Showing papers on "Plane wave published in 2007"

Wave fields in real media : wave propagation in anisotropic, anelastic, porous and electromagnetic media

Abstract: Contents Preface Acknowledgments About the author Basic notation Glossary of main symbols 1 Anisotropic elastic media 2 Viscoelasticity and wave propagation 3 Isotropic anelastic media 4 Anisotropic anelastic media 5 The reciprocity principle 6 Reflection and transmission of plane waves 7 Biot's theory for porous media 8 Numerical methods

Carbon Nanotubes as Optical Antennae

Abstract: Light scattering from an array of aligned multiwall carbon nanotubes (MWCNTs) has previously been investigated, and shown to be consistent with that from an array of antennae. Two basic antenna effects have been demonstrated: 1) the polarization effect, which suppresses the response of an antenna when the electric field of the incoming radiation is polarized perpendicular to the dipole antenna axis, and 2) the antenna-length effect, which maximizes the antenna response when the antenna length is a multiple of the radiation half wavelength in the medium surrounding the antenna. In these previous experiments a random nanotube array was chosen to eliminate the intertube diffraction effects. In this communication, we provide compelling evidence of the antenna action of an MWCNT, by demonstrating that its directional radiation characteristics are in an excellent and quantitative agreement with conventional radio antenna theory and simulations. According to conventional radio antenna theory, a simple “thin” wire antenna (a metallic rod of diameter d and length l >> d) maximizes its response to a wavelength k when l = mk/2, where m is a positive integer. Thus, an antenna acts as a resonator of the external electromagnetic radiation. An antenna is a complex boundary value problem; it is a resonator for both the external fields, and the currents at the antenna surface. In a long radiating antenna, a periodic pattern of current distribution is excited along the antenna, synchronized with the pattern of fields outside. The current pattern consists of segments, with the current direction alternating from segment to segment. Thus, a long antenna can be viewed as an antenna array consisting of smaller, coherently driven antennae (segments) in series. Therefore, the resulting radiation pattern, as a function of the angle with respect to the antenna axis, consists of lobes of constructive interference, separated by radiation minima due to destructive interference. Consider a simple antenna as shown in Figure 1a. The radiation pattern produced by this antenna is rotationally symmetric about the z axis. For a center-fed antenna, or one excited by an external wave propagating perpendicular to the antenna axis (i.e., with the glancing angle hi = 90°), the pattern is also symmetric with respect to the x–y plane. For an antenna excited by an incoming wave propagating at an angle (hi < 90°), the relative strengths of the radiation lobes are expected to shift towards the specular direction. This follows from a qualitative argument based on the single-photon scattering picture, and conservation laws for scattered photons from an antenna. Since such scattering is elastic, the energy of each scattering photon x (where is the reduced Planck constant and x is the angular frequency) and its total momentum k = ki= ks (where k is the wave number, ki is the incident wave vector, and ks is the scattered wave vector) must be conserved. Due to the cylindrical symmetry, K, the length of the momentum vector component perpendicular to the antenna, must also be conserved. Thus, the momentum components parallel to the antenna for the incoming, and scattered photons, k (s) and k (i) respectively, satisfy the following condition k (i) = k 2 – K = k (s), or finally k (s) = ±k (i). This immediately leads to a formula for the angle of scattering hs = 180° – hi, since for a “thin” antenna with diameter d << l, the back scattering is suppressed, and therefore the negative sign is unlikely. Thus, scattering is dominated by the specular reflection. We have confirmed this effect, by measuring the scattered microwave radiation from a simple wire antenna; the forward radiation was about one order of magnitude more intense than for the backward scattered wave. To complete the preliminary antenna studies, we performed computer simulations of the detailed antenna response to an external plane wave. Figure 1b shows a polar coordinate plot of the radiation pattern (field intensity versus h), in the y–z plane, from a thin antenna (d = 0.001l) with l = 7k. The incoming wave of fixed wavelength k, struck the antenna at different incidence angles hi = 30°, 45°, and 60°. This produced radiation patterns dominated by lobes at hs = 180 – hi = 150°, 135°, and 120°, respectively: the scattering was dominated by the specular reflection with respect to the 90° line, in agreement with the simple analysis above. Due to cylindrical symmetry in the x–y plane, the response was mirror-image symmetric about the z axis in the x–z plane. The dependence of C O M M U N IC A IO N

