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The pressure field produced by two dimensional nonlinear time and space periodic standing waves was calculated as a series expansion in the wave height. The high order series was summed by the use of Pade approximants. Calculations included the pressure variation at great depth, which was considered to be a likely cause of microseismic activity, and the pressure distribution on a vertical barrier or breakwater.
For a nearly Gaussian transmitted pulse shape scattered from a nearly Gaussian distributed sea surface, a small argument series expansion of one term lead to a several term power series expression for the mean return waveform. Specific expressions are given for the first four terms. These results, which require less computer time than numerical convolution, are useful for data analysis from current or past radar altimeters and for design studies of future systems. Several representative results are presented for an idealized Seasat-1 radar altimeter.
Problems of wave phenomena in fields of acoustics, electromagnetics and elasticity are often reduced to an integration of the inhomogeneous Helmholtz equation. Results are presented for volume integrals associated with the Helmholtz operator, nabla(2) to alpha(2), for the case of an ellipsoidal region. By using appropriate Taylor series expansions and multinomial theorem, these volume integrals are obtained in series form for regions r 4' and r r', where r and r' are distances from the origin to the point of observation and source, respectively. Derivatives of these integrals are easily evaluated. When the wave number approaches zero, the results reduce directly to the potentials of variable densities.
Results are presented for volume integrals associated with the Helmholtz operator, nabla(2) + alpha(2), for the cases of a finite cylindrical region and a region of rectangular parallelepiped. By using appropriate Taylor series expansions and multinomial theorem, these volume integrals are obtained in series form for regions r r' and r 4', where r and r' are distances from the origin to the point of observation and source, respectively. When the wave number approaches zero, the results reduce directly to the potentials of variable densities.
This paper presents a technique for the partial-fraction expansion of functions which are ratios of polynomials with real coefficients. The expansion coefficients are determined by writing the polynomials as Taylor's series and obtaining the Laurent series expansion of the function. The general formula for the inverse Laplace transform is also derived.
The acoustic fields and transmission losses produced in semi-infinite circular ducts with peripherally segmented liners are analyzed using a series expansion of hard-wall duct modes. The coefficients of the series are computed using Galerkin's method. Unlike finite element approaches, this analysis includes the effects of realistic sources and the number of peripheral strips need not be small. It is shown that peripherally segmented liners redistribute the acoustic energy in waves composed of only a single circumferential mode at the source into other waves which contain a multitude of circumferential modes in the lined section. The accuracy of eigenfunctions computed from the analysis was observed to increase as either the frequency or radial mode order increased. The transmission losses were found to be accurate at frequencies above the cut-on value of the first-order radial mode in a hard-wall duct. The results show that for plane wave sources, peripherally segmented liners may attenuate as much sound as an optimized uniform liner at the optimal point while giving more noise suppression at most other frequencies.
Problems of wave phenomena in the fields of acoustics, electromagnetics and elasticity are often reduced to an integration of the inhomogeneous Helmholtz equation. Results are presented for volume integrals associated with the inhomogeneous Helmholtz equation, for an ellipsoidal region. By using appropriate Taylor series expansions and the multinomial theorem, these volume integrals are obtained in series form for regions r greater than r-prime and r less than r-prime, where r and r-prime are the distances from the origin to the point of observation and the source. Derivatives of these integrals are easily evaluated. When the wavenumber approaches zero the results reduce directly to the potentials of ellipsoids of variable densities.
A new method for propagating the solution of the radial Schroedinger equation is derived from a Taylor series expansion of the wavefunction and partial re-summation of the infinite series. Truncation of the series yields an approximation to the exact propagator which is applied to a model calculation and found to be highly convergent.
The orbits of particles undergoing gravitational instability are parameterized by generalizing the Zel'dovich approximation to a series expansion of arbitrary accuracy in the nonlinear regime. The coefficients of this series are determined from an action principle, or, more generally, from moments of the equation of motion. It is shown that the series is more rapidly convergent than previous nonlinear approximations. The method therefore provides a practical means of determining particle orbits, even for highly nonlinear perturbations. As an alternative, we also outline how the nonlinear dynamics may be computed as a field theory in which the evolution of the density and the velocity is determined in fixed comoving Eulerian coordinates.
