Phase Speed Changes Research Articles

Ultrasonic wave transmission in periodically undulated plates

This study focuses on quantifying the change in phase speed of waves transmitting through periodically undulated plates under pass band interaction. A perturbation technique is used to analyze the transmission of horizontally polarized guided waves in elastic plates with sinusoidal periodicity at their outerfaces. Phase speed of transmitting modes is presented as a function of various parameters, including outerface wavenumber, undulation amplitude, degree of undulations symmetry about the periodically undulated plate midplane, plate average thickness, and frequency of propagation.

Observation of the gravity waves from GPS/MET radio occultation data

We show that the amplitude of the Global Positioning System (GPS) signals in the radio occultation (RO) experiments is sensitive to the atmospheric wave structures. Earlier the phase of the GPS occultation signals have been used for statistical investigation of the gravity waves (GW) activity in the height interval 10–40 km on a global scale. Analysis of the RO amplitude data revealed wave clusters (quasi-regular structures) with the vertical size of about 10 km and interior vertical period ∼0.8–2 km in the tropopause and lower stratosphere. The amplitude RO data may be utilized to determine the temperature vertical profiles and its vertical gradient in the upper troposphere and stratosphere. In the considered RO events variations of the vertical temperature gradient dT(h)/dh corresponding to the amplitude clusters are in the range from −9 K/km to 6 K/km with vertical scales ∼1–3 km. We show that these variations can be linked to the GW propagation in the atmosphere. We use the polarization and dispersion relationships and Hilbert transform to find the 1-D GW image in the atmosphere by analyzing the vertical temperature gradient d T( h)/d h. The GW image consists of the phase and amplitude of the GW as functions of height. The GW amplitude is non-uniformly distributed with main contribution associated with the tropopause and the secondary maximums in height interval 18–35 km. Using our method we find vertical profiles of the horizontal wind perturbations and their vertical gradient associated with the GW influence. The horizontal wind perturbations are changing in the interval v∼2–12 m/s with vertical gradients d v/d h∼4–25 m s −1 km −1) in the tropopause area and v∼ 3–9 m/s, dv/dh ∼ 2–15 m/(s km) in the stratosphere for the considered events. For one RO event we compared the estimated values of the horizontal wind perturbations with aero-logical data and found fairly good agreement. The height dependence of the GW vertical wavelength was inferred through the differentiation of the GW phase. Analysis of this dependence using the dispersion relationship for the GW gives estimation of the GW intrinsic phase speed. For the considered events the magnitude of the intrinsic phase speed changes between 1.5–15 m/s at heights 10–40 km. We conclude that the amplitude of the GPS occultation signals contain important information about the GW propagation in the atmosphere on a global scale.

Liquid flows and vibration characteristics of straight-tube cylindrical shells

The influence of a moving fluid confined by a solid circular cylindrical shell on the propagation of acoustic waves generated by sources located on the circular cylindrical shell is examined. An expression for the acoustic pressure in a moving fluid is derived including azimuthal asymmetry effects in the general case, where the fluid velocity points along the cylindrical shell axis and can be written as an infinite power series expansion in the radial co-ordinate. Secondly, continuity of pressure and normal velocity at the liquid–shell interface is imposed to (a) derive a set of coupled differential equations governing the possible vibrational modes of the shell and (b) determine dispersion relations, i.e., mode propagation constants β as a function of frequency as well as changes in β values accomodated by flow. In the remaining part of the paper, phase speed changes with flow and transit-time differentials of circular cylindrical shell vibrations are discussed with special emphasis to flow measurement properties.

SOUND PROPAGATION IN A MOVING FLUID CONFINED BY CYLINDRICAL WALLS—A COMPARISON BETWEEN AN EXACT ANALYSIS AND THE LOCAL-PLANE-WAVE APPROXIMATION

A discussion of sound propagation in a moving fluid confined by cylindrical walls is presented. Based on the continuity equation and the Euler equation, a single “exact” ordinary differential equation in the acoustic pressure is derived for the case where the medium flow v0(r) depends on the radial co-ordinate only and points in the axial direction. This “exact” pressure wave equation is solved semi-analytically by means of the Frobenius method and compared with the conventional approximative wave equation known as the local-plane-wave (LPW) approximation for a range of flow values. In this way, information about mode phase-speed changes with flow and flow-meter performance is obtained. It is found that the LPW approximation works well only for mode propagation parallel or nearly parallel to the direction of flow. Based on the “exact” acoustic pressure wave equation, it is also concluded that flow-meter errors become independent of ultrasound frequency and cylinder radius, a point that the LPW approximation fails to predict. Furthermore, an “exact” procedure shows that flow-meter errors depend on the Reynolds number and the mode number only. In actual fact, it is found that flow measurement based on the fundamental mode is approximately free of errors while all other modes are characterized by the same (and, generally, non-vanishing) deviation of measurement.

