Plane Parallel Boundaries Research Articles

Fluctuations of Sound Transmission in the Straits of Florida

Sound waves transmitted across the Florida Current between fixed transducers are used to investigate relationships between acoustic and environmental variations. Year-long measurements of the amplitude and phase of 420-Hz sound waves reveal fluctuations with periods from seconds to weeks and longer-term trends. The variations of phase correspond to changes in the medium and increase in magnitude with increase in period. The amplitude fluctuations are not readily identifiable. The slowly changing phases of the multipath arrivals result in varying patterns of amplitude interference and fading which have noiselike spectra. The Florida Current is characterized by a cross-stream temperature-depth structure related to the transport of water. Variations of transport from tidal and wind forces induce fluctuations of temperature, sound speed, and phase. Temperature and phase are also modulated by internal waves and components of current parallel to the sound path. The observed characteristics of the amplitude and phase variations appear in computations with ray-propagation models in which plane parallel boundaries and measured time variations of sound speed versus depth are used.

Fixed-System Studies of Underwater Acoustic Propagation

Sound waves of 420 Hz continuously transmitted across the Florida Current from a fixed source to fixed hydrophones are used to investigate relationships between temporal fluctuations of the waves and changes of Florida Current transport and water-column temperatures. The amplitude and phase of the received waves show variations with periods in the range from seconds to weeks and longer-term trends. Characteristically, variations of acoustic phase correspond to changes of transport and temperature, but fluctuations of acoustic amplitude are noiselike and not readily identifiable. Long period changes of transport and temperature result in slowly changing phases of the multipath arrivals and varying patterns of interference and amplitude fading. Spectra of the amplitude fluctuations show components over a wide frequency range which mask components corresponding to transport and temperature changes. The above characteristics also appear in calculations with ray-propagation models in which sound speed versus depth was varied with time in accordance with measured variations. Since the models have plane parallel boundaries, characteristic features of the variations appear to be independent of surface and bottom irregularities. This raises questions of modeling amplitude fluctuations from transport and temperature changes obtained from acoustic phase measurements. The paper summarizes relevant observations and analyses.

Fixed System Acoustic Measurements in the Straits of Florida

Relationships between variations of underwater sound transmitted across the Florida Current and variations in the medium are under study with fixed sound transducers and environmental sensors. Continuous-wave and pseudorandom-sequence signals of 420 Hz are transmitted on a continuous basis, and both the amplitude and the phase of the received signals are measured. Temperatures at two locations in the sound path and current transport across a 5.5-mile section of the path are also measured. Relationships between the measured quantities, which exhibit variations with periods from seconds to weeks and longer term trends, are investigated by correlation and coherence techniques. Variations of acoustic phase correspond to variations of temperature and transport but acoustic amplitude variations are noise-like without identifiable periods. Calculations with ray path models having plane parallel boundaries confirm the observations and indicate that their characteristic features arise from multipath interference relatively independent of effects of surface and bottom irregularities. For time intervals of a few minutes or less, phase variations are small and sound transmission is phase-coherent. Propagation in the Florida Straits, which has a maximum depth of 800 m, takes place primarily by refracted bottom-reflected rays. Tests with fixed instrumentation in the deep ocean off Bermuda indicated features of propagation by surface-reflected bottom-reflected rays which were similar to the features of the Florida Straits observations. [Work supported principally by the Office of Naval Research and partially by the Naval Ship Systems Command.]

Dielectric Response of a Semi-Infinite Degenerate Electron Gas

The dielectric response at frequency $\ensuremath{\omega}$ (neglecting retardation) of an idealized metal with two plane parallel surfaces is calculated in the Hartree approximation, with special emphasis on the limit of infinite thickness. It is assumed that the unperturbed system may be regarded as consisting of free particles confined between plane parallel boundaries. The validity of this approximation is discussed. The response is expressed in terms of the perturbing source potential in Fourier representation, and involves the inverse of an infinite matrix E; E is calculated in the limit of infinite interfacial separation at $\ensuremath{\omega}=0$ and at $\ensuremath{\omega}\ensuremath{\ne}0$ for one-dimensional potentials. A dielectric function ${\ensuremath{\epsilon}}_{\mathrm{Q}}(\ensuremath{\omega})$, where Q is a wave vector parallel to the surface, is introduced, which is inversely proportional to the sum of all elements in ${\mathbf{E}}^{\ensuremath{-}1}$. The surface-plasmon dispersion relation is given implicitly by ${\ensuremath{\epsilon}}_{\mathrm{Q}}(\ensuremath{\omega})+1=0$. The classical image theorem for a semi-infinite dielectric medium is obeyed at a given Q and $\ensuremath{\omega}$ if ${\ensuremath{\epsilon}}_{\mathrm{Q}}(\ensuremath{\omega})$ replaces the classical dielectric constant. The Fermi-Thomas approximation, the lowest-order correction to it, and the classical (high-frequency) approximation are derived. In agreement with earlier work, the corrections to the classical approximation, which relate to the damping and dispersion of surface plasmons, are found to be linear in the wave vector. Numerical results are obtained for a uniform static electric field normal to the surface. The calculated screening length $d$ is well approximated by $d\ensuremath{\simeq}{\ensuremath{\lambda}}^{\ensuremath{-}1}+\frac{1}{8}{\ensuremath{\lambda}}_{F}$, where ${\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the Fermi-Thomas screening length and ${\ensuremath{\lambda}}_{F}$ the Fermi wavelength, the second term representing the lowest quantum correction; this result is 2-3 times ${\ensuremath{\lambda}}^{\ensuremath{-}1}$ in the metallic density range, owing almost entirely to the low electron density in the surface region. The results are compared with experiment. The long-range Friedel oscillations are discussed in an appendix.