Multipole Plasmons in Metal Nanorods: Scaling Properties and Dependence on Particle Size, Shape, Orientation, and Dielectric Environment

Abstract: T-matrix formalism was used to study the multipole resonances excited by electromagnetic plane waves in gold and silver nanorods whose shape was modeled by prolate spheroids and cylinders with flat or semispherical ends (s-cylinders). The particle diameters and aspect ratio were varied from 20 to 80 nm and from 2 to 20, respectively. By using extended precision T-matrix codes, we calculated the extinction, absorption, and scattering spectra for random and fixed orientations of the particle axis with respect to the incident transverse magnetic (TM) and transverse electric (TE) polarized light, where the reference plane is defined by the particle axis and the incident wave vector. We found that the parity of a given spectral resonance number n coincides with the parity of their multipole contributions l, where I is equal to or greater than n, and the total resonance magnitude is determined by the lowest multipole contribution. The random-orientation resonances are excited most effectively by the TM scattering configurations, except for the short-wavelength resonance, which equals the sum of the dominant dipole TE resonance and the other multipole contributions. The even multipole resonances are maximal at intermediate orientations, whereas the odd multipoles can effectively be excited at both perpendicular and intermediate orientations of the rod axis with respect to the TM incident wave. In particular, the quadrupole resonance can be excited only by the TM incident wave, and the resonance magnitude is maximal for orientation of the particle symmetry axis near 54° with respect to the incident light. Finally, we found that the multipole resonance wavelengths obey a universal linear scaling when plotted versus the particle aspect ratio divided by the resonance number. This remarkable property of multipole resonances can be understood in terms of a simple concept based on plasmon standing waves excited in metal nanowires by an electric field of incident light (Schider et al. Phys. Rev. B 2003, 68, 155427). The refractive index sensitivity of the multipole resonance wavelength to a dielectric environment also exhibits linear scaling properties. Specifically, the relative shift of the resonance wavelength is proportional to the relative refractive index increment with a universal angular slope coefficient.

An Investigation Into the Hydrodynamic Efficiency of an Oscillating Water Column

Abstract: An oscillating water column device enables the conversion of wave energy into electrical energy via wave interaction with a semi-submerged chamber coupled with a turbine for power take off. This present work concentrates on the wave interaction with the semi-submerged chamber, whereby a shore based oscillating water column (OWC) is studied experimentally to examine energy efficiencies for power take-off. The wave environment considered comprises plane progressive waves of steepnesses ranging from kA=0.01 to 0.22 and water depth ratios varying from kh=0.30 to 3.72, where k, A, and h denote the wave number, wave amplitude, and water depth, respectively. The key feature of this experimental campaign is a focus on the influence of front wall geometry on the OWC’s performance. More specifically, this focus includes: front wall draught, thickness, and aperture shape of the submerged front wall. We make use of a two-dimensional inviscid theory for an OWC for comparative purposes and to explain trends noted in the experimental measurements. The work undertaken here has revealed a broad banded efficiency centered about the natural frequency of the OWC. The magnitude and shape of the efficiency curves are influenced by the geometry of the front wall. Typical peak magnitude resonant efficiencies are in the order of 70%.

Foundations of Stress Waves

Abstract: Chapter 1. Introduction Chapter 2. Elementary Theory of One-Dimensional Stress Waves in Bars Chapter 3. Interaction of Elastic Longitudinal Waves Chapter 4. Interaction of Elastic-Plastic Longitudinal Waves in Bars Chapter 5. Rigid Unloading Approximation Chapter 6. One-Dimensional Visco-Elastic Waves and Elastic-Visco-Plastic Waves Chapter 7. One-Dimensional Strain Plane Waves Chapter 8. Spherical Waves and Cylindrical Waves Chapter 9. Elastic Plastic Waves Propagating in Flexible Strings Chapter 10. Elastic Plastic Waves Propagating in Beams under Transverse Impact (Bending Wave Theory) Chapter 11. General Theory for Linear Elastic Waves Chapter 12. Numerical Methods for Stress Wave Propagation

Compact broad-band admittance tunnel incorporating gaussian beam antennas

Abstract: A plane wave antenna including: a horn antenna; a waveguide at least partially inside the horn antenna, wherein the waveguide includes: a central dielectric slab increasing in width toward the horn antenna and with a first dielectric constant, an upper slab above the central dielectric slab with a second dielectric constant, and a lower slab below the central dielectric slab with the second dielectric constant; wherein the central dielectric slab has a substantially constant thickness less than a quarter of a wavelength at a highest frequency of operation of the plane wave antenna.

