Undistorted Waves Research Articles

Propagation of weak waves in a dusty, van der Waals gas

In this paper, the propagation of weak disturbances in a non-ideal gas with dust particles in which the equation of state is of Mie–Gruneisen type are studied. The method of relatively undistorted waves is used to study the wave profile and its amplitude in the medium. The influence of the non-ideal parameter, the effect of dust particles and the non-planar configuration on the wavefront and on non-linear distortion are discussed.

Nonparaxial wave beams and packets with general astigmatism

We present exact solutions of the wave equation involving an arbitrary wave form with a phase closely similar to the general astigmatic phase of paraxial wave optics. Special choices of the wave form allow general astigmatic beamlike and pulselike waves with a Gaussian-type unrestricted localization in space and time. These solutions are generalizations of the known Bateman-type waves obtained from the connection existing between beamlike solutions of the paraxial parabolic equation and relatively undistorted wave solutions of the wave equation. As a technical tool, we present a full description of parametrizations of $2\ifmmode\times\else\texttimes\fi{}2$ symmetric matrices with positive imaginary part, which arise in the theory of Gaussian beams.

On the Kudryashov–Sinelshchikov equation for waves in bubbly liquids

A nonlinear evolution equation for wave propagation in bubbly liquids, taking into account viscosity and heat transfer, has been derived by Kudryashov and Sinelshchikov. In the case of no dissipation the authors have provided analytical solutions representing undistorted waves. These results are cast into a simpler form and studied in more detail. In addition to the wave profiles the corresponding phase curves are presented. Depending on some parameter the solutions represent solitary or periodic waves. Some of the periodic waves exhibit peaks or cusps. From the periodic waves a new type of “meandering” solutions is constructed.

Gegenbauer expansion to model the incident wave-field of a high-order Bessel vortex beam in spherical coordinates

The aim of this short communication is to report that Gegenbauer’s (partial-wave) expansion, that may be used (under some specific conditions) to represent the incident field of an acoustical (or optical) high-order Bessel beam (HOBB) in spherical coordinates, anticipates earlier expressions for undistorted waves. The incident wave-field is written in terms of the spherical Bessel function of the first kind, the gamma function as well as the Gegenbauer or ultraspherical functions given in terms of the associated Legendre functions when the order m of the HOBB is an integer number. Expressions for high-order and zero-order Bessel beams as well as for plane progressive waves reported in prior works can be deduced from Gegenbauer’s partial-wave expansion by appropriate choice of the beams’ parameters. Hence the value of this note becomes historical. In addition, Gegenbauer’s expansion in spherical coordinates may be used to advantage to model the wave-field of a fractional HOBB at the origin (i.e. z = 0).

Relatively undistorted waves and waveforms in nonlinear acoustics: Exact solutions of equations

We demonstrate the physical conditionality and efficiency of using relatively undistorted waves as an anzats for determining exact solutions to equations of nonlinear acoustics.

RELATIVISTIC LAGUERRE POLYNOMIALS AND SPLASH PULSES

New solutions of the homogeneous wave equation of the type usually referred to as relatively undistorted waves are presented. Such solutions relate to the so-called splash modes, from which indeed they can be generated by applying the Laguerre polynomial operator. Accordingly, the solutions here presented resort to the relativistic Laguerre polynomials — introduced about one decade ago within a purely mathematical context — which in fact appear as modulating factor of the basic splash mode waveform. Similar solutions of the homogeneous spinor wave equation are also suggested.

Propagation and Interaction of Waves in a Non-ideal Gas

The propagation of disturbances through a non-ideal medium, where the solid behaves like a fluid with an equation of state of Mie-Grüneisen type is considered. The method of relatively undistorted waves is used to study the situations when the disturbance amplitude is finite, arbitrarily small, and not so small. The effects of the solid particles and that of the non-planar wave front configuration on the distortion attenuation, and shock formation of propagating pulses are described in detail. The resonant multiwave theory of weakly nonlinear geometrical optics, which takes into consideration systems where many waves coexist and interact with one another resonantly, is applied to derive transport equations for signals having all possible wave modes which are mutually coupled and interact resonantly among themselves.