The plane-wave method in compressible media with plane-parallel boundaries

The plane-wave method in compressible media with plane-parallel boundaries

Lifshitz- van der Waals Forces between Graphite Masses –Deviations from Pairwise Additivity–

Attractive forces between graphite masses separated by 100A are calculated to be 4.76×10 4 dyne/cm 2 by making use of Lifshitz' formula for interaction forces between macroscopic neutral bodies. The two semi-infinite blocks are supposed to be put side by side with plane-parallel boundaries (perpendicular to the c -axis). For dielectric constants needed, results by Taft and Philipp evaluated from reflectance data are used. A contribution due to π-electron to the force is about 67%. The total force is compared with that evaluated from an interlayer potential function of graphite given semiempirically by Crowell under assumptions of pairwise additivity. The latter is 4.4 times as large as the former. An origin of this discrepancy may be attributed mainly to a shielding effect due to the basal planes, which reduces the force, obtained without regard to the effect, to one third.

Reflection and transmission of elastic waves in a system of corrugated layers

abstract The scattered field generated by normally incident body waves in a system of layers having small, but otherwise arbitrary, periodic deviations from plane parallel boundaries is shown to consist of superposed plane body and surfacetype waves. Results of numerical computations for two like half-spaces separated by a sinusoidally corrugated single layer, and by two layers, reveal the variation of the amplitude of the field with ratios of velocities, densities, impedances, and with those of depth of layers and wavelength of the boundary corrugations to the wavelength of the incident wave.

Quenching of excited alkali atoms and related effects in flames: Part I. theoretical analysis

An a.c. photoelectric detection device has been used to determine the yield factor p of resonance fluorescence of the Na(5890/96) Å and K(7665/99) Å and (4044/47) Å resonance doublets in atmospheric flames as a function of the composition and temperature of the flame. From these measurements the value (and temperature dependence) of the specific effective quenching cross-sections of the considered lines for N 2, CO 2, CO, O 2, H 2, Ar and H 2O could be derived. The experimental procedure and the results obtained will be reported and discussed in Part II of this paper. In Part I a theoretical analysis is given of the effect of radiative non- equilibrium on the occupation of the excited state, as a function of the p-value, the metal concentration in the flame, and the distance from the flame surface. In particular, an approximate expression for the population (or source) function in the high density case is derived by an iteration procedure for assumed rectangular, Doppler and Lorentzian spectral line profiles respectively. It is assumed that the source function is independent of frequency and that the excited level combines with the ground state only. In the analysis a homogeneous and isothermal slab of flame gases between two plane parallel boundaries is considered. Diffusion effects of the atomic metal vapour at the flame border are considered semi-quantitatively. Also the effect of radiative non- equilibrium on the intensity of the outgoing radiation, the line-reversal temperature, the shape of the curve of growth, and the intensity ratio of the doublet lines is analysed. Finally, the expected influence of quenching collisions on the broadening of spectral lines in flames is calculated.

Effect of Initial Velocity on One-Dimensional, Bipolar, Space-Charge Currents

Oppositely directed flows of positive and negative charges constitute a bipolar current. A general case is analyzed in which the charges traverse the evacuated space between plane-parallel boundaries and all particles of a given charge species are monoenergetic. Moreover, it is assumed that the charges may possess nonvanishing initial kinetic energy. This represents an extension of analyses by Langmuir and Müller-Lübeck for vanishing initial kinetic energy. Sample calculations of dimensionless current densities, electric and potential fields, and charge-density distributions are exhibited for cases where the electric field is assumed to vanish at one boundary but the species initial velocities may not. It is shown that resulting ion currents may be several times the Child's law limit if initial kinetic energies are of the order of the potential energy.

Wave propagation phenomena at an irregular infinite interface: Part I: Theory

Abstract Much wave propagation theory is not generally useful because plane parallel boundaries are assumed. In the present paper, the theory for wave phenomena at an irregular infinite interface is developed. As most interfaces in the earth are irregular, such a theory should prove valuable. The Weber integral solution to the wave equation is interpreted using a dual limiting procedure to define a principal value. Numerical formulation and applications are given in following articles.

