Case Of Long Waves Research Articles

Shear-horizontal waves in a rotated Y-cut quartz plate in contact with a viscous fluid

We study shear-horizontal (SH) waves in a crystal plate of rotated Y-cut quartz in contact with a semi-infinite viscous fluid. The crystal plate and the fluid are governed by the equations of anisotropic elasticity and the theory of Newtonian fluids. A transcendental equation that determines the dispersion relations of the waves is obtained. Approximate analytical solutions to the equation are presented for the case of low viscosity fluids and the case of long waves whose wavelength is much larger than the plate thickness. The effects of the fluid viscosity and density on the dispersion relations of the waves are examined. The results obtained are fundamental and useful to the understanding and design of acoustic wave fluid sensors for measuring fluid viscosity or density.

Long waves in erodible channels and morphodynamic influence

Like most media, open channel flows propagate information through waves. When the channel boundary is fixed, the vectors of information consist typically of surface gravity waves. In the less straightforward case of channels with cohesionless bed and possibly erodible banks, other types of waves arise from the erodible nature of the boundaries and the ability of the stream to transport sediments. In this paper we restrict our attention to the important case of long waves, which can be described by employing the shallow water approximation for the flow field and a quasi‐equilibrium assumption for sediment transport on weakly sloping beds. We focus on a major issue: In which direction is information propagated? This is a problem raised and partially solved by de Vries in the context of one‐dimensional morphological modeling as early as 1965. We review some of the available knowledge on this subject, viewed in a more general context where vectors of information can be a variety of waves: purely longitudinal one‐dimensional sediment waves, two‐dimensional waves driven by large‐scale bed forms (bars), and plan form waves carrying information related to the planform shape of the channel. Both linear and nonlinear, migrating and stationary waves are considered. It turns out that the role played by the Froude number in determining the direction of one‐dimensional perturbations of bed topography is somewhat taken by the aspect ratio of the channel when large‐scale two‐dimensional bed forms as well as planform waves are considered.

On the mass transport of water waves in a turbulent boundary layer

Abstract An analytical theory describing the mass transport velocity induced by water waves in a turbulent boundary layer is presented. A one-layer time-invariant eddy viscosity model, which is more realistic under natural conditions, is specified to calculate steady streaming within boundary layer. A simplified analysis is developed based on the wave friction factor relating the bottom shear stress and free stream velocity. A predicted reversal of the mass transport is obtained for the case of long waves. The results are similar to those based on time-dependent eddy viscosity models. The present theory is much simpler mathematically than existing theories. The numerical results are compared favorably with available data.

Long waves in two superposed layers of magnetic fluid

The equations and boundary conditions describing plane-parallel potential motions of two superposed layers of stably stratified magnetic fluid are formulated. The fluid is assumed to fill entirely a horizontal plane channel in the presence of a uniform longitudinal magnetic field induced by external sources. With reference to the case of long waves propagating over the interface between the upper and lower layers, it is shown that the action of the field may be interpreted as the result of an increase in the nondimensional surface tension by an amount proportional to the square of the undisturbed field. In the linear formulation the effect of the field on the evolution of a long-wave perturbation of the initially plane interface is investigated. Korteweg-de Vries equations with quadratic and cubic nonlinearities are derived and the action of the field on the internal solitary waves is analyzed.

The motion of a vortex filament with axial flow

Infinitesimal waves on a uniform vortex with axial flow are studied. The equation for the frequency of helical waves is obtained, and solved for the case of long waves which leave the internal structure almost unaltered. A method is developed to obtain results for vortices of non-uniform structure and for displacements which are not necessarily small compared with the core radius. The approach consists of balancing the Kutta—Joukowski lift force, the momentum flux due to the axial motion, and the ‘tension’ of the vortex lines. A general equation for the motion of a vortex filament is obtained, valid for arbitrary shape and internal structure, and in the presence of an external irrotational velocity field. When the axial flow vanishes, the method is equivalent to using the Biot—Savart law for the self-induced velocity, with a suitable cutoff. The impulse of a vortex filament is discussed and its rate of change is given.

