Green Function of Steady Motion in Finite Water Depth

2006 ◽  
Vol 50 (02) ◽  
pp. 120-137
Author(s):  
Qinzheng Yang ◽  
Odd M. Faltinsen ◽  
Rong Zhao
Keyword(s):  
Green Function ◽  
Shallow Water ◽  
Wave Resistance ◽  
Partial Sums ◽  
Vertical Force ◽  
Slender Body Theory ◽  
Local Disturbance ◽  
Wave Part ◽  
Pitch Moment ◽  
The Green Function

The Green function associated with a steady translating source on a straight horizontal course in water with finite constant depth and infinite horizontal extent satisfying the classical free surface condition is studied by decomposing it into three parts: an array of Rankine singularities A, local disturbance D, and downstream wave part S. Each of the three parts is studied by several methods. This is used to verify the numerical scheme and find the most time-efficient procedure. The method of repeated averaging of partial sums for oscillating series is efficiently used to evaluate the infinite sum of Rankine singularities and the downstream wave part. The local disturbance needed in vertical force and pitch moment calculations is most demanding. The Green function is used in combination with thin ship theory to calculate wave resistance, vertical force, pitch moment, and far-field wash for a Wigley hull. The results are compared with Tuck's (1966) slender body theory for shallow water and experimental and theoretical results of wave resistance by Everest and Hogben (1970). The agreement is satisfactory. A shallow water wave resistance ratio r expressing the ratio between wave resistance in finite depth and infinite depth is introduced as an indirect way to minimize wash. It is demonstrated that a large influence of critical depth Froude number requires the ratio between fluid depth and ship length to be small.

Fourier integral representation of the Green function for an anisotropic elastic half-space

1993 ◽  
Vol 443 (1918) ◽  
pp. 367-389 ◽  
Keyword(s):  
Green Function ◽  
Integral Representation ◽  
Half Space ◽  
Isotropic Material ◽  
Fourier Integral ◽  
Elastic Half Space ◽  
Material Symmetry ◽  
Vertical Force ◽  
Anisotropic Elastic ◽  
The Green Function

This paper describes the development of a Fourier integral representation of the Green function for an anisotropic elastic half-space. The representation for an isotropic material is integrated in closed form and shown to reduce to Mindlin’s solution. An application of the anisotropic representation is made to deduce the exact displacement caused by a two-dimensional periodic vertical force distribu­tion applied to the interior of a half-space with cubic material symmetry.

Shallow-Water Effect on the Motions of Three-Dimensional Bodies in Waves

1987 ◽  
Vol 31 (01) ◽  
pp. 34-40
Author(s):  
Hideichi Endo
Keyword(s):  
Green Function ◽  
Shallow Water ◽  
Three Dimensional ◽  
Source Distribution ◽  
Linear Wave ◽  
Water Effect ◽  
Surface Source ◽  
Distribution Method ◽  
The Green Function ◽  
Shallow Water Effect

The motions of three-dimensional bodies of arbitrary shape freely floating in waves in shallow water are studied. The wave loads on and hydrodynamic forces of a rigid body are calculated by applying the surface source distribution method (Green's function method) in the framework of linear wave potential theory. Special attention is paid to the numerical evaluation of the Green function for finite water depth; namely, an improper integral containing a singularity in the Green function is obtained by Gauss-Laguerre quadrature, and the ∫1 lr* ds term obtained is by numerical quadrature. Computational results of wave exciting forces, hydrodynamic coefficients, and motions of freely floating structures in shallow and deep water are compared with those obtained in the literature. Furthermore, the shallow-water effect on the motions of a large structure is examined.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

The Green function method for the two-surface problem

1970 ◽  
Vol 8 (13) ◽  
pp. 1069-1071 ◽  
Author(s):  
F. Flores ◽  
F. Garcia-Moliner ◽  
J. Rubio
Keyword(s):  
Green Function ◽  
Function Method ◽  
Green Function Method ◽  
The Green Function

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

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