Kirchhoff Formula Research Articles

The Complexity of Octopus Graph, Friendship Graph, and Snail Graph

Graphs are basic structures that represent objects with nodes and relationships between objects with edges. Trees are one of the parts studied in graph theory along with finding the number of spanning trees of a graph such as octopus graph, friendship graph, and snail graph. The complexity of an octopus graph is strongly dependent on the number and length of tentacles, the complexity of a friendship graph is dependent on the number of triangle cycles, and the complexity of a snail graph is dependent on the number of edges and vertices located in the shell-like part of the snail. To calculate the number of spanning trees (τ(G)) of a graph, various calculations can be used, such as the extension of Kirchhoff's formula. The extension of Kirchhoff's formula uses the determinant of the adjacency matrix and degree matrix of the graph complement of a graph. Therefore, this research applies the extension of Kirchhoff's formula to obtain the complexity of octopus graph, friendship graph, and snail graph. From the analysis, it is obtained that for any n≥2, the number of spanning trees of octopus graph and friendship graph are τ(On )=1/5 √5 [((3+√5)/2)^n-((3-√5)/2)^n ] and τ(Fn )=3^n and the number of spanning trees of snail graph is τ(SIn )=2^(n+2)+3n∙2^(n-1) for n≥1.

A simple formula for coherent specular reflection loss of underwater sound from rough ocean surfaces

The interaction of underwater sound with rough sea surfaces affects its propagation. The interaction may involve several coupled mechanisms: (1) a rough air-sea interface scatters and spreads the incident sound energy over a range of angles, (2) entrapped sub-surface air bubbles absorb and scatter sound energy, and (3) medium of bubble clouds refract sound energy upwards. In this paper, we ignore the effect of bubbles and consider the effect of rough air-sea interface only. We further limit our consideration to the coherent component in the specular direction. Two simple closed-form formulas are often used to model the coherent reflection loss from rough ocean surfaces. One is derived from perturbation approximation and suitable for smaller grazing angles. The other is from Kirchhoff approximation and suitable for larger grazing angles. More accurate approximations over a wide-range of grazing angles require numerical evaluation of integrals or complex special functions. In this work, we simplify and adapt earlier results by other authors to obtain a simple expression that transits from the perturbation to the Kirchhoff formulas as grazing angle increases. The new simple analytical formula is easy to use and compares well with results from complex numerical methods.

Biquaternion generalizations of Maxwell's and Dirac's equations and properties of their solutions

The differential algebra of biquaternions is used to construct generalized solutions of the biquaternion wave (biquaternion) equation, special cases of which are the systems of equations of Maxwell and Dirac. The method of generalized functions is used to construct solutions and study their properties. Fundamental and generalized solutions are constructed, an analogue of the Kirchhoff formula for the wave equation, which gives a solution to the Cauchy problem. A dynamic analog of the Gauss formula is obtained, which gives a solution to the biwave equation in a limited region under known boundary and initial conditions for the desired biquaternion. Using the constructed analogues of the Kirchhoff and Green's formula, one can obtain analogues of the Green's formula under nonzero initial conditions by decomposing the solution of Eq. (1) into two biquaternions, one of which satisfies the initial conditions. Conditions on the boundary for the second biquaternion will be obtained using the boundary values of the original biquaternion and the first constructed one. After that, formulas similar to (30) give an integral representation of the second biquaternion. Since the equations of Maxwell and Dirac are a special case of biwave equations, the constructed solutions can be used for problems of electrodynamics and field theory. They can be used in experiments, so the field characteristics of EM fields at the boundary can be measured experimentally without solving the SHIE. The constructed solutions can be used in field theory, elementary particles and electro-gravimagnetic interactions.

Ring shaped fs supercontinuum with a thermally induced self-diffraction effect.

