Interfacial Wave Modes Research Articles

Investigation of water wave scattering by an elastic sea-bed of varying depth in two superposed fluids covered by an ice-sheet

A hydroelastic model is taken up in this work to investigate the propagation of normal incident waves over a small undulation located on an elastic sea-bed in a fluid flow for two superposed fluids where a floating ice-sheet covers the upper fluid. Here, both the floating ice-sheet and elastic sea-bed are approximated as thin elastic plates which follow the Euler–Bernoulli beam equation, familiar for such studies. Due to its minimal contribution, the surface tension at the interface of the fluid layers is not accounted for. The behavior of the roots of the associated complex dispersion relation is investigated through the application of a counting argument along with a contour plot. The time-harmonic waves, for a given frequency, are observed to propagate with exactly two wavenumbers such that the one corresponding to the smaller wavenumber propagates along the ice-sheet while the other corresponding to the higher wavenumber along the interface. By assuming the fluid to be incompressible and inviscid following an irrotational motion, and then applying a perturbation technique, the first-order contribution to the velocity potentials in each layer is evaluated with the help of Fourier transform. Following that, the associated reflection and transmission coefficients are computed by carrying out appropriate integration for integrals containing a shape function depicting the sea-bed undulation. In order to validate the present model developed in this work, one obtained result for reflection coefficient is validated against an available result which points towards a good agreement. This validation establishes the model and the method followed in this work to be effective and allows subsequent investigation for this problem to be taken up. For verification of the theory developed here, one specific undulating bed topography is selected to assess the reflection as well as transmission coefficients which are depicted by graphical representations to examine and establish the energy conversion between the flexural wave mode and the interfacial wave mode. Present investigation demonstrates that the reflection and transmission of the waves exhibit an oscillatory behavior, the reason for which is the outcome of multiple interaction of the incident wave with the elastic bottom undulation, interface and the ice-sheet. It further shows that flexural wave moment is significantly affected as compared to the interfacial wave influence. It is noted that significant variation in the elastic specification of the ice-sheet and the elastic sea-bed has a considerable influence for wave scattering by the undulating elastic sea-bed. It is observed that, when water waves propagate over a deformable bottom, the wave characteristics undergo a significant change. This occurs due to the elasticity of the sea-bed which immensely influences the kinematics and dynamics of the wave over and above providing the demonstration of the elastic behavior of the soil located below the elastic sea-bed. The approach adopted here is likely to be of immense significance in the design and construction of various types of floating breakwaters and very large floating structures (VLFS) for the purpose of various scientific activities mainly in polar regions. Further, the solutions of problems like these dealing with a deformable sea-bed will be beneficial in providing the required background to the researchers with respect to the incorporation of the characteristics of the infinite depth soil located below the sea-bed.

Scattering of linear oblique water waves by an elastic bottom undulation in a two-layer fluid

A hydrodynamic model, with incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid, with the upper layer exposed to a free surface. Following the Euler–Bernoulli beam equation, the elastic bed is approximated as a thin elastic plate. The surface tension at the interface of the layers is completely ignored since its contribution will be minimal. While considering water waves passing over a deformable bottom, a significant change in the wave characteristics is observed due to the elasticity of the bottom which has an immense impact on the water wave kinematics and dynamics in addition to demonstrating the elastic behavior of the soil beneath. Time-harmonic waves propagate over an elastic bed with two different modes: the one corresponding to the smaller wavenumber propagates along the interface and the other one corresponding to the higher wavenumber along the free surface for any given frequency. Considering an irrotational motion in an incompressible and inviscid fluid, and applying perturbation technique, the first-order corrections to the velocity potentials are evaluated by an appropriate application of Fourier transform and, subsequently, the corresponding reflection and transmission coefficients are computed through integrals containing a shape function which depicts the bottom undulation. To validate the theory developed, one particular undulating bottom topography is taken up as an example in order to evaluate the hydrodynamic coefficients which are represented through graphs to establish the water wave energy conversion between those modes. The observation is that when the oblique wave is incident on the interface, energy transfer takes place to the free surface, but for free-surface oblique incident waves, no such energy transfer to the interface takes place because of the parameter ranges. It is noticed that reasonable changes in the elasticity of the bed have a significant impact when the propagating wave encounters a small elastic bottom undulation. Further, the values of reflection and transmission coefficients obtained for both the interfacial wave mode as well as the free-surface wave mode in the fluid are found to satisfy the important energy balance relations almost accurately. Such problems with a deformable bed, to be precise elastic here, will enable researchers to take up problems which take into account the characteristics of the infinite depth of soil beneath the bed, and the present study is expected to provide the necessary background.