Propagation and Interaction of Waves in a Non-ideal Gas

The propagation of disturbances through a non-ideal medium, where the solid behaves like a fluid with an equation of state of Mie-Gruneisen type is considered. The method of relatively undistorted waves is used to study the situations when the disturbance amplitude is finite, arbitrarily small, and not so small. The effects of the solid particles and that of the non-planar wave front configuration on the distortion attenuation, and shock formation of propagating pulses are described in detail. The resonant multiwave theory of weakly nonlinear geometrical optics, which takes into consideration systems where many waves coexist and interact with one another resonantly, is applied to derive transport equations for signals having all possible wave modes which are mutually coupled and interact resonantly among themselves.

Electromagnetic fields produced by sources on a spherical wavefront

Transient solutions of the inhomogeneous Maxwell equations are obtained. The specific current source belongs to the spherical surface expanding with the velocity of light. Representations of the electromagnetic field components in the spherical coordinates in terms of the Fourier series are found. The obtained solutions are correct near the wavefront. The general expressions are applied to a description of electromagnetic waves produced by different sources and to a discussion of formation of directional waves and the family of relatively undistorted waves.

Buoyant convection in the Belousov–Zhabotinsky reaction. II. Chemically driven convection and instability of the wave structure

A numerical study of the complete system of reaction-diffusion-convection equations that govern the dynamics of Belousov–Zhabotinsky (BZ) reactive fluid has been conducted in order to understand the interaction of chemical waves with fluid convection induced by chemical reactions. In the regime studied, it was found that although the heat of reaction alone could drive convection, it was too weak to destabilize the wave. When the influence of the isothermal chemical reactions was combined with that of the heat of reaction, however, convective instability of the wave structure could result for both up-going and down-going waves. The regime in the parameter space was defined within which instability of the up-going wave and stable propagation of the down-going wave is expected, as observed in a magnetic resonance imaging (MRI) experiment. The velocity of an undistorted wave front in the presence of convection was found to be independent of the lateral scale of the system and the direction of wave propagation, as was also observed in the MRI experiment.

Propagation and interaction of waves in a relaxing gas

Propagation of disturbances through a uniform region of a relaxing gas in a duct with spatially varying cross section is analysed using the methods of relatively undistorted waves and weakly nonlinear geometrical optics. Particular attention is focused on sit­uations when the disturbance amplitude is finite, arbitrarily small and not so small. In certain situations a complete history of the evolutionary behaviour of waves in­cluding weak shocks can be traced out. The asymptotic decay laws for weak shocks in a non-relaxing gas are exactly recovered. The damping effects of relaxation and non-planar wavefront configurations on the distortion, attenuation and shock for­mation of pulses, as they propagate, are described in detail. In the small-amplitude high-frequency limit, a solution up to the second order is obtained and numerical computations are carried out for typical values of the physical parameters involved in the solution. Transport equations are derived for signals having all possible wave modes which are mutually coupled and interact resonantly among themselves. The progressive wave approach describes the far field behaviour which is governed by the generalized Burger’s equation.

Space charge measurement in thick dielectric materials by pulsed electroacoustic method

The new application of pulsed electroacoustic (PEA) method to measure space charge profiles in several millimeters thick polymers such as power transmission cable was proposed. In this method, space charge profiles are detected as acoustic wave signals that are generated by the space charge profiles existing in the sample under stimulation of an electric pulse. In most polymer samples, such as cross-linked polyethylene (XLPE) widely used in XLPE cable (CV cable), however, the acoustic wave form is obviously distorted as it propagates through the thick polymer. The acoustic wave distortion cannot be negligible when applying this method directly to very thick samples (several millimeters thick). To recover the wave form and obtain the accurate space charge profiles, it is necessary to study the attenuation and dispersion characteristics of the sample and remove their effects to the propagating acoustic wave. So-called ‘‘deconvolution processing’’ using ‘‘Jacobi method’’ was employed in this report to obtain the undistorted wave form. The principle of the data processing method is introduced and some examples for recovering the wave form are given to verify the effectiveness of the data processing method.