Perturbation theory applied to sound propagation in flowing media confined by a cylindrical waveguide.

First-order perturbation theory is employed to examine sound propagation in flowing media confined by a cylindrical waveguide. The use of perturbation theory allows examination of mode phase-speed changes due to any radially dependent flow w(r) as long as the flow magnitude is sufficiently small. The condition to be fulfilled is satisfied in the flow range: 0-0.3 m/s for the specific values of cylinder radius, ultrasound frequency, and sound speed analyzed in the present work [in the general case, however, the condition in Eq. (1) of the present work must be fulfilled]. This freedom of choice, i.e., the possibility to handle any radial flow profile, is used to analyze two flow profile cases: (1) where w(r) is a linear combination of a laminar flow profile and a flat profile corresponding to turbulent flow, and (2) where w(r) is a linear combination of a laminar flow profile and a more realistic logarithmic-dependent turbulent flow profile. In both cases, it is shown that large errors may result in ultrasound flow measurements if several modes are excited by the transmitting transducer, and that a logarithmic flow profile in the turbulent regime leads to somewhat larger measurement errors at high flow values as compared to assuming a simple flat profile in the turbulent regime.

The Speed of Observed and Theoretical Long Extratropical Planetary Waves

Planetary or Rossby waves are the predominant way in which the ocean adjusts on long (year to decade) timescales. The motion of long planetary waves is westward, at speeds ≥1 cm s−1. Until recently, very few experimental investigations of such waves were possible because of scarce data. The advent of satellite altimetry has changed the situation considerably. Curiously, the speeds of planetary waves observed by TOPEX/Poseidon are mainly faster than those given by standard linear theory. This paper examines why this should be. It is argued that the major changes to the unperturbed wave speed will be caused by the presence of baroclinic east–west mean flows, which modify the potential vorticity gradient. Long linear perturbations to such flow satisfy a simple eigenvalue problem (related directly to standard quasigeostrophic theory). Solutions are mostly real, though a few are complex. In simple situations approximate solutions can be obtained analytically. Using archive data, the global problem is treated. Phase speeds similar to those observed are found in most areas, although in the Southern Hemisphere an underestimate of speed by the theory remains. Thus, the presence of baroclinic mean flow is sufficient to account for the majority of the observed speeds. It is shown that phase speed changes are produced mainly by (vertical) mode-2 east–west velocities, with mode-1 having little or no effect. Inclusion of the mean barotropic flow from a global eddy-admitting model makes only a small modification to the fit with observations; whether the fit is improved is equivocal.

Displacement-based finite element method for guided wave propagation problems: Application to poroelastic media

Poroelastic materials are used extensively in the literature for sound absorption and attenuation. Guided wave propagation in these materials is therefore of considerable interest to researchers. In the present paper, a finite element method is presented for guided wave propagation in poroelastic media. For harmonic vibrations, the finite element method gives rise to a quadratic eigenvalue problem in the phase speed. Solution of the standard finite element procedure is shown to excite modes with spurious phase speeds. It is shown that the real and spurious modes can be identified using the penalty constraint approach. For this, one needs to compute the rate of change of phase speed with the penalty constraint constant α. An expression for computing this rate of change of phase speed has been derived. It is shown that in most cases the rate of change of phase speed needs to be evaluated only at α=0. The results apply to a pure fluid and also to poroelastic media. In the process, analytical expressions are also derived for wave propagation in poroelastic cylinders. The use of reduced order integration to eliminate spurious modes is also discussed in this paper. Dispersion curves for the phase speeds are presented for the first few breathing modes of a poroelastic cylinder. Finally, results are presented for the case of a rigid, lined ducted system.

Determination of a steady velocity field in a rotating frame of reference at the surface of the Earth's core

SUMMARY We relax the steady-motions theorem by solving for a steady velocity field at the surface of the core in a frame of reference drifting at a linear rate with respect to an observer fixed in the mantle frame of reference. We make the frozen-flux approximation, and compare the misfit of the secular variation (SV) predicted by the drifting velocity field with that from a steady velocity field fixed to the mantle frame of reference. The decrease of the misfit to the geomagnetic SV across the period 1960-80 is substantial, but is marginal across the interval 1930-60. The drift rate changes sign at the 1970 geomagnetic ‘jerk’ epoch, indicating a change in phase speed between the mantle and core flow. The marginal decrease in misfit prior to 1960 is inadequate to fit the SV data, necessitating a more complex drift function or perhaps a fully time-dependent flow. The results suggest that the SV is driven by deep-seated convection rather than from the core-mantle boundary.

On the changes in phase speed of one train of water waves in the presence of another

We present calculations of the change in phase speed of one train of water waves in the presence of another. We use a general method, based on Zakharov's (1968) integral equation. It is shown that the change in phase speed of each wavetrain is directly proportional to the square of the amplitude of the other. This generalizes the work of Longuet-Higgins & Phillips (1962) who considered gravity waves only.In the important case of gravity-capillary waves, we present the correct form of the Zakharov kernel. This is used to find the expressions for the changes in phase speed. These results are then checked using a perturbation method based on that of Longuet-Higgins & Phillips (1962). Agreement to 6 significant digits has been obtained between the calculations based on these two distinct methods. Full numerical results in the form of polar diagrams over a wide range of wavelengths, away from conditions of triad resonance, are provided.

Flow across reefs or between islands, and effects on shelf-sea motions

Oscillatory (e.g., tidal) flow over and around shallow reefs and islands is considered. Overall matching conditions (between sea levels and flows across the reef) are derived, to represent reef effects in models of larger regions. Analyses include various reef forms of uniform cross-section and water depth h(x), both narrow (uniform cross-reef transport) and broad (decoupled motions on either side). Simple island chains, or reefs broken by passages conducting all the flow, are also considered. In all cases, cross-reef transport is “down” the sea-surface slope, phase-shifted by 0 to 1/4 cycle. The transport is sufficient to render broken reefs effectively transparent to tides, unless the reefs are unusually far from the coast (> 100km, say), dense (> 90%) and broad (> 25km). The matching conditions are consistent with (expected) energy losses for unforced trapped waves propagating along a continental shelf with a long-shelf reef. There is a corresponding change of phase speed, if these effects are small, as is generally so for Kelvin waves. Continental shelf waves, particularly near their maximum frequency, may be strongly retarded and damped by a shelf-edge reef.

External Rossby Modes on a β–Plane with Realistic Vertical Wind Shear

Abstract Free oscillations of small amplitude and low frequency in an isothermal atmosphere at rest have the vertical structure of an edge wave, energy density decaying exponentially away from the ground. In this paper we treat these external Rossby modes on a middle latitude β-plane to see how their vertical structure and phase speed are changed by the presence of a winter zonal wind with realistic vertical structure. We find that for planetary-scale modes the basic state interior potential vorticity gradient and the temperature gradient at the lower boundary produce roughly equal but opposite changes in phase speed. Energy density decays less rapidly with altitude when the basic state temperature gradient is included in the lower boundary condition. As wavenumber is increased, the phase speed of the external Rossby mode is found to approach the speed of the zonal wind at the surface, and there is a limiting wavenumber beyond which the mode does not exist. At this limiting value of wavenumber, the mode b...

The propagation of planetary waves over a random topography

The purpose of this paper is to consider the effect of one-dimensional random depth variation on the propagation of planetary waves in a homogeneous layer of fluid having a free upper surface. We begin by determining the dispersion relation for the coherent part of the wave using the vorticity equation for the transport stream function and a previously described perturbation method. Then, from the resulting first-order expressions for the wavenumber, we obtain the phase speeds for the two possible planetary-wave solutions. These are compared with the corresponding phase speeds of planetary waves over a smoothly varying topography; the validity limits of the approximations are discussed. For the most physically realizable situation, of random depth correlation lengths much shorter than a typical wavelength, we find that the phase speed of the shorter (longer) wave component is less (greater) over a randomly varying topography than over a smoothly varying topography. In the case of the shorter waves, greatest relative changes in phase speed occur when the associated fluid motions are at right angles to the ‘strike’ of the roughness elements, while for both long and short waves there is no relative change in phase speed if fluid motions are parallel to the roughness contours. Moreover, both types of waves are shown to lose energy in the direction of energy propagation as a result of scattering. Numerical values are then obtained using hydrographic charts of the western North Pacific, and show that the randomness may significantly decrease the phase speed of the shorter planetary-wave component. Finally, we give a brief descriptive explanation of the results based on the effect of the topography on the wave restoring mechanism.

Internal waves in a randomly stratified fluid

Abstract We discuss the propagation of internal waves in a rotating stratified unbounded fluid with randomly varying stability frequency, N. The first order smoothing approximation is used to derive the dispersion relation for the mean wave field when N is of the form N 2 = N o 2(1 + eμ), where μ is a centered stationary random function of either depth (z) or time (t), N o = constant and O < e2 ≦ 1. Expressions are then derived for the change in phase speed and growth rate due to the random fluctuations μ; in particular, attention is focused on the behaviour of these expressions for short and long correlation lengths (case μ = μ(z)) and times (case μ = μ(t)). For the case μ = μ(z), which represents a model for the temperature and salinity fine-structure in the ocean, the appropriate statistics of the fluctuations observed at station P (50°N, 145°W) have been incorporated into the theory to estimate the actual importance of the effects due to these random fluctuations. It is found that the phase speed of t...