SH wave propagation in joined half-spaces composed of elastic metamaterials

Based on the effective-medium theory, the propagation of a shear horizontal (SH) wave in joined half-spaces composed of elastic metamaterials (EMMs) is investigated. From the dispersion relations, the effects of negative effective-medium parameters on the properties of a SH wave traveling near the interface are analyzed in detail. It is found that a SH wave can always appear and travel along the interface under specific effective-parameter combinations no matter whether the effective transverse wave velocity is imaginary or real. This is significantly different from the classical case (joined half-spaces composed of natural media), and the existence of these SH interfacial wave modes may have important impacts on EMM-based SH wave manipulation, especially wave isolation and object protection.

Large-eddy simulations of dense-gas dispersion within a high-Reynolds number turbulent boundary layer

Large-eddy simulations of flows over a backward-facing step with release of four different gases with increasing densities have been performed. The results have been analysed with particular emphasis on the part of the flow field sufficiently far downstream where the local effect of the source dynamics is no longer dominating the flow field. The dense gas plumes maintain high concentrations close to the ground and become approximately twice as shallow in comparison with a neutrally buoyant plume. The vertical mixing is significantly reduced close to the ground, and both momentum and scalar flux reversal are found in the two densest cases, indicating negative shear and buoyancy production of turbulence kinetic energy. Examination of the gradient Richardson number indicates that all dense gases are significantly affected by buoyancy. Interestingly, a narrow layer within the plumes is identified where the impact of stratification is expected to be reduced. It is found that relaminarization is most likely occurring and that fluctuations found might originate from interfacial wave modes rather than from turbulence. The large-scale structures are seen to resemble those found in the plane channel flow close to the wall and further away from it. In the middle layer, significant differences are found due to buoyancy effects. The large-scale structures become more pancake-shaped with large-scale vorticity almost exclusively about the spanwise direction in the dense gas cases.

Hydroelastic response of a circular plate in waves on a two-layer fluid of finite depth

The hydroelastic response of a circular, very large floating structure (VLFS), idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the plate-covered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presented to investigate the effects of different physical quantities, such as the thickness of the plate, Young’s modulus, the ratios of the densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid. Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.

Nonlinear generation of harmonics through the interaction of an internal wave beam with a model oceanic pycnocline

The interaction of an internal wave beam (IWB) with an idealized oceanic pycnocline is examined using two-dimensional fully nonlinear direct numerical simulations based on a spectral multidomain penalty method in the vertical direction. The phenomenon of focus is the nonlinear generation of harmonics. A total of 24 simulations have been performed, varying the normalized pycnocline thickness and the ratio of peak pycnocline Brunt-Väisälä frequency to that of the stratified lower layer. Harmonics at the point of IWB entry into the pycnocline increase in amplitude and number with a measure of the maximum gradient of the Brunt-Väisälä frequency, suggesting refraction as an important factor in harmonic generation. Among the simulations performed, two distinct limits of pycnocline thickness are identified. For thin pynoclines, whose thickness is 10% of the incident IWB's horizontal wavelength, harmonics trapped within the pycnocline have maximum amplitude when their frequency and wavenumber match those of the natural pycnocline interfacial wave mode. Results in this case are compared with weakly nonlinear theory for harmonic generation by plane wave refraction. For thicker pycnoclines, whose thickness is equal the incident IWB's horizontal wavelength, IWB refraction results in harmonic generation at multiple locations in addition to pycnocline entry, giving rise to complex flow structure inside the pycnocline.

Water wave diffraction by a bottom-mounted circular cylinder clamped to an elastic plate floating on a two-layer fluid

The wave diffraction by a bottom-mounted circular cylinder, which is clamped to the center of a floating circular thin elastic plate, in the two-layer fluid of finite depth is investigated for the time-harmonic incident waves of the surface and interfacial wave modes. Each fluid layer is inviscid, incompressible and of constant density. The flexural–gravity waves are composed of the propagating, decaying propagating and evanescent wave modes. Within the framework of the linear potential flow theory, a closed system of simultaneous linear equations is derived to solve the undetermined expansion coefficients with the methods of the angular eigenfunction expansion and the inner product. Explicit numerical computations are employed to test the convergence of the two series for the angular expansions and the evanescent wave modes. The horizontal forces and the moments exerted on the circular cylinder due to different wave modes are discussed in the case of the incident waves of either the surface or interfacial wave mode. It is obtained that the evanescent wave modes are appreciable parts for a high frequency.