Non-linear wave propagation in a hot-electron plasma

The theory of relatively undistorted waves is used to study the finite-amplitude waves in a hot-electron model of plasma. A simple asymptotic expansion accounting for the non-linear effects has been used in the small amplitude limit to calculate the first and the second order solutions. The speed of propagation and the location of weak shocks has been determined through solutions of high-frequency waves.

Nonlinear waves and shock calculations for hyperelastic fluid-filled tubes

The propagation of both finite amplitude and weakly nonlinear waves in fluid-filled hyperelastic tethered tubes subjected to axial strain is investigated in some detail. Procedures based upon the methods of characteristics and relatively undistorted waves are deployed to compute the time and location of first shock formation in tubes having both constant and variable properties ahead of the wave. The influence of wall thickness changes in shock formation is explored and it is further found that if the transmural pressure ahead of the wave is zero then no shock can form on the lead characteristic for any model based on a rational theory of finite elasticity. This latter result is in disagreement with several earlier studies.

Pulse distortion and Hilbert transformation in multiply reflected and refracted body waves

abstract Many seismic body waves are associated with rays which are not minimum travel-time paths. Such arrivals contain pulse deformation due to a phase shift in each frequency component. For sufficiently high frequencies, the phase shift each time a ray touches an internal caustic is π/2 and frequency-independent. The distorting effect of a frequency-independent phase shift is successfully observed in seismograms from events in several regions. The data examined are long-period (T > 9 sec). They include deep earthquakes (depth > 500 km), in which a series of well-separated S phases (S, sS, SS and sSS) are available. These show that the wave form of SS, which has been distorted in propagation through the Earth, can be derived from the wave form of sS, which is not distorted. Shallow events, in which multiple S phases overlap, also exhibit behavior predicted by phase distortion. Rays supercritically reflected or refracted at a discontinuity in the Earth also suffer a constant phase shift, which in general can have any value. An important case is SKKS: its undistorted wave form resembles that of SKS, which has a minimum travel-time path. Without exception, all the distorted wave forms bear little or no resemblance to the original wave form. That is, neither the first arrival of energy nor the subsequent relative position of peaks and troughs on a distorted wave form appear at the ray theoretical times. Thus, T-Δ curves constructed by choosing arrival times to correspond to the first arrival of energy may be biased. Similarly, doubt is cast on differential travel times chosen from first motions, or from averaging several points on what appear to be corresponding peaks and troughs of two wave forms. Some of the rays most important to seismology, in which the distortion phenomenon occurs, include P and S (where d2T/dΔ2 > 0), PKPAB, PP, SS, and SKKS. Removal of phase distortion in the data is computationally straightforward. By exploiting the resulting wave forms to full advantage in correctly picking arrival times, we may hope to improve velocity models of the Earth. It is shown that matched filtering to obtain differential travel times is appropriate for certain pairs of body waves if they are phase-corrected.

Effects of viscosity on long waves

A study of effects of viscosity on non-linear long waves is made. Beginning with the Navier-Stokes equations of motion, the long wave approximation is achieved by an expansion scheme similar to Friedrichs'. Non-linear solutions are obtained by applying the theory of relatively undistorted waves. It is found that shock formation is delayed by the viscous effect. Various conditions are obtained in determining the viscous, non-linear and radial decay effects on the solution for a shockless expansion wave-front propagating over large distances.