On a Class of Singular Nonlinear Traveling Wave Equations

Abstract: The existence of solitary wave, kink wave and periodic wave solutions of a class of singular reaction–diffusion equations is obtained using some effective methods from the dynamical systems theory. Specially, for a class of nonlinear wave equations, fundamental properties of profiles of traveling wave solutions determined by some bounded orbits of the traveling wave systems are rigorously proved. Parametric conditions that guarantee the existence of the aforementioned solutions are derived and given explicitly.

Binding energy of a hydrogenic donor impurity in a rectangular parallelepiped-shaped quantum dot : Quantum confinement and Stark effects

Abstract: We calculate the binding energy of a hydrogenic donor impurity in a rectangular parallelepiped-shaped quantum dot (QD) in the framework of effective-mass envelope-function theory using the plane wave basis. The variation of the binding energy with edge length, position of the impurity, and external electric field is studied in detail. A finite potential model is adopted in our calculations. Compared with the infinite potential model [C. I. Mendoza et al., Phys. Rev. B 71, 075330 (2005)], the following results are found: (1) if the impurity is located in the interior of the QD, our results give a smaller binding energy than the infinite potential model; (2) the binding energies are more sensitively dependent on the applied electric field in the finite potential model; (3) the infinite potential model cannot give correct results for a small QD edge length for any location of the impurity in the QD; (4) some degeneracy is lifted when the dot is no longer cubic.

Model GW band structure of InAs and GaAs in the wurtzite phase

Abstract: We report quasiparticle calculations of the newly observed wurtzite polymorph of InAs and GaAs. The calculations are performed in the GW approximation (based on a model dielectric function) using plane waves and pseudopotentials. For comparison we also report the study of the zinc-blende phase within the same approximations. In the InAs compound the In 4d electrons play a very important role: whether they are frozen in the core or not leads either to a correct or a wrong band ordering (negative gap) within the local-density appproximation (LDA). We have calculated the GW band structure in both cases. In the first approach, we have estimated the correction to the pd repulsion calculated within the LDA and included this effect in the calculation of the GW corrections to the LDA spectrum. In the second case, we circumvent the negative gap problem by first using the screened exchange approximation and then calculating the GW corrections starting from the so obtained eigenvalues and eigenfunctions. This approach, that can be thought of as a step towards self-consistency, leads to a more realistic band structure and was also used for GaAs. For both InAs and GaAs in the wurtzite phase we predict an increase of the quasiparticle gap with respect to the zinc-blende polytype.

Study on band gaps of elastic waves propagating in one-dimensional disordered phononic crystals

Abstract: Band gaps of elastic waves, both in-plane and anti-plane waves, propagating along arbitrary direction in one-dimensional disordered phononic crystals are studied in this paper. The localization of wave propagation due to random disorder is discussed by introducing the concept of the localization factor. As a special case between ordered and disordered structures, we analyze the properties of the band gaps of phononic crystals with quasi-periodicity (i.e. phononic quasicrystals). Compared with the periodic structure, phononic quasicrystals involve more bands with localization of wave motion. The transmission coefficients are also calculated and the results show the same behaviors as the localization factor does. Therefore, the localization factor may act as an accurate and efficient parameter to characterize band structures of both ordered and disordered (including quasi-periodic) phononic crystals.

Graded index photonic crystals.

Abstract: We explore two-dimensional triangular lattice photonic crystals composed of air holes in a dielectric background which are subject to a graded-index distribution along the direction transverse to the propagation. The proper choice of the parameters such as the input beam width, gradient coefficient, and the operating frequency allow the realizations of the focusing (lens) and guiding (waveguide) effects upon which more complex optical devices such as couplers can be designed. Numerical results obtained by the finite-difference time-domain and planewave expansion methods validate the application of Gaussian optics within a range of parameters where close agreement between them are observed.

Dynamics of light propagation in spatiotemporal dielectric structures.