Hydroelastic interaction between obliquely incident waves and a semi-infinite elastic plate on a two-layer fluid

The hydroelastic response of a semi-infinite thin elastic plate floating on a two-layer fluid of finite depth due to obliquely incident waves is investigated. The upper and lower fluids with different densities separated by a sharp and stable interface are assumed to be inviscid and incompressible and the motion to be irrotational. Simply time-harmonic incident waves of the surface and interfacial wave modes with a given angular frequency are considered within the framework of linear potential flow theory. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, a closed system of simultaneous linear equations is derived for the reflection and transmission coefficients of the series solutions. Based on the dispersion relations for the gravity waves and the flexural–gravity waves in a two-layer fluid and Snell’s law for refraction, we obtain a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves. Graphical representations of the series solutions show the interaction between the water waves and the plate. The effects of several physical parameters, including the density and depth ratios of the fluid and the thickness of the plate, on the wave scattering and the hydroelastic response of the plate are studied. It is found that the variation of the thickness of the plate may change the wave numbers and the critical angles. The density ratio is the main factor to influence the wave numbers of the interfacial wave modes. Finally, the stress state is considered.

Transient flexural- and capillary-gravity waves due to disturbances in two-layer density-stratified fluid

Generation of the transient flexural- and capillary-gravity waves by impulsive disturbances in a two-layer fluid is investigated analytically. The upper fluid is covered by a thin elastic plate or by an inertial surface with the capillary effect. The density of each of the two immiscible layers is constant. The fluids are assumed to be inviscid and incompressible and the motion be irrotational. A point force on the surface and simple mass sources in the upper and lower fluid layers are considered. A linear system is established within the framework of potential theory. The integral solutions for the surface and interfacial waves are obtained by means of the Laplace-Fourier transform. A new representation for the dispersion relation of flexural- and capillary-gravity waves in a two-layer fluid is derived. The asymptotic representations of the wave motions are derived for large time with a fixed distance-to-time ratio with the Stokes and Scorer methods of stationary phase. It is shown that there are two different modes, namely the surface and interfacial wave modes. The wave systems observed depend on the relation between the observer's moving speed and the intrinsic minimal and maximal group velocities.

Spatial attenuation rates of interfacial waves: Field and numerical tests of Sommerfeld theory using ground-penetrating radar pulses

We tested the geometric amplitude attenuation rates predicted by classic Sommerfeld theory for horizontally polarized interfacial waves propagating over dielectric ground. We used ground-penetrating radar pulses, the brief time duration of which allowed different interfacial wave modes to separate. We tested rates in the intermediate range of tens of wavelengths, and for azimuthal and radial polarizations. For azimuthal polarization, a closed form solution predicts inverse range-squared rates, and for radial polarization, calculations suggest an inverse range exponent between 1 and 2. Over low loss frozen ground having a dielectric constant of 6.8 azimuthally polarized air waves centered at 46 MHz attenuated nearly in proportion to the square of range, as predicted, while the radial rate at 37 MHz was close to the 1.6 power of range, as generally expected. At 360–390 MHz, air wave rates were higher than expected and likely caused by scattering losses. Three D time domain modeling at 37 MHz confirmed the rate for azimuthal polarization and the qualitative difference in rates between the two polarizations, but the exponent may be about 26% too high for the radial case. Not readily extractable from Sommerfeld theory are rates for subsurface direct waves, for which our models show that both polarizations attenuate in proportion to the square of range after about 5 subsurface wavelengths. This suggests that geometric rates for all horizontally polarized subsurface interfacial waves spatially attenuate in proportion to range-squared in both intermediate and far field ranges, and so could be subtracted from actual rates to determine loss rates caused by intrinsic attenuation and scattering.

Surface and Interfacial Gravity Waves Induced by an Impulsive Disturbance in a Two-Layer Inviscid Fluid

The transient gravity waves due to an impulsive source in a two-layer fluid system are investigated analytically. The fluid is assumed to be inviscid and incompressible. The density of each of the two layers is constant. Five different boundary conditions are considered. The depth of each of the two layers is infinite or finite. The upper fluid of finite depth is covered by a rigid lid or a free surface. Based on the assumption of small-amplitude waves, a linear system is established. The integral solutions for the free-surface and interfacial waves are obtained by means of the Fourier-Laplace transform. The corresponding asymptotic representations are derived for large time with a fixed distance-to-time ratio by the Stokes and Scorer methods of stationary phase. The analytical solutions show that there are two different modes, namely the free-surface and interfacial wave modes. The wave profiles observed depend on the relation between the distance-to-time ratio and the maximal group velocities and on the limiting values of the second derivatives of the frequencies as well.