The far‐field behaviour of Green's function for a triangular lattice and radiation conditions

2021 ◽  
Author(s):  
David Kapanadze
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Triangular Lattice ◽  
Far Field ◽  
Radiation Conditions ◽  
Field Behaviour

scholarly journals Far-field Approximations to the Derivatives of the Green's Function for the Ffowcs Williams and Hawkings Equation

2021 ◽  
Author(s):  
Zhiteng Zhou ◽  
Zhengyu Zang ◽  
Hongping Wang ◽  
Shizhao Wang
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Source Term ◽  
Vortex Pair ◽  
High Order ◽  
Far Field ◽  
Two Dimensional ◽  
Multiple Integrals ◽  
Surface Correction ◽  
Derivatives Of

Abstract The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of the Green's function. The far-field approximations to the derivatives of the Green's function have been used without derivation and verification in the previous work. In this work, we provide the detailed derivations of the far-field approximations to the derivatives of the Green's function. The binomial expansions for the derivatives of the Green's function and the far-field condition are employed during the derivations to circumvent the difficulties in computing the high-order derivatives. The approximations to the derivatives of the Green's function are systemically verified by using the benchmark two dimensional convecting vortex and the co-rotating vortex pair. In addition, we provide the derivations of the approximations to the multiple integrals of the Green's function by using the far-field approximations to the derivatives.

Far-Field Solution of Circular Lining and Linear Crack by SH-Wave

2009 ◽  
Vol 79-82 ◽  
pp. 1447-1450
Author(s):  
Hong Liang Li ◽  
Rui Zhang
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Crack Detection ◽  
Scattered Wave ◽  
Structure Design ◽  
Far Field ◽  
Elastic Space ◽  
Sh Wave ◽  
Field Solution ◽  
Out Of Plane

Circular lining is used widely in structure design. In this paper, the method of Green’s function is used to investigate the problem of far field solution of circular lining and linear crack impacted by incident SH-wave. Firstly, a Green’s function is constructed, which is a fundamental solution of displacement field for an elastic space possessing a circular lining while bearing out-of-plane harmonic line source force at any point; Secondly, in terms of the solution of SH-wave’s scattering by an elastic space with circular lining, anti-plane stresses which are the same in quantity but opposite in direction to those mentioned before, are loaded at the region where the linear crack is in existent actually; Finally, the expressions of displacement and stress are given when the circular lining and linear crack exist at the same time. Then, the far field of scattered wave is studied. The results can be applied in the study of fracture, and undamaged frame crack detection.

The Green' function forwaves in a homogeneous anisotropic absorbing plasma

1976 ◽  
Vol 15 (1) ◽  
pp. 133-150 ◽  
Author(s):  
J. A. Bennett
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Fourier Integral ◽  
Two Dimensions ◽  
Saddle Point Method ◽  
Generalized Function ◽  
Far Field ◽  
Dispersion Surface ◽  
The Matrix ◽  
The Green Function

The Green's function (or matrix) for a source of sinusoidal time dependence in an infinite homogeneous absorbing magneto-ionic plasma is written as a Fourier integral over wavenumber space. It is shown that this Fourier integral solution exists, and is unique as a generalized function. By extending the Fourier integral to complex wavenumbers, it is shown that the far-field expression for the Green's function may be written as an integral over sections of the dispersion surface, which in this case is a complex sub-manifold of the space of three complex variables. Use of the saddle-point method in two dimensions allows a further simplification of the far-field result. The matrix coefficients in the resulting expression are shown to represent a decomposition into modes. Corresponding results are also obtained for sources with spatial dependence, described by either functions of compact support or rapidly decreasing functions.

Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics

1994 ◽  
Vol 266 ◽  
pp. 33-68 ◽  
Author(s):  
H. Ardavan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Radial Distance ◽  
Stationary Points ◽  
Source Distribution ◽  
Far Field ◽  
Acoustic Analogy ◽  
Time Intervals ◽  
Conservation Of Energy ◽  
Retarded Time

The application of Lighthill's acoustic analogy to the generation of sound by rotating surfaces with supersonic speeds results in radiation integrals in which the stationary points of the phase function – that describes the sapce-time distance between each source point and a fixed observation point – have variable positions and coalesce at a caustic in the space of source points. Here, the leading term in the asymptotic expansion of the corresponding Green's function at this caustic is calculated, both in the time and the frequency domains, and it is shown that the radiation generated by volume sources, which are steady in the uniformly rotating blade-fixed frame, has an amplitude that does not obey the spherical spreading law, i.e. does not fall off like RP–1 with the radial distance RP away from the source. Within a finite solid angle, depending on the extent of the source distribution, the amplitude of this newly identified sound decays like RP–½, so that it is stronger in the far field than any previously studied element. That this is not incompatible with the conservation of energy is because the emission time intervals associated with the volume elements of the source distribution which contribute towards the non-spherically decaying component of the radiation are by a large (RP-dependent) factor greater than the time intervals during which the signals generated by these elements are received. The contributing source elements are those whose positions at the retarded time coincide witht the locus of singularities of the Green's function, i.e. with the one-dimensional locus of points, fixed in the rotating frame, which approach the observer with the wave speed and zero acceleration along the radiation direction. Because the signals received at two neighbouring instants in time arise from distinct, coherently radiating filamentary parts of the source which have both different extents and different strengths, the resulting overall waveform in the far zone consists of the superposition of a (continuous) set of narrow subpulses with uneven amplitudes. These subpulses are narrower the larger the distance from the source.The differences between the new results and those of the earlier works in the literature are shown to arise from the error terms in the far-field and high-frequency approximations, approximations that are inappropriate for volume sources moving supersonically.

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