Oscillatory flow of linear viscoelastic fluids in thin-walled elastico-viscous tubes

The oscillatory flow of a linear visco-elastic fluid in a thin-walled elastico-viscous tube is investigated for the case of long waves. It is found that the elasticity of the fluid as well as its viscosity, affect the flow mainly at low frequencies whereas at high frequencies the inertia forces become dominant. The elasticity of the fluid in creases the wave velocity, affects very little the transmission of the wave, may cause the average velocity to “lead” the pressure gradient, increases the amplitude of the velocity of the wall, decreases the phase lag between the velocity of the wall and the average velocity of the fluid and increases the effect of the annular distribution of the amplitude of the velocity in the cross section.

Stabilization of drift waves by a non-uniform high-frequency field

The paper considers the interaction of a high-frequency electromagnetic field with a magnetized inhomogene-ous plasma. From kinetic and Maxwellian equations a system of equations is obtained which describes the linear oscillations of an inhomogeneous plasma. The dispersion relation is given for the potential oscillations. The results are obtained for the limiting case of long waves k⊥ρΛ→0. Three types of drift instability are considered: (a) electronic drift instability; (b) slow ion-acoustic waves, and(c) drift-temperature instability.It is shown that, by applying an h.f. field having a certain frequency and intensity it is possible to enlarge the region of stability against drift oscillations in an inhomogeneous magnetized plasma.When an h.f. field is applied, the stability region for a fast ion-acoustic wave grows considerably in the direction of negative values of ηe. Stabilization occurs as a result of the increased oscillation frequency which accompanies the application of the h.f. field, and this in turn intensifies Landau damping by the electrons.The stabilizing effect of the h.f. field pressure is associated with a Doppler shift resulting from plasma particle drift with velocity g/ωHα (g is the effective ‘acceleration due to gravity’ with allowance for the potential forces in a non-perturbed state). The sign of the growth rate of the slow ion-acoustic wave does not change when the h. f. field is applied, but its value decreases considerably. The h.f. magnetic field stabilizes the drift-temperature instability for negative values of ηi.

Localized Vibrational Modes at Plane Faces of Defects in Crystal Lattices

AbstractThe localized modes of vibrations are dicussed for the planes (001), (011), and (111) of impurity atoms in an infinitely extended simple cubic lattice. The frequencies show a band structure according to the translational invariance of the lattice in the impurity plane. It is found that optical and accoustical bands exist, the former lying completely, the latter only partially in the band of the unperturbed lattice. A free surface can be regarded as a special case of these models. Choosing appropriate coupling parameters one obtains localized surface states which have to coincide with the well‐known Rayleigh waves in the limiting case of long waves.

The instability to long waves of unbounded parallel inviscid flow

Formulas for the determination of the instability characteristics of unbounded parallel flow are obtained for the case of long waves, and applied, together with some general results, to give a qualitative description of the different modes of instability of such flows. It is found that there is a finite number of different modes unstable to long waves, essentially one for each relative maximum and minimum of the velocity profile. These modes appear to become stable when the wavelength is sufficiently small, reducing to neutral solutions associated with inflexion points as stability is approached. The formulas are also useful for quantitative calculation of instability characteristics.

Application of the phase integral method to the analysis of the diffraction and refraction of wireless waves round the earth

The increasing use of short waves of less than 10 m. has given a new stimulus to the problem of calculating the signal strength to be expected at a distance from a transmitter, and especially of determining the gain of signal strength with height above the ground. For short waves the predominant factor quite near to the transmitter is the diffraction of energy round the curve of the earth with heavy earth losses, so that the problem has to be approached by considering the complete solution for propagation over an imperfectly conducting curved earth. The solution of the problem was first put on an unimpeachable basis by Watson (1918), who expressed the expansion for the potential function as a contour integral leading to a more rapidly convergent series, and his work has been the starting-point for subsequent writers. His solution is formally complete, but the application of it to practical cases involves considerable mathematical difficulties. He gave the solution explicitly only for the case of long waves where the earth was so highly conducting that, for mathematical purposes, it approached the limit given by a perfect conductor. Also, without further reduction, the solution only expressed the potential function, and hence the electromagnetic field, at points on the surface of the earth.