scholarly journals A second-order asymptotic model for Rayleigh waves on a linearly elastic half plane

2020 ◽  
Vol 85 (1) ◽  
pp. 113-131 ◽  
Author(s):  
Peter T Wootton ◽  
Julius Kaplunov ◽  
Danila Prikazchikov
Keyword(s):  
Boundary Condition ◽  
Rayleigh Waves ◽  
Correction Term ◽  
Point Load ◽  
Second Order ◽  
Asymptotic Model ◽  
Surface Boundary ◽  
Leading Order ◽  
Order Model ◽  
Surface Boundary Condition

Abstract We derive a second-order correction to an existing leading-order model for surface waves in linear elasticity. The same hyperbolic–elliptic equation form is obtained with a correction term added to the surface boundary condition. The validity of the correction term is shown by re-examining problems which the leading-order model has been applied to previously, namely a harmonic forcing, a moving point load and a periodic array of compressional resonators.

scholarly journals An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. T1-T9 ◽  
Author(s):  
Chong Zeng ◽  
Jianghai Xia ◽  
Richard D. Miller ◽  
Georgios P. Tsoflias
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Rayleigh Waves ◽  
Surface Boundary ◽  
Near Surface ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Grid Nodes

Rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography. Modeling of Rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics. For simulation of Rayleigh waves, the traction-free boundary condition needs to be satisfied on the free surface. A vacuum formulation naturally incorporates surface topography in finite-difference (FD) modeling by treating the surface grid nodes as the internal grid nodes. However, the conventional vacuum formulation does not completely fulfill the free-surface boundary condition and becomes unstable for modeling using high-order FD operators. We developed a stable vacuum formulation that fully satisfies the free-surface boundary condition by choosing an appropriate combination of the staggered-grid form and a parameter-averaging scheme. The elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes. The improved vacuum formulation can accurately and stably simulate Rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high Poisson’s ratios ([Formula: see text]). This method requires fewer grid points per wavelength than the stress-image-based methods. Internal discontinuities in a model can be handled without modification of the algorithm. Only minor changes are required to implement the improved vacuum formulation in existing 2D FD modeling codes.

Analyses of Waves and the Wave Resistance due to Transom-Stern Ships

1969 ◽  
Vol 13 (02) ◽  
pp. 155-167
Author(s):  
Bohyun Yim
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Boundary ◽  
Exact Formulation ◽  
Wave Problem ◽  
Ship Wave ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

Waves and the wave resistance due to a ship with a transom stern are analyzed starting with an exact formulation of the general ship wave problem. Green's theorem is utilized together with the well-known Green's function which satisfies the linear free-surface boundary condition on the free surface. Here it is required to take at least the second-order terms in the formal development. It is found that the immersed transom stern acts to cancel stern waves. Using a limited form of Michell's wave-resistance formula, the wave resistance of the transom-stern ship is analyzed

scholarly journals Relating the Diffusive Salt Flux just below the Ocean Surface to Boundary Freshwater and Salt Fluxes

2019 ◽  
Vol 49 (9) ◽  
pp. 2365-2376 ◽  
Author(s):  
A. J. George Nurser ◽  
Stephen M. Griffies
Keyword(s):  
Boundary Condition ◽  
Mass Conservation ◽  
Ocean Surface ◽  
Boussinesq Approximation ◽  
Specific Form ◽  
Salt Flux ◽  
Surface Boundary ◽  
Flux Balance ◽  
Surface Boundary Condition ◽  
Diffusive Fluxes

AbstractWe detail the physical means whereby boundary transfers of freshwater and salt induce diffusive fluxes of salinity. Our considerations focus on the kinematic balance between the diffusive fluxes of salt and freshwater, with this balance imposed by mass conservation for an element of seawater. The flux balance leads to a specific balanced form for the diffusive salt flux immediately below the ocean surface and, in the Boussinesq approximation, to a specific form for the salinity flux. This balanced form should be used in specifying the surface boundary condition for the salinity equation and the contribution of freshwater to the buoyancy budget.

On the Suspended Sediment Concentration Profile

2012 ◽  
Vol 212-213 ◽  
pp. 20-24 ◽  
Author(s):  
Chen Cheng ◽  
Zhi Yao Song ◽  
Yi Gang Wang ◽  
Jin Shan Zhang
Keyword(s):  
Boundary Condition ◽  
Suspended Sediment ◽  
Suspended Sediment Concentration ◽  
Order Approximation ◽  
Sediment Concentration ◽  
Suspended Sediment Transport ◽  
Surface Boundary ◽  
Sediment Diffusion Coefficient ◽  
Surface Boundary Condition ◽  
First Order Approximation

After analyzing the surface-boundary condition of suspended sediment concentration (SSC), Cheng et al.[7] further improved the sediment diffusion coefficient which was proposed by Bose and Dey[6]. Then an improved Rouse law (IRL) was developed. This equation, which has a similar form as Rouse law, not only overcomes the zero concentration at the free surface, but also behaves generally better than Rouse law and van Rijn equation over the whole water depth in the verification analysis. In this paper, the surface-boundary condition of SSC is further analyzed. It is elucidated that IRL satisfies the surface-boundary condition more reasonably than Rouse law. In addition, a first-order approximation of IRL is developed. From this approximation, we can easily get the explicit expression of the depth-averaged SSC without any implicit integrals to be solved numerically or by the help of a chart. This is very useful in the further study of non-equilibrium suspended sediment transport (SST).

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