scholarly journals The All-Source Green's Function of Linear Shallow Water Dynamic System: Its Numerical Constructions and Applications to Tsunami Problems

10.5772/14062 ◽  
2010 ◽  
Author(s):  
Zhigang Xu
Keyword(s):  
Dynamic System ◽  
Green’S Function ◽  
Shallow Water ◽  
Green's Function

scholarly journals A Green’s Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method

2021 ◽  
Vol 11 (6) ◽  
pp. 2722
Author(s):  
Zhiwen Qian ◽  
Dejiang Shang ◽  
Yuan Hu ◽  
Xinyang Xu ◽  
Haihan Zhao ◽  
...  
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Plane Waves ◽  
Spherical Wave ◽  
Sound Field ◽  
Efficient Computation ◽  
Image Source ◽  
Source Method ◽  
Pekeris Waveguide

The Green’s function (GF) directly eases the efficient computation for acoustic radiation problems in shallow water with the use of the Helmholtz integral equation. The difficulty in solving the GF in shallow water lies in the need to consider the boundary effects. In this paper, a rigorous theoretical model of interactions between the spherical wave and the liquid boundary is established by Fourier transform. The accurate and adaptive GF for the acoustic problems in the Pekeris waveguide with lossy seabed is derived, which is based on the image source method (ISM) and wave acoustics. First, the spherical wave is decomposed into plane waves in different incident angles. Second, each plane wave is multiplied by the corresponding reflection coefficient to obtain the reflected sound field, and the field is superposed to obtain the reflected sound field of the spherical wave. Then, the sound field of all image sources and the physical source are summed to obtain the GF in the Pekeris waveguide. The results computed by this method are compared with the standard wavenumber integration method, which verifies the accuracy of the GF for the near- and far-field acoustic problems. The influence of seabed attenuation on modal interference patterns is analyzed.

The effects of measurement errors in determining the Green's function for shallow water propagation

1985 ◽  
Vol 78 (S1) ◽  
pp. S71-S71
Author(s):  
James F. Lynch ◽  
George V. Frisk ◽  
James A. Doutt ◽  
Christopher E. Dunn
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Measurement Errors

Nonlinear landslide tsunami run-up

2011 ◽  
Vol 691 ◽  
pp. 440-460 ◽  
Author(s):  
M. Sinan Özeren ◽  
Nazmi Postacioglu
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Shallow Water Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Submarine Landslides ◽  
Partial Derivatives ◽  
Nonlinear Shallow Water Equations ◽  
Run Up

AbstractInhomogeneous nonlinear shallow-water equations are studied using the Carrier–Greenspan approach and the resulting equations are solved analytically. The Carrier–Greenspan transformations are commonly used hodograph transformations that transform the nonlinear shallow-water equations into a set of linear equations in which partial derivatives with respect to two auxiliary variables appear. Yet, when the resulting initial-value problem is treated analytically through the use of Green’s functions, the partial derivatives of the Green’s functions have non-integrable singularities. This has forced researchers to numerically differentiate the convolutions of the Green’s functions. In this work we remedy this problem by differentiating the initial condition rather than the Green’s function itself; we also perform a change of variables that renders the entire problem more easily treatable. This particular Green’s function approach is especially useful to treat sources that are extended in time; we therefore apply it to model the run-down and run-up of the tsunami waves triggered by submarine landslides. Another advantage of the method presented is that the parametrization of the landslide using sources is done within the integral algorithm that is used for the rest of the problem instead of treating the landslide-generated wave as a separate incident wave. The method proves to be more accurate than the techniques based on Bessel function expansions if the sources are very localized.

QUICK EVALUATION OF RUNUP HEIGHT AND INUNDATION AREA FOR EARLY WARNING OF TSUNAMI

2012 ◽  
Vol 06 (01) ◽  
pp. 1250005 ◽  
Author(s):  
GUAN-YU CHEN ◽  
CHIA-HAO LIN ◽  
CHIN-CHU LIU
Keyword(s):  
Flow Field ◽  
Green’S Function ◽  
Shallow Water ◽  
Wave Height ◽  
Green's Function ◽  
Early Warning ◽  
Shallow Water Equations ◽  
Bottom Slope ◽  
In Situ Investigation

A methodology combining the offshore tsunami calculated by using numerical reciprocal Green's function (RGF) and the runup flow field calculated by using an analytical Green's function (AGF) is proposed to quickly estimate a tsunami hazard. For a vulnerable city, the RGF is computed previously via linear shallow water equations over real bathymetry and the offshore tsunami can be obtained promptly once the initial rupture is known. A transformation from time to space is then applied to obtain an equivalent waveform. With an integral with the AGF, derived by Carrier et al. [2003] based on 1D fully nonlinear shallow water equations over a uniform constant slope, the runup flow field is calculated. Thus, besides saving computation time and reducing the memory requirement, the desired initial condition for the AGF can also be generate by RGF. In this approach, the max wave height and the inundation distance are estimated very quickly and can be applied to broadcast an early warning of tsunami. To verify the method, data obtained during the 2004 Indian Ocean Tsunami from Sri Lanka and Phuket, Thailand is applied. The offshore condition is first verified by comparing with the record at Maldives. The accuracy of RGF is also tested. Then, by taking the nearshore shelf slope as the constant bottom slope for the analytical solution, the max tsunami height agrees reasonably well with the in-situ measurement. Therefore, this method is a useful tool for tsunami early warning by quickly estimating if the max wave height is higher than the seawall or the breakwater. The max inundation distance calculated by the analytic integral solution also has reasonable agreement with the field survey, but the value from the in-situ investigation scatters widely, which suggests that the detailed local topography plays an important role. A different method for the determination of the bottom slope is also tested and the result shows that the slope should be based on the bathymetry nearshore.

Bayesian source localization with uncertain Green's function in an uncertain shallow water ocean

2016 ◽  
Vol 139 (3) ◽  
pp. 993-1004 ◽  
Author(s):  
Yann Le Gall ◽  
Stan E. Dosso ◽  
François-Xavier Socheleau ◽  
Julien Bonnel
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Source Localization
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