Propagation of transient leaking modes in a stratified elastic waveguide

By using a technique first applied to seismological problems by Haskell, it is a simple matter to obtain the secular function for any given elastic waveguide, when the waveguide is composed of homogeneous layers with plane parallel boundaries. Roots of the secular function are usually exhibited as dispersion curves. In this paper the independent variable is dimensionless wave number, K, and the dependent variable is dimensionless frequency, F. A plot of complex F versus K yields the diagnostic (F, K) diagram—a dispersion curve. The procedure employed is to expand the secular function about K = 0 and then to trace the behavior of each root, F(K), for increasing values of K. In each of the specific problems treated the initial position of each root, F(0), admits of a simple physical description. For example, in a simple continental waveguide (solid layer/solid half‐space), there are three sets of roots: the Lamb roots, consisting of and modes; the shear ‘Organ pipe’ roots; and the compressional ‘organ pipe’ roots. There is an infinite number of the two kinds of organ pipe roots. The low‐frequency PL wave arises from one of the roots, and the normal shear modes arise from transitions of all three kinds of roots. In a simple oceanic waveguide (fluid layer/solid half‐space) the shear organ pipe roots are absent. The low‐frequency PL wave again arises from one of the roots, and the normal modes arise from transitions of the Lamb roots and the compressional organ pipe roots. In a simple acoustic waveguide (fluid layer/fluid half‐space) only the compressional organ pipe roots are present, and the normal modes arise from transitions of these roots. The PL wave is absent. Its disappearance is clearly traced as the half‐space of the oceanic waveguide approaches a Poisson ratio of 0.5.The treatment of waveguides composed of more than one layer offers only the additional difficulty of finding the initial positions of the organ pipe roots. When these positions have been found, the analysis proceeds in a manner similar to that for simple waveguides. The initial positions of the organ pipe roots are complex, a situation that may be interpreted physically as radiation into the half‐space of the waveguide. It is the presence of such radiation that leads one to speak of leaking modes. The roots also have complex initial positions, if the half‐space is soft enough. In addition to being complex, the dispersion curves for the leaking modes sometimes have regions of negative group velocity. If one equates group velocity to velocity of energy transport along the waveguide, then one must conclude that there is an inward energy flux when the group velocity is negative. But when the group velocity is demonstrably not the velocity of energy transport, a simple physical picture of negative group velocity is sometimes unavailable. Plots of group velocity versus F show a banded structure in some cases but are generally quite complex. The situation can be clarified somewhat by rejecting those modes with weak excitation functions, large decay parameters, or both. Still, the (F, K) diagram is clearer. Now that seismic array processing procedures are being developed, one hopes that experimental (F, K) diagrams will become a standard tool in seismic analysis, leading to a clearer picture of seismic dispersion and propagation characteristics.

Influence of Fluid Motion Past a Plane Boundary on Sound Reflection, Absorption, and Transmission

It is shown that the effect of fluid motion past a plane boundary on the reflection and absorption of sound is equivalent to an increase of the normal acoustic impedance of the boundary by a factor (1 + M sinφ), where φ is the angle of incidence of the sound wave, and M is the Mach number of the flow velocity component in the incidence reflection plane of the wave. Similarly, the acoustic energy flux perpendicular to the boundary and the flow is shown to be increased by the same factor. Reflection and transmission coefficients of a thin solid interface between a fluid in motion and one at rest are given. Furthermore, some comments on the problem of transmission in duets are given. For propagation between two plane parallel boundaries with the same acoustic admittance, we find, for sufficiently small values of the admittance, that the sound pressure attenuation constant of the fundamental mode is modified approximately by the factors (1 + M)−2 and (1 − M)−2 for downstream and upstream propagation, where M is the flow Mach number.

The Acoustic Resonator Interferometer: I. The Acoustic System and its Equivalent Electric Network

The steady state of motion of a fluid between two infinite plane parallel boundaries is found for the case in which one of the boundaries is given a prescribed periodic motion normal to its surface, the other boundary being infinitely rigid or being assigned a coefficient of reflection. The excess pressure at any point in the fluid is found, being of particular interest at the boundary of the source where it has a term in phase with the velocity of the source and one in phase with its displacement. These terms pass through cyclical values as the distance between the source and reflector is increased, the first passing through sharp maxima, the second changing rapidly from negative to positive values at reflector distances of an integral number of half wave-lengths in the fluid. Application is made to the case where the source is the surface of a piezoelectric plate maintained in forced vibration. The equivalent electric network of the plate and coupled fluid column is found to be the same as that for the plate alone, with modified resistance and capacity coefficients, making possible consideration of the theory of the acoustic resonator interferometer in conjunction with driving and measuring circuits.