and compared with high-energy models.
TL;DR: In this paper, a study of finite-amplitude baroclinic instability for a two-layer system with small but non-zero dissipation is presented, and it is shown that the presence of dissipation, however slight, allows the existence of steady finite amplitude wave solutions.
Abstract: A study of finite-amplitude baroclinic instability for a two-layer system with small but non-zero dissipation is presented. The presence of dissipation, however slight, allows the existence of steady finite-amplitude wave solutions. For sufficiently small friction, however, the steady wave may be unstable if a certain criterion, presented in this paper, is satisfied. Calculations indicate that in such cases a continuous, slow, periodic amplitude pulsation exists which is independent of the initial conditions.
Patent•
08 Nov 1971TL;DR: In this article, an in-jection well and at least one producing well are produced by passing flooding fluid at a preselected rate into the formation via the injection well while transmitting oscillatory pressure waves from the injectionwell outwardly through the formation for forming a wave zone in the formation and moving the wave zone outwardly.
Abstract: A hydrocarbon-containing formation penetrated by at least one in-jection well and at least one producing well is produced by passing flooding fluid at a preselected rate into the formation via the injection well while transmitting oscillatory pressure waves from the injection well outwardly through the formation for forming a wave zone in the formation and moving the wave zone outwardly through the formation by altering at least one of the frequency or amplitude of the oscillatory pressure wave transmissions.
TL;DR: In this article, a model of the acoustic emission source wave is developed, and arguments are given why this wave should be a pulselike function, rather than an oscillatory function of stress.
Abstract: Illustrations are given of information carried in the waveforms and frequency spectra of acoustic emissions. The effects of multiple reflections and resonances are discussed. A model of the acoustic‐emission source wave is developed, and arguments are given why this wave should be a pulselike function, rather than an oscillatory function of stress. Further use of this model may allow more quantitative treatment of emission amplitudes, energies, and spectra. Experimental results show the use of two instruments for the evaluation of emission spectra. The feasibility of using frequency analysis to obtain information about source events is demonstrated.
TL;DR: In this article, the dislocation arrangement in the stress-applied state was investigated extensively by transmission electron microscopy and compared with the results of corresponding investigations on unloaded crystals in stage II dislocation braids and dislocation grids (sheets).
Abstract: Copper single crystals orientated for single glide were deformed in tension at 78°K and irradiated before load-removal with fast neutrons at 4°K or 20°K. The dislocation arrangement which was thus stabilized in the stress-applied state was investigated extensively by transmission electron microscopy and compared with the results of corresponding investigations on unloaded crystals. In stage II dislocation braids and dislocation grids (sheets) are observed both in the stress-removed and the stress-applied states. Groups of primary dislocations of the same sign, however, are a representative feature of the dislocation arrangement in the stress-applied state only. It is concluded from the observations that the grids are formed by secondary slip at the head of piled-up groups and are capable of growth. The observed spatial change in the curvature of free primary dislocations in the stress-applied state points directly to the existence of a quasi-periodic long-range internal stress field whose amplitude is of ...
TL;DR: In this paper, the stability of the Taylor-vortex flow is calculated using fifth-order terms in amplitude, and using the full equations rather than the small-gap approximation, and qualitative confirmation is found of instabilities to the Taylor vortex flow to non-a.xisymmetric disturbances at about 10 % above the first critical Taylor number.
Abstract: Davey, Di Prima & Stuart's (1968) double amplitude expansion for disturbances in flow between concentric cylinders is formulated in matrix notation. The stability of the secondary equilibrium (Taylor-vortex) flow is calculated using fifth-order terms in amplitude, and using the full equations rather than the small-gap approximation. Qualitative confirmation is found of instabilities to the Taylor-vortex flow to non-a.xisymmetric disturbances at about 10 % above the first critical Taylor number.
TL;DR: In this article, nonlinear damping of electromagnetic waves propagating along a uniform magnetic field has been calculated by computing nonlinear trajectories for trapped and resonant untrapped particles while using linear theory to describe the rest.
Abstract: Nonlinear damping of electromagnetic waves propagating along a uniform magnetic field has been calculated by computing nonlinear trajectories for trapped and resonant untrapped particles while using linear theory to describe the rest. This procedure parallels O'Neil's calculation for electrostatic modes, and the results are qualitatively similar. After an initial linear damping, amplitude oscillations set in and the amplitude quickly approaches a finite constant value. The frequency of the amplitude oscillations is temperature dependent. Phase mixing results from the spread in both parallel and perpendicular velocities, giving rise to a more rapid approach to the asymptotic amplitude than for electrostatic modes.
TL;DR: In this article, two forms of wind-induced instability of long bodies with bluff cross-sections are described, including vortex-induced bending oscillation and transverse galloping, and the effects of body and wind parameters on both forms of oscillation are considered, and methods of analysis and suppression for susceptible structures are described.
Abstract: Forms of wind-induced instability of structures are described, and two of these, typical of long bodies with bluff cross-sections, are selected for more detailed consideration. The first is vortex-induced bending oscillation, a type of resonant response to the periodic surface pressure loading caused by the discrete wake vortex street formed from the shear layers separating from the bluff cross-section. Oscillation phenomena are described, including capture of the vortex frequency by the structural response frequency over a discrete wind speed range and amplification and phase shift of the loading over this range. The second form is transverse galloping, arising from aerodynamic instability of the bluff cross-sectional shape, so that small-amplitude oscillations generate forces which increase the amplitudes to large values. Oscillation phenomena are described, including the occurrence at very nearly natural frequencies, and the relatively large amplitudes (compared to vortex-induced oscillations) increasing with wind speed beyond a critical wind speed dependent on the level o fstructural damping. Effects of body and wind parameters on both forms of oscillation are considered, and methods of analysis and suppression for susceptible structures are described. Some probable future requirements and prospects are considered.
TL;DR: The method of multiple scales was used to analyze three non-linear physical systems which support dispersive waves in this paper, and it was found that the partial differential equations that govern the temporal and spatial variations of the wave numbers, amplitudes, and phases have the same form for all three systems.
Abstract: The method of multiple scales is used to analyze three non-linear physical systems which support dispersive waves. These systems are (i) waves on the interface between a liquid layer and a subsonic gas flowing parallel to the undisturbed interface, (ii) waves on the surface of a circular jet of liquid, and (iii) waves in a hot electron plasma. It is found that the partial differential equations that govern the temporal and spatial variations of the wave-numbers, amplitudes, and phases have the same form for all of these systems. The results show that the non-linear motion affects only the phase. For the constant wave-number case, the general solution for the amplitude and the phase can be obtained.