A STUDY ON ACOUSTIC PROBE AUDIOMETRY BY MEANS OF A NYLON BAR WITH A NOTCHED HEAD

A nylon bar, with the head notched to make easy adequete contact with the short process of the malleus, was used for an acoustic probe.The first part of this paper deals with the acoustic properties of the nylon probe in comparison with those of a piano wire probe and a silver wire probe. The following results are obtained :(1) Clinically the most suitable length of the nylon probe was 8cm.(2) The nylon probe proved to be capable of transmitting undistorted wave forms of test pure tones at the intensity level necessary for clinical measurment at the frequencies of 500, 1000, 2000 and 4000cps. Amplitude of the output through the nylon probe was a linear function of the intensity of input signals.(3) The intensity of the output through the nylon probe was kept almost constant to the test signals of 500, 1000, 2000 and 4000cps, when the fixing pressure of the nylon probe was in the range between 0.02gr and ibgr.(4) The nylon probe with a notched head showed the best fixation with the short process of the malleus in a corpse as compaired to the piano and silver wire probes.(5) When auditory threshold was measured at the short process of the malleus in a normal subject by the acoustic probe audiometry, the nylon probe showed more reliable values than the other probes.The second part deals with the clinical appli. cation of the acoustic probe audiometry with this nylon probe. A total of 82 ears with middle ear involvements were measured before surgical treatment. Auditory threshold test was carried out by fixing the notched head of the nylon probe at the short process of the malleus.The following results are obtained:(1) It was concluded that the ossicle hearing loss of less than 21db should be indicative of the existence of the normal ossicular chain, more than 25db indicative of interruption of the ossicular chain, whereas the ossicule hearing loss from 22db to 24db would indicate no definite pathological change in the ossicular chain.(2) Function of the ossicular chain might be represented by the value of auditory threshold measured by the nylone probe at the short process of the malleus, (3) It is not possible by the present test to detect the existence of the slightly interrupted ossicular chain, in which conductive connection was maintained to some degree by granulations.(4) The ossicle hearing loss was elevated with an increase in the degree of stapes fixation, however, very slight stapes fixation shown by slight negativity of Tsunoda's liquid conduction test could not be identified by the nylon probe audiometry.

Scattering and stripping of 27.0 MeV deuterons on nitrogen

The angular distributions of protons from the reaction N 14(d, p)N 15 and of the elastically scattered deuterons were measured. The experimental results of the first reaction were analyzed in terms of the theory of Butler with undistorted waves. The reaction takes place between the ground states of N 14 and N 15, and their spins and parities are known. In order to fit the experimental points it was necessary to superimpose the theoretical curves of l n = 1 and l n = 3, l n being the orbital angular momentum of the captured nucleon, with interaction radii a = 3.7 fm and a = 3.2 fm. No appreciable contribution from compound nucleus, exchange, or heavy particle stripping processes seems to be present in the angular distribution. The stripping reduced width is θ 2 = 0.31. The l n = 3 value of the orbital angular momentum implies the spin flip of the outcoming nucleon. The elastic scattering angular distribution exhibits a typical diffraction structure. Using a simple optical analogy that implies the scattering of waves from a black disk, an interaction radius was calculated for the deuteron-nucleus system: r 0 = 6.5 fm.

On the Lens Interferometer

The interferometer designed by Michelson has been adapted in recent years to the testing of various types of optical systems, one form being used for the testing of prisms and another form—the “lens interferometer”—for the examination of the symmetrical optical system. The details of the instrument may be found in a paper by Mr F. Twyman, in which are given some photographs of the results of its application to certain lenses of known aberration. Essentially the interferometer ensures that a plane and undistorted wave of light falls upon the lens or system to be examined, after refraction through which it falls upon a convex mirror whose centre is at the focus of the converging wave; by which, therefore, the wave is returned through the optical system and emerges as a plane wave if the system under examination be ‘perfect,’ or free from aberration. Otherwise the final wave is not plane, but is distorted by the aberrations impressed upon it by the system under test—through which, it will be noticed, it passes twice; and this emergent wave is superposed upon a portion of the incident (plane) wave, obtained by previous passage through a partly silvered mirror; with which, therefore, interference effects are observed. And these effects depend only upon the aberrations produced by the system under examination, for they measure, in fact, the departure of the finally emergent wave from the ideal plane form. Various interference contours are obtained in this way, and their form and size enable the observer to deduce the aberrations of the system and to measure their magnitude.