scholarly journals Modal Analysis of Internal Wave Propagation and Scattering over Large-Amplitude Topography

2020 ◽  
Vol 50 (2) ◽  
pp. 305-321 ◽  
Author(s):  
Noé Lahaye ◽  
Stefan G. Llewellyn Smith
Keyword(s):  
Wave Propagation ◽  
Internal Wave ◽  
Large Amplitude ◽  
Analytical Form ◽  
Orthogonality Condition ◽  
Coupled Mode ◽  
Simple Set ◽  
Standard Set ◽  
Coupling Terms ◽  
Theoretical Insight

AbstractCoupled-mode equations describing the propagation and scattering of internal waves over large-amplitude arbitrary topography in a two-dimensional stratified fluid are derived. They consist of a simple set of ordinary differential equations describing the evolution of modal amplitudes, based on an orthogonality condition that allows one to distinguish leftward- and rightward-propagating modes. The coupling terms expressing exchange of energy between modes are given in an analytical form using perturbation theory. This allows the derivation of a weak-topography approximate solution, generalizing previous linear solutions for a barotropic forcing that were described in 2002 by Llewellyn Smith and Young . In addition, the orthogonality condition derived is valid for a different set of eigenmodes defined on a sloping bottom, which shows a better convergence rate when compared with the standard set of modes. The work presented here provides a useful and simple framework for the investigation of internal wave propagation in an inhomogeneous ocean, along with theoretical insight.

A multi-layer model for nonlinear internal wave propagation in shallow water

2012 ◽  
Vol 695 ◽  
pp. 341-365 ◽  
Author(s):  
Philip L.-F. Liu ◽  
Xiaoming Wang
Keyword(s):  
Wave Propagation ◽  
Internal Wave ◽  
Shallow Water ◽  
Internal Waves ◽  
Bottom Topography ◽  
Laboratory Data ◽  
Constant Density ◽  
Layer Model ◽  
Fluid System ◽  
Nonlinear Internal Wave

AbstractIn this paper, a multi-layer model is developed for the purpose of studying nonlinear internal wave propagation in shallow water. The methodology employed in constructing the multi-layer model is similar to that used in deriving Boussinesq-type equations for surface gravity waves. It can also be viewed as an extension of the two-layer model developed by Choi & Camassa. The multi-layer model approximates the continuous density stratification by an $N$-layer fluid system in which a constant density is assumed in each layer. This allows the model to investigate higher-mode internal waves. Furthermore, the model is capable of simulating large-amplitude internal waves up to the breaking point. However, the model is limited by the assumption that the total water depth is shallow in comparison with the wavelength of interest. Furthermore, the vertical vorticity must vanish, while the horizontal vorticity components are weak. Numerical examples for strongly nonlinear waves are compared with laboratory data and other numerical studies in a two-layer fluid system. Good agreement is observed. The generation and propagation of mode-1 and mode-2 internal waves and their interactions with bottom topography are also investigated.

Internal wave propagation in an inhomogeneous fluid of non-uniform depth

1969 ◽  
Vol 38 (2) ◽  
pp. 365-374 ◽  
Author(s):  
Joseph B. Keller ◽  
Van C. Mow
Keyword(s):  
Wave Propagation ◽  
Internal Wave ◽  
Experimental Result ◽  
Bottom Surface ◽  
Radius Of Curvature ◽  
Constant Density ◽  
Trapped Waves ◽  
Inhomogeneous Fluid ◽  
Sloping Beach ◽  
Scale Length

An asymptotic solution is obtained to the problem of internal wave propagation in a horizontally stratified inhomogeneous fluid of non-uniform depth. It also applies to fluids which are not stratified, but in which the constant density surfaces have small slopes. The solution is valid when the wavelength is small compared to all horizontal scale lengths, such as the radius of curvature of a wavefront, the scale length of the bottom surface variations and the scale length of the horizontal density variations. The theory underlying the solution involves rays, a phase function satisfying the eiconal equation, and amplitude functions satisfying transport equations. All these equations are solved in terms of the rays and of the solution of the internal wave problem for a horizontally stratified fluid of constant depth. The theory is thus very similar to geometrical optics and its extensions. It can be used to treat problems of propagation, reflexion from vertical cliffs or from shorelines, refraction, determination of the frequencies and wave patterns of trapped waves, etc. For fluid of constant density, it reduces to the theory of Keller (1958). The theory is applied to waves in a fluid with an exponential density distribution on a uniformly sloping beach. The predicted wavelength is shown to agree well with the experimental result of Wunsch (1969). It is also applied to determine edge waves near a shoreline and trapped waves in a channel.

scholarly journals Non-reciprocal wave propagation in modulated elastic metamaterials

2017 ◽  
Vol 473 (2202) ◽  
pp. 20170188 ◽  
Author(s):  
H. Nassar ◽  
H. Chen ◽  
A. N. Norris ◽  
M. R. Haberman ◽  
G. L. Huang
Keyword(s):  
Wave Propagation ◽  
Elastic Wave ◽  
Effective Mass ◽  
Mass Operator ◽  
High Amplitude ◽  
Coupled Mode Theory ◽  
Coupled Mode ◽  
Transient Regimes ◽  
Elastic Metamaterials ◽  
Large Frequency

Time-reversal symmetry for elastic wave propagation breaks down in a resonant mass-in-mass lattice whose inner-stiffness is weakly modulated in space and in time in a wave-like fashion. Specifically, one-way wave transmission, conversion and amplification as well as unidirectional wave blocking are demonstrated analytically through an asymptotic analysis based on coupled mode theory and numerically thanks to a series of simulations in harmonic and transient regimes. High-amplitude modulations are then explored in the homogenization limit where a non-standard effective mass operator is recovered and shown to take negative values over unusually large frequency bands. These modulated metamaterials, which exhibit either non-reciprocal behaviours or non-standard effective mass operators, offer promise for applications in the field of elastic wave control in general and in one-way conversion/amplification in particular.

Internal waves in a horizontally inhomogeneous flow

1984 ◽  
Vol 142 ◽  
pp. 233-249 ◽  
Author(s):  
A. Ya. Basovich ◽  
L. Sh. Tsimring
Keyword(s):  
Wave Propagation ◽  
Internal Wave ◽  
Internal Waves ◽  
Short Wave ◽  
Wave Transformation ◽  
Wave Trapping ◽  
Wave Amplification ◽  
Normal Surfaces ◽  
Inhomogeneous Flow ◽  
Horizontally Inhomogeneous

The effect of horizontally inhomogeneous flows on internal wave propagation in a stratified ocean with a constant Brunt-Väisälä frequency is analysed. Dispersion characteristics of internal waves in a moving fluid and kinematics of wave packets in smoothly inhomogeneous flows are considered using wave-normal surfaces. It is shown that internal-wave blocking and short-wave transformation may occur in longitudinally inhomogeneous flows. For parallel flows internal-wave trapping is possible in the vicinity of the limiting layer where the wave frequency in the locally comoving frame of reference coincides with the Brunt-Väisälä frequency. Internal-wave trapping also takes place in jet-type flows in the vicinity of the flow-velocity maximum. WKB solutions of the equation describing internal-wave propagation in a parallel horizontally inhomogeneous flow in the linear approximation are obtained. Singular points of this equation and the related effect of internal-wave amplification (overreflection) under the action of the flow are investigated. The spectrum and the growth rate of internal-wave localized modes in a jet-type flow are obtained.

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