The Asymptotic Waveform Evaluation (AWE) technique is applied to a generalized frequency domain electromagnetic problem. Most of the frequency domain techniques in computational electromagnetics result in a matrix equation, which is solved at a single frequency. In the AWE technique, the Taylor series expansion around that frequency is applied to the matrix equation. The coefficients of the Taylor's series are obtained in terms of the frequency derivatives of the matrices evaluated at the expansion frequency. The coefficients hence obtained will be used to predict the frequency response of the system over a frequency range. The detailed derivation of the coefficients (called 'moments') is given along with an illustration for electric field integral equation (or Method of Moments) technique. The radar cross section (RCS) frequency response of a square plate is presented using the AWE technique and is compared with the exact solution at various frequencies.
A fractional-step splitting scheme breaks the full Navier-Stokes equations into explicit and implicit portions amenable to the calculus of variations. Beginning with the functional forms of the Poisson and Helmholtz equations, we substitute finite expansion series for the dependent variables and derive the matrix equations for the unknown expansion coefficients. This method employs a new splitting scheme which differs from conventional three-step (nonlinear, pressure, viscous) schemes. The nonlinear step appears in the conventional, explicit manner, the difference occurs in the pressure step. Instead of solving for the pressure gradient using the nonlinear velocity, we add the viscous portion of the Navier-Stokes equation from the previous time step to the velocity before solving for the pressure gradient. By combining this 'predicted' pressure gradient with the nonlinear velocity in an explicit term, and the Crank-Nicholson method for the viscous terms, we develop a Helmholtz equation for the final velocity.
An iterative technique which recovers density profiles in a nonhomogeneous absorbing atmosphere is derived. The technique is based on the concept of factoring a function of the density profile into the product of a known term and a term which is not known, but whose power series expansion can be found. This series converges rapidly under a wide range of conditions. A demonstration example of simulated data from a high resolution infrared heterodyne instrument is inverted. For the examples studied, the technique is shown to be capable of extracting features of ozone profiles in the troposphere and to be particularly stable.
Some methods are described for the expansion of the disturbing function in planetary theory. One method uses the classical binomial expansion theorem or a successive approximation process derived from it. Another method is a direct application of the Laplace series expansions. For both methods it is proposed to first prepare the series to be manipulated by a scaling operation. These methods can be applied either in a literal or in a numerical form, or any combination of both, but they are especially designed for use on a large scale digital computer with standard Poisson series programs. No usage is made of Newcomb operators or derivatives of Laplace coefficients.
This paper presents a notation system to facilitate to solution of differential equations via Taylor series expansions and applies it to solve the circular restricted three body problem. Unlike previous Taylor series methods in the astrodynamics literature, computer algebra solvers are not used. Instead the notation system allows one to solve a system of differential equations analytically “by hand” without resorting to computer algebra software. This method produces recurrence relations explicitly in terms of a sequence of derivatives of the state with respect to time for the coefficients of Taylor Series solutions that can be evaluated numerically or manipulated further to investigate properties of the solution. For example, additional derivatives with respect to other parameters may also be found, including those that describe the dependence of the solution on initial conditions.
Computation of fields radiated by parabolic reflectors and planar apertures is a problem of prime importance, and conventional approaches to performing these computations are often very time-consuming. The paper demonstrates the usefulness of a series expansion technique for efficient computation of radiated fields in both the Fraunhofer and Fresnel regions. The series approach is based on the use of the Jacobi polynomials. The application of the series method to determining the far-field pattern of the antenna from the near-field measurements for an arbitrary aperture antenna is also discussed.
A locally Gaussian model of turbulence velocities is postulated which consists of the superposition of a slowly varying strictly Gaussian component representing slow temporal changes in the mean wind speed and a more rapidly varying locally Gaussian turbulence component possessing a temporally fluctuating local variance. Series expansions of the probability density and exceedance rate functions of the turbulence velocity model, based on Taylor's series, are derived. Comparisons of the resulting two-term approximations with measured probability density and exceedance rate functions of atmospheric turbulence velocity records show encouraging agreement, thereby confirming the consistency of the measured records with the locally Gaussian model. Explicit formulas are derