Modified Saddle Point Method Research Articles

An asymptotic expansion of the hyberbolic umbilic catastrophe integral

We obtain an asymptotic expansion of the hyperbolic umbilic catastrophe integral Psi ^{(H)}(x,y,z):= int _{-infty }^{infty }int _{-infty }^{infty }exp (i(s^3+t^3+zst+yt+xs))mathrm{{d}}s,mathrm{{d}}t for large values of |x| and bounded values of |y| and |z|. The expansion is given in terms of Airy functions and inverse powers of x. There is only one Stokes ray at arg x=pi . We use the modified saddle point method introduced in (López et al. J Math Anal Appl 354(1):347–359, 2009). The accuracy and the asymptotic character of the approximations are illustrated with numerical experiments.

Asymptotic solution for the electromagnetic scattering of a vertical dipole over plasmonic and non‐plasmonic half‐spaces

A new asymptotic solution for the scattered electromagnetic fields of a vertical Hertzian dipole antenna in the presence of an imperfectly conducting half-plane for ordinary and plasmonic media is proposed. The scattered electric and magnetic field components are calculated from the intermediate Hertz potential expressed in terms of the Fourier-Bessel transforms associated with the Sommerfeld-type integral, which is difficult to evaluate due to the singularities of the integrand near the integration path and its oscillatory and slowly decaying integrand. Using the modified saddle point method, an approximate closed-form solution of the far-zone scattered electromagnetic fields including surface waves is presented. The new formulations are applied to calculate radiation patterns of different impedance half-planes for both ordinary and plasmonic media, that is, seawater, silty clay soil, silty loam soil and lake water as ordinary, and silver and gold as plasmonic media. Furthermore, a numerical evaluation of the proposed solution at various frequencies and comparisons with two alternative state of the art solutions shows that the proposed solution has higher accuracy in terms of the normalised root-mean-square error and the normalised maximum absolute error for plasmonic and non-plasmonic structures.

The swallowtail integral in the highly oscillatory region III

We consider the swallowtail integral for large values of and bounded values of and . The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al., A systematization of the saddle point method application to the Airy and Hankel functions. J Math Anal Appl. 2009;354:347–359. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of for large and fixed x and y. The asymptotic analysis requires the study of three different regions for separated by three Stokes lines in the sector . The asymptotic approximation is a certain combination of two asymptotic series whose terms are elementary functions of x, y and z. They are given in terms of an asymptotic sequence of the order when , and it is multiplied by an exponential factor that behaves differently in the three mentioned sectors. The accuracy and the asymptotic character of the approximations are illustrated with some numerical experiments.

On the far‐zone electromagnetic field of a vertical Hertzian dipole over an imperfectly conducting half‐space with extensions to plasmonics

AbstractNew asymptotic formulas for the electromagnetic fields of a vertical Hertzian dipole over an imperfectly conducting half‐space are derived using two variants of the modified saddle point method, referred to as subtractive and multiplicative. The accuracy of the two far‐zone formulations is assessed by comparisons with the state‐of‐the‐art Norton‐Bannister theory and with the rigorous results of numerical integration for problems involving seawater in the microwave range and gold in the visible range. It is found that the performance of the three approximate methods is similar, except at the surface of plasmonic media, where the saddle point methods are superior to the Norton‐Bannister formulation.

Sound radiation of a line source moving above an absorbing plane with a frequency-dependent admittance

This paper is concerned with the derivation of an analytical solution for the acoustic pressure field generated by a line source moving at a constant speed and height above an absorbing plane and summarizes the results obtained in a recent article [Dragna and Blanc-Benon, J. Sound Vib. 349, 259-275, 2015]. As an extension of previous studies, the frequency dependence of the admittance is accounted for. First, a Lorentz transformation is used to obtain an equivalent stationary problem. A special attention is paid to the translation of the admittance boundary condition in the Lorentz space. An analytical solution is obtained as a Fourier transform. Excitation of surface waves is investigated depending on the Mach number. In the far field, an asymptotic expression is sought using the modified saddle point method. The solution is expressed under the form of a Weyl-Van der Pol formula, in which the admittance is evaluated at the Doppler frequency. Comparison of the pressure field obtained with the analytical s...

The Pearcey integral in the highly oscillatory region

We consider the Pearcey integral P(x, y) for large values of |y| and bounded values of |x|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then we consider here the modified saddle point method introduced in [Lopez, Pérez and Pagola, 2009] [4]. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of P(x, y) for large |y|. The asymptotic analysis requires the study of three different regions for argy separately. In the three regions, the expansion is given in terms of inverse powers of y2/3 and the coefficients are elementary functions of x. The accuracy of the approximation is illustrated with some numerical experiments.