Abstract: Propagation, transmission and reflection properties of linearly polarized plane waves and arbitrarily short electromagnetic pulses in one-dimensional dispersionless dielectric media possessing an arbitrary space-time dependence of the refractive index are studied by using a two-component, highly symmetric version of Maxwell's equations. The use of any slow varying amplitude approximation is avoided. Transfer matrices of sharp nonstationary interfaces are calculated explicitly, together with the amplitudes of all secondary waves produced in the scattering. Time-varying multilayer structures and spatiotemporal lenses in various configurations are investigated analytically and numerically in a unified approach. Several effects are reported, such as pulse compression, broadening and spectral manipulation of pulses by a spatiotemporal lens, and the closure of the forbidden frequency gaps with the subsequent opening of wave number band gaps in a generalized Bragg reflector.

Design of a mode converter for efficient light-atom coupling in free space

Abstract: In this article, we describe how to develop a mode converter that transforms a plane electromagnetic wave into an inward-moving dipole wave. The latter one is intended to bring a single atom or ion from its ground state to an excited state by absorption of a single photon wave packet with near-100% efficiency.

Magnetic-field-induced helical and stripe phases in Rashba superconductors

Abstract: Due to the lack of both parity and time-reversal symmetries, the Rashba superconductors $\mathrm{Ce}{\mathrm{Pt}}_{3}\mathrm{Si}$, $\mathrm{Ce}\mathrm{Rh}{\mathrm{Si}}_{3}$, and $\mathrm{Ce}\mathrm{Ir}{\mathrm{Si}}_{3}$, in the presence of a magnetic field, are unstable to helical (single plane wave) order. We develop a microscopic theory for such superconductors and examine the stability of this helical phase. We show that the helical phase typically occupies most of the magnetic field--temperature phase diagram. However, we also find that this phase is sometimes unstable to a multiple-$q$ phase (loosely called a stripe phase), in which both the magnitude and the phase of the order parameter are spatially varying. We find the position of this helical to multiple-$q$ phase transition. We further examine the density of states and identify features unique to the helical phase.

Long-time asymptotics of the nonlinear Schrödinger equation shock problem

Abstract: The long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrodinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in space-time: a region with the original wave modified by a phase perturbation, a residual region with a one-phase wave, and an intermediate transition region with a modulated two-phase wave. The leading-order terms for the three regions are computed with error estimates using the steepest-descent method for Riemann-Hilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the plane-wave state well beyond the shock front at large times. © 2007 Wiley Periodicals, Inc.

Theoretical prediction of the nondiffractive propagation of sonic waves through periodic acoustic media

Abstract: We predict theoretically the nondiffractive propagation of sonic waves in periodic acoustic media (sonic crystals) by expansion into a set of plane waves (Bloch mode expansion) and by finite-difference time-domain calculations of finite beams We also give analytical evaluations of the parameters for nondiffractive propagation, as well as the minimum size of the nondiffractively propagating acoustic beams

On the use of transmission eigenvalues to estimate the index of refraction from far field data

Abstract: We consider the scattering of time harmonic electromagnetic plane waves by a bounded inhomogeneous medium and show that under certain assumptions a lower bound on the index of refraction can be obtained from a knowledge of the smallest transmission eigenvalue corresponding to the medium. It is then shown by numerical examples that this eigenvalue can be determined from a knowledge of the far field pattern of the scattered wave, thus providing a practical method for estimating the index of refraction from far field data.

Bloch–Floquet bending waves in perforated thin plates

Abstract: This paper presents a mathematical model describing propagation of bending waves in a perforated thin plate. It is assumed that the holes are circular and form a doubly periodic square array. A spectral problem for the biharmonic operator is formulated in a unit cell containing a single defect, and its analytical solution is constructed using a multipole method. The overall system for the coefficients in the multipole expansion is then solved numerically. We generate dispersion diagrams for the two cases where the boundaries of holes are either clamped or free. We show that in the clamped case, there is a total low-frequency band gap in the limit of inclusions of zero radius, and give a simple formula describing the corresponding band diagram in this limit. We show that in the free-edge case, the band diagram of the vibrating plate is much closer to that of plane waves in a uniform plate than for the clamped case.

Negative axial radiation forces on solid spheres and shells in a Bessel beam.