A broadband femtosecond supercontinuum (SC) light with a dark core has been generated in a pump-probe type geometry. It has been achieved by introducing a small thermal perturbation in a highly absorbing medium with a continuous wave pump laser. The SC was generated with the help of a photonic crystal fiber and a nJ femtosecond oscillator. Numerical simulations for the system were done by employing the Fresnel-Kirchhoff formula. The power dependence of the dark core diameter was also studied experimentally, and the results agree well with numerical simulations. The dark core size can be controlled by varying the input power of the pump beam.

Trees and Forests for Nonequilibrium Purposes: An Introduction to Graphical Representations

Using local detailed balance we rewrite the Kirchhoff formula for stationary distribution of Markov jump processes in terms of a physically interpretable tree-ensemble. We use that arborification of path-space integration to derive a McLennan-tree characterization close to equilibrium, as well as to obtain response formula for the stationary distribution in the asymptotic regime of large driving. Graphical expressions of currents and of traffic are obtained, allowing the study of various asymptotic regimes. Finally, we present how the matrix-forest theorem gives a representation of quasi-potentials, as used e.g. for computing excess work and heat in nonequilibrium thermal physics. A variety of examples illustrate and explain the graph elements and constructions.

Testing scalar diffraction theory: Gaussian beam on a slit.

We recorded diffraction patterns using a commercially available slit and sensor over a wide range of experimental circumstances, including near- and far-field regimes and oblique incidence at large angles. We then compared the measured patterns with theoretical intensity curves calculated via the numerical integration of formulas derived within the framework of scalar diffraction theory. Experiment and theory show particularly good agreement when the first Rayleigh-Sommerfeld (R-S) formula is used. The Kirchhoff formula, though problematic in the context of mathematical consistency, agrees with the first R-S formula, even for large incidence angles, whereas the second R-S formula differs visibly. To obtain such a good agreement, we replaced the assumption of an incident plane wave with that of a Gaussian beam and implemented geometric corrections to account for slit imperfections. These results reveal how the scope of scalar diffraction theory can be extended with a small set of auxiliary assumptions.

Separating and imaging diffractions of seismic waves in the full-azimuth dip-angle domain

Abstract Seismic diffractions are ideal carriers of information on small-scale, discontinuous objects and can therefore be used to detect these geologic objects. However, recognizing diffractions is difficult because specular reflection with strong energy masks the weak diffraction. In this study, we propose a diffraction separation and imaging method based on a Mahalanobis-based and phase-based attenuation function used to modify the Kirchhoff migration formula in the full-azimuth dip-angle domain. In this domain, reflections are restricted to within the first Fresnel zone and are distributed in the vicinity of the stationary point, while diffractions are located across a wide range of azimuth and dip angles. Synthetic and field data applications suggest that this new method can effectively separate and image diffractions. The results also demonstrate the efficiency of the new method in clarifying subsurface small-scale objects, which can provide finer information about these structures for seismic interpretation.

Research on quantitative mechanism of Kirchhoff diffraction acoustical holography

In the method of far-field acoustic holographic noise identification, the Kirchhoff formula has been widely used. Achievement of reconstruction is the outstanding advantage of acoustical holography over the widely-used beam-forming method. In this paper, the mechanism of acoustical holography based on finite Kirchhoff diffraction is established, the relationship between holographic aperture angle and the quantitative accuracy is revealed. With simulations for known sound source, the quantitative mechanism is validated successfully.

Development of acoustic target strength near-field equation for underwater vehicles

For modern weapon systems, the most important factor in survivability is detection capability. Acoustic target strength is a major parameter of the active sonar equation. The traditional target strength equation used to predict the re-radiated intensity for the far field is derived with a plane-wave assumption. In this study, a near-field target strength equation was derived without a plane-wave assumption for a polygonal plate. The target strength equation for polygonal plates, which is applicable to the near field, is provided by the Helmholtz–Kirchhoff formula that is used as the primary equation for solving the re-radiated pressure field. A generalized definition of the sonar cross section is suggested that is applicable to the near field. In comparison experiments for a cylinder, the target strength equation for polygonal plates in near field was executed to verify the validity and accuracy of the analysis. In addition, an underwater vehicle model was analyzed with the developed near-field equation to confirm various parameter effects such as distance and frequency.

A DFT study on the enthalpies of thermite reactions and enthalpies of formation of metal composite oxide

The standard thermite reaction enthalpies (ΔrHmθ) for seven metal oxides were theoretically analyzed using density functional theory (DFT) under five different functional levels, and the results were compared with experimental values. Through the comparison of the linear fitting constants, mean error and root mean square error, the Perdew-Wang functional within the framework of local density approximation (LDA-PWC) and Perdew-Burke-Ernzerhof exchange-correlation functional within the framework of generalized gradient approximation (GGA-PBE) were selected to further calculate the thermite reaction enthalpies for metal composite oxides (MCOs). According to the Kirchhoff formula, the standard molar reaction enthalpies for these MCOs were obtained and their standard molar enthalpies of formation (ΔfHmθ) were finally calculated. The results indicated that GGA-PBE is the most suitable one out of the total five methods to calculate these oxides. Tungstate crystals present the maximum deviation of the enthalpies of thermite reactions for MCOs and these of their physical metal oxide mixtures, but ferrite crystals are the minimum. The correlation coefficients are all above 0.95, meaning linear fitting results are very precise. And the molar enthalpies of formation for NiMoO4, CuMoO4, PbZrO3 (Pm/3m), PbZrO3 (PBA2), PbZrO3 (PBam), MgZrO3, CdZrO3, MnZrO3, CuWO4 and Fe2WO6 were first obtained as −1078.75, −1058.45, −1343.87, −1266.54, −1342.29, −1333.03, −1210.43, −1388.05, −1131.07 and − 1860.11 kJ·mol−1, respectively.

Wavefield interpolation in 3D large‐step Fourier wavefield extrapolation

ABSTRACTExtrapolating wavefields and imaging at each depth during three‐dimensional recursive wave‐equation migration is a time‐consuming endeavor. For efficiency, most commercial techniques extrapolate wavefields through thick slabs followed by wavefield interpolation within each thick slab. In this article, we develop this strategy by associating more efficient interpolators with a Fourier‐transform‐related wavefield extrapolation method. First, we formulate a three‐dimensional first‐order separation‐of‐variables screen propagator for large‐step wavefield extrapolation, which allows for wide‐angle propagations in highly contrasting media. This propagator significantly improves the performance of the split‐step Fourier method in dealing with significant lateral heterogeneities at the cost of only one more fast Fourier transform in each thick slab. We then extend the two‐dimensional Kirchhoff and Born–Kirchhoff local wavefield interpolators to three‐dimensional cases for each slab. The three‐dimensional Kirchhoff interpolator is based on the traditional Kirchhoff formula and applies to moderate lateral velocity variations, whereas the three‐dimensional Born–Kirchhoff interpolator is derived from the Lippmann–Schwinger integral equation under the Born approximation and is adapted to highly laterally varying media. Numerical examples on the three‐dimensional salt model of the Society of Exploration Geophysicists/European Association of Geoscientists demonstrate that three‐dimensional first‐order separation‐of‐variables screen propagator Born–Kirchhoff depth migration using thick‐slab wavefield extrapolation plus thin‐slab interpolation tolerates a considerable depth‐step size of up to 72 ms, eventually resulting in an efficiency improvement of nearly 80% without obvious loss of imaging accuracy. Although the proposed three‐dimensional interpolators are presented with one‐way Fourier extrapolation methods, they can be extended for applications to general migration methods.

Forward Scatter Radar for Air Surveillance: Characterizing the Target-Receiver Transition from Far-Field to Near-Field Regions

A generalized electromagnetic model is presented in order to predict the response of forward scatter radar (FSR) systems for air-target surveillance applications in both far-field and near-field conditions. The relevant scattering problem is tackled by developing the Helmholtz–Kirchhoff formula and Babinet’s principle to express the scattered and the total fields in typical FSR configurations. To fix the distinctive features of this class of problems, our approach is applied here to metallic targets with canonical rectangular shapes illuminated by a plane wave, but the model can straightforwardly be used to account for more general scenarios. By exploiting suitable approximations, a simple analytical formulation is derived allowing us to efficiently describe the characteristics of the FSR response for a target transitioning with respect to the receiver from far-field to near-field regions. The effects of different target electrical sizes and detection distances on the received signal, as well as the impact of the trajectory of the moving object, are evaluated and discussed. All of the results are shown in terms of quantities normalized to the wavelength and can be generalized to different configurations once the carrier frequency of the FSR system is set. The range of validity of the proposed closed-form approach has been checked by means of numerical analyses, involving comparisons also with a customized implementation of a full-wave commercial CAD tool. The outcomes of this study can pave the way for significant extensions on the applicability of the FSR technique.

Diffraction imaging method by Mahalanobis-based amplitude damping

Clarifying and locating small-scale discontinuities or inhomogeneities in the subsurface, such as faults and collapsed columns, plays a vital role in safe coal mining because these discontinuities or inhomogeneities may destroy the continuity of layers and result in dangerous mining accidents. Diffractions carry key information from these objects and therefore can be used for high-resolution imaging. However, diffracted/scattered waves are much weaker than reflected waves and consequently require separation before being imaged. We have developed a Mahalanobis-based diffraction imaging method by modifying the classic Kirchhoff formula with an exponential function to account for the dynamic differences between reflections and diffractions in the shot domain. The imaging method can automatically account for destroying of reflected waves, constructive stacking of diffracted waves, and strengthening of scattered waves. The method can overcome the difficulties in handling Fresnel apertures, and it is suitable for high-resolution imaging because of the consistency of the waveforms in the shot domain. Although the proposed method in principle requires a good migration velocity model for calculating elementary diffraction traveltimes, it is robust to an inaccurate migration velocity model. Two numerical experiments demonstrate the feasibility of the proposed method in removing reflections and highlighting diffractions, and one field application further confirms its efficiency in resolving masked faults and collapsed columns.

A time-domain Kirchhoff formula for the convective acoustic wave equation.

Kirchhoff's integral method allows propagated sound to be predicted, based on the pressure and its derivatives in time and space obtained on a data surface located in the linear flow region. Kirchhoff's formula for noise prediction from high-speed rotors and propellers suffers from the limitation of the observer located in uniform flow, thus requiring an extension to arbitrarily moving media. This paper presents a Kirchhoff formulation for moving surfaces in a uniform moving medium of arbitrary configuration. First, the convective wave equation is derived in a moving frame, based on the generalized functions theory. The Kirchhoff formula is then obtained for moving surfaces in the time domain. The formula has a similar form to the Kirchhoff formulation for moving surfaces of Farassat and Myers, with the presence of additional terms owing to the moving medium effect. The equation explicitly accounts for the influence of mean flow and angle of attack on the radiated noise. The formula is verified by analytical cases of a monopole source located in a moving medium.

An integral formula adapted to different boundary conditions for arbitrarily high-dimensional nonlinear Klein-Gordon equations with its applications

In this paper, we are concerned with the initial boundary value problem of arbitrarily high-dimensional Klein-Gordon equations, posed on a bounded domain Ω ⊂ ℝd for d ≥ 1 and equipped with the requirement of boundary conditions. We derive and analyze an integral formula which is proved to be adapted to different boundary conditions for general Klein-Gordon equations in arbitrarily high-dimensional spaces. The formula gives a closed-form solution to arbitrarily high-dimensional homogeneous linear Klein-Gordon equations, which is totally different from the well-known D’Alembert, Poisson, and Kirchhoff formulas. Some applications are included as well.

Scattering by an elliptic cylinder with a strongly elongated cross section

The paper continues an earlier study devoted to the asymptotic behavior of the field in the boundary layer near the surface of an elliptic cylinder. Using the Kirchhoff formula, the previously derived asymptotics are recalculated to the far field asymptotics, which are considered within a narrow sector near the major axis of the cylinder’s cross section. The resulting asymptotic expressions are uniform with respect to the parameter characterizing the elongation of the ellipse and make it possible to trace the variation of the far field pattern from the limiting case of a strip to the case of a circular cylinder.

Acoustic Velocity Formulation for Kirchhoff Data Surfaces

This paper deals with the derivation of an analytical time-domain formulation for the prediction of the acoustic velocity field generated by moving bodies in a medium at rest, according to the Kirchhoff method. The present formulation can be implemented in acoustic pressure codes based on the Farassat's Kirchhoff formula for arbitrary moving bodies, thus allowing direct and fast calculation of the acoustic velocity field in scattering problems. For validation purposes, four test cases are considered, namely a three-dimensional monopole, dipole and quadrupole source, as well as a monopole in uniform flow. Comparison of the results with the analytical solutions proves the remarkable accuracy of the present formulation.

Derivation and implementation of the boundary integral formula for the convective acoustic wave equation in time domain.

Kirchhoff's formula for the convective wave equation is derived using the generalized function theory. The generalized convective wave equation for a stationary surface is obtained, and the integral formulation, the convective Kirchhoff's formula, is derived. The formula has a similar form to the classical Kirchhoff's formula, but an additional term appears due to a moving medium effect. For convenience, the additional term is manipulated to a final form as the classical Kirchhoff's formula. The frequency domain boundary integral can be obtained from the current time domain boundary integral form. The derived formula is verified by comparison with the analytic solution of source in the uniform flow. The formula is also utilized as a boundary integral equation. Time domain boundary element method (BEM) analysis using the boundary integral equation is conducted, and the results show good agreement with the analytical solution. The formula derived here can be useful for sound radiation and scattering by arbitrary bodies in a moving medium in the time domain.

Solutions of the wave equation on hybrid spaces of constant curvature

In this paper, we study the wave equation on the simplest hybrid spaces of constant curvature, namely, on Euclidean space or a sphere with a glued ray. We obtain explicit formulas for solutions of the Cauchy problem, which are the simplest nontrivial analogs of Kirchhoff or Herglotz-Petrovsky formulas; especially simple formulas are obtained in the case of three-dimensional Euclidean space with a glued ray. The solutions depend on the boundary conditions at the point of gluing, and these conditions determine the choice of the domain of the Laplace operator; the conditions ensuring the full reflection or full passage of waves are described separately.

Reconstructing surface wave profiles from reflected acoustic pulses using multiple receivers.

Surface wave shapes are determined by analyzing underwater reflected acoustic signals collected at multiple receivers. The transmitted signals are of nominal frequency 300 kHz and are reflected off surface gravity waves that are paddle-generated in a wave tank. An inverse processing algorithm reconstructs 50 surface wave shapes over a length span of 2.10 m. The inverse scheme uses a broadband forward scattering model based on Kirchhoff's diffraction formula to determine wave shapes. The surface reconstruction algorithm is self-starting in that source and receiver geometry and initial estimates of wave shape are determined from the same acoustic signals used in the inverse processing. A high speed camera provides ground-truth measurements of the surface wave field for comparison with the acoustically derived surface waves. Within Fresnel zone regions the statistical confidence of the inversely optimized surface profile exceeds that of the camera profile. Reconstructed surfaces are accurate to a resolution of about a quarter-wavelength of the acoustic pulse only within Fresnel zones associated with each source and receiver pair. Multiple isolated Fresnel zones from multiple receivers extend the spatial extent of accurate surface reconstruction while overlapping Fresnel zones increase confidence in the optimized profiles there.