A modified saddle point method for predicting sound fields above a non-locally reacting porous medium

This paper examines an asymptotic analysis for predicting sound fields above a rigid-frame porous medium, the so-called non-locally reacting porous medium. Their solutions can be represented by a direct wave term, a reflected wave term and a diffraction wave term. Exact and analytical solutions are available for the direct wave and the reflected wave from a perfectly hard ground. In the contrary, the diffraction wave term can only be cast in an integral form that is amenable to approximate analysis. A modified saddle-point method is explored to evaluate the diffraction integral asymptotically. Three different types of non-locally reacting surfaces, which are an extended reaction, a hard-backed layer, and an impedance-backed layer, were considered. The sound fields above these porous surfaces have the same form but they are different by an augmented diffraction term in the solutions. The analytical formula for the total sound fields, which can be stated in a closed form, offer a physically interpretable solution comprising of a direct wave and ground reflected wave terms. This latter term can further be split into a specularly reflected plane wave and ground wave components. A series of numerical comparisons have been conducted to validate the asymptotic analyses presented in this study. It has been demonstrated that the overall sound fields can be predicted well by the formula for all incidence angles and for a wide range of non-locally reacting porous surfaces.

On the asymptotic solution of sound penetration into a rigid porous half-plane: A modified saddle point method

An approximate analytical formula has been derived for the prediction of sound penetration into a semi-infinite rigid porous medium due to a monopole source. The sound fields can be expressed in an integral form that is amenable to analytical and numerical analyses. A modified saddle point method is applied to evaluate the integral asymptotically, which leads to a closed-form expression. The validity of the asymptotic formula is confirmed by comparison with the numerical results computed by the fast field formulation and the direct evaluation of the integral. The present analytical formula has been demonstrated to be sufficiently accurate to predict the penetration of sound into the semi-infinite rigid porous medium.

A modified saddle point method for predicting sound penetration into a rigid-porous ground

An approximate analytical formula has been derived for the prediction of sound penetration into a semi-infinite rigid porous ground due to a monopole source. The sound fields can be expressed in an integral form that is amenable to analytical and numerical analyses. A modified saddle point method is applied to evaluate the integral asymptotically that leads to a closed form expression. The validity of the asymptotic formula is confirmed by comparing with the numerical results computed by the fast field formulation and the direct evaluation of the integral. It has been demonstrated that the present analytical formula is sufficiently accurate to predict the penetration of sound into the semi-infinite rigid porous ground. Furthermore, the sound fields predicted by modified saddle point are compared with that computed by the double saddle point method. It is found that these two asymptotic schemes give precise solutions at far fields but the modified saddle method is more accurate at the near field especially when both the source and receiver are close to the air/ground interface. [Research is partially funded by the China Research Scholarship Council.]

Propagation of wave packets through resonant quantum systems

We use a previously proposed modified saddle point method to describe the tunneling of a rectangular wave packet through a resonant quantum system. We calculate the shape of the wave packet obtained at the quantum system output analytically for different values of the level width. The result of propagation is a wave packet that is the superposition of complementary error functions. Comparing the result with the exact numerical solution obtained without using any asymptotic methods shows a rather good coincidence. We study the propagation of Gaussian and rectangular wave packets in detail for large values of the resonance level width.

Sound propagation above an impedance boundary

An asymptotic solution of the scalar wave field due to a point source above a locally reacting plane surface is obtained by a modified saddle point method. Numerical calculation indicates that the relatively simple solution is more accurate than the Thomasson asymptotic solution and the Chien-Soroka solution. Furthermore, it is shown that another solution, simplified still more, is also in good agreement with the exact solution.

Field of an antenna submerged in a dissipative dielectric medium

The field components due to a horizontal dipole antenna submerged in a lake are represented in integral forms. The integrals are then evaluated using three techniques: 1) numerical integration along branch cuts where the integrands are smoothly varying as a function of the integration variable, 2) integration along the real axis with the fast Fourier transform technique, and 3) analytical integration with the modified saddle point method. Both the transmitter and the receiver are placed very near the interface. While the analytical result is good only at the larger distances from the transmitter, the numerical integration methods provide savings in computer time for small distances from the transmitter. Thus the analytical and numerical methods are good complements of each other. All three approaches yield essentially identical results over the distance ranges of interest. The theoretical results are compared with recent experimental measurements.