Abstract: Prior computations predict that fluid spheres illuminated by an acoustic Bessel beam can be subjected to a radiation force directed opposite the direction of beam propagation. The prediction of negative acoustic radiation force is extended to the cases of a solid poly(methylmethacrylate) PMMA sphere in water and an empty aluminum spherical shell in water. Compared with the angular scattering patterns for plane wave illumination, the scattering into the back hemisphere is suppressed when the radiation force is negative. This investigation may be helpful in the development of acoustic tweezers and in the development of methods for manipulating objects during space flight.

Control analysis of the tunable phononic crystal with electrorheological material

Abstract: The elastic band structure of a two-dimensional phononic crystal (PC) with electrorheological (ER) material is investigated and the plane-wave-expansion (PWE) method is adopted to solve the problem in this paper. The ER material is used to act as a tunable elastic composite inserts and its material parameter can be changed by the applied electric fields. Variations of the band gaps for the phononic composite system are also calculated with various electric fields and it is found to have a significant effect on the band gaps of the phononic system with ER material inserts. The band gaps of the smart system can be controlled and the acoustic characteristics of the system are also changed by applied different electric fields.

Liquid sloshing and wave breaking in circular and square-base cylindrical containers

Abstract: Near resonance sloshing in containers, filled with a liquid to a given depth h, depends on three parameters, which are the viscous damping, the frequency offset that contains the forcing amplitude and the fluid depth. Experiments have been conducted with low-viscosity liquids mainly in circular cylindrical containers of radius R subjected to harmonic horizontal forcing; complementary experiments on wave breaking have been performed in a square-base container. The fluid depth was kept large (h/R > 1) so that it was no longer a variable parameter. The bounds of existence of the different wave regimes, namely planar waves, swirling waves, chaotic sloshing as well as breaking waves, have been determined as a function of forcing frequencies relative to the lowest natural frequency ω1 and for a wide range of forcing amplitudes. It is shown that when the forcing frequency ω is slightly larger than the lowest natural frequency ω1, planar wave motion bifurcates to a swirling wave mode at finite wave amplitude, the value of which depends on the offset parameter. The swirl wave amplitude grows exponentially and saturates at a certain value. The swirl has a hard-spring behaviour, is very robust and can generate a vortical flow of the liquid column. Chaotic sloshing and wave breaking occur quasi-periodically: growth of planar wave amplitude at a rate depending on the forcing amplitude, collapse, irregular swirl and again growth of planar wave amplitude. The details and periodicity of the chaotic behaviour and breaking depend on the frequency-offset parameter. Close to the natural frequency, chaotic wave motion is possible without breaking. Planar wave breaking is, in general, associated with spilling caused by the encounter of nearly freely falling lumps of fluid with the upward moving wave crest, in a way demonstrated previously in two-dimensional wave breaking. In three dimensions, the wave crest is destabilized in the crosswise direction so that spilling is not uniform along the wave crest and an irregular swirl is generated following breaking; free fall of fluid lumps occurs over many wave periods. The complementary experiments, performed in a square-base container of base dimension L, show four different wave patterns of wavelengths L and L/2 crosswise to the primary wave. This cross-wave instability is interpreted in terms of parametric instability.

Subsurface Sensing of Buried Objects Under a Randomly Rough Surface Using Scattered Electromagnetic Field Data

Abstract: This paper proposes a new inverse method for microwave-based subsurface sensing of lossy dielectric objects embedded in a dispersive lossy ground with an unknown rough surface. An iterative inversion algorithm is employed to reconstruct the geometry and dielectric properties of the half-space ground as well as that of the buried object. B-splines are used to model the shape of the object as well as the height of the rough surface. In both cases, the control points for the spline function represent the unknowns to be recovered. A single-pole rational transfer function is used to capture the dispersive nature of the background. Here, the coefficients in the numerator and denominator are the unknowns. The approach presented in this paper is based on the state-of-the-art semianalytic mode matching forward model, which is a fast and efficient algorithm to determine the scattered electromagnetic fields. Numerical experiments involving two-dimensional geometries and TM incident plane waves demonstrate the accuracy and reliability of this inverse method

Application of correlation-induced spectral changes to inverse scattering

Abstract: It is shown how the phenomenon of correlation-induced spectral changes generated on scattering of a polychromatic plane wave on a spatially homogeneous random medium may be used to determine the correlation function of the scattering potential of the medium.

Near-field excitation of nanoantenna resonance

Abstract: An array of paired elliptic nanoparticles designed to enhance local fields around the particle pair is fabricated with gold embedded in quartz. Light excites a coupled plasmon resonance in the particle pair and the system acts like a plasmonic nanoantenna providing an enhanced electromagnetic field. Near-field scanning optical microscopy and finite element modeling are used to study the local field effects of the nanoantenna system. Local illumination shows similar resonant properties as plane wave illumination: a strong, localized optical resonance for light polarized parallel to the main, center-to-center axis.

Comparison of Methods for Calculating the Field Excited by a Dipole Near a 2-D Periodic Material

Abstract: A comparison of methods for calculating the field from a single dipole source in the proximity of an infinite periodic artificial material (PAM) is given. A direct plane-wave expansion method (PWM) is compared with the ldquoarray scanning methodrdquo (ASM). The ASM is shown to be the most efficient method for sources that are close to the PAM, since it only requires integration in the wavenumber plane over the Brillouin zone rather than over the entire wavenumber plane. It is also shown how the ASM may be used to efficiently calculate the Fourier transform of the field at any aperture plane, which is useful for asymptotic analysis. The ASM is shown to be more efficient than the PWM or the reciprocity method for calculating the transform of the aperture field. A discussion of the nature of the singularities in the complex wavenumber plane is given, and numerical issues associated with the implementation of the ASM are discussed.

Inner-shell spectroscopy by the Gaussian and augmented plane wave method

Abstract: We present an approach for calculating near-edge X-ray absorption spectra at the density functional theory level, which is suited for condensed matter simulations. The method is based on the standard solution of the all-electron KS equations with a modified core-hole potential, which reproduces the relaxation of the orbitals induced by the promotion of the core electron to an unoccupied valence level. The all-electron description of the charge density is based on the Gaussian and augmented plane wave formalism. The reliability of the proposed method is assessed by comparing the computed spectra of some small molecules in the gas phase to the experimental spectra reported in literature. The sensitivity of the computed spectra to the local environment, i.e. the specific bonds formed by the absorbing atom or the presence of hydrogen bonds, open promising perspective for this technique as a predictive tool in the investigation of a more complex system of an unknown structure. The straightforward extension of the method to condensed matter is demonstrated by the calculation of the C K-edge in diamond.

Evolution of wide-spectrum unidirectional wave groups in a tank: an experimental and numerical study

Abstract: Evolution of unidirectional nonlinear wave groups with wide spectra is studied experimentally and numerically. As an example of such an evolution, focusing of an initially wide wave train that is modulated both in amplitude and in frequency, to a single steep wave at a prescribed location along the laboratory wave tank is investigated. When numerous frequency harmonics arrive at the focusing location in phase, a very wave steep single emerges. The experimental study was carried out in two wave flumes that differ in size by an order of magnitude: a 330 m long Large Wave Channel in Hanover, and in 18 m long Tel-Aviv University wave tank. The spatial version of the Zakharov equation was applied in the numerical simulations. Detailed quantitative comparison is carried out between the experimental results and the numerical simulations. Spectra of the 2nd order bound waves are calculated using the theoretical model adopted. It is demonstrated that with the contribution of bound waves accounted for, a very good agreement between experiments and simulations is achieved.

The interaction between an internal gravity wave and the planetary boundary layer. Part II: Effect of the wave on the turbulence structure

Abstract: An atmospheric internal gravity wave was measured over a two-hour period by a microbarograph array and a series of fast response wind and temperature sensors deployed along a 300m tower. the particularly monochromatic nature of the pressure signal at the ground enabled an explicit separation of the velocity field into mean, wave, and turbulent components. Large wave-frequency fluctuations were observed in the turbulent Reynolds stresses. Their significance is discussed at length with special regard to their role in the budget of wave kinetic energy. It is shown that the quadrature relationship which they maintain with the components of wave-shear, limits their effectiveness in reducing the wave amplitude. Analysis of the important time scales in the budget of turbulent kinetic energy reveals that the energy containing eddies have time scales longer than the wave period, and consequently the turbulence cannot remain in equilibrium with large, wave-frequency fluctuations in shear production. the result is a boundary layer which never attains a true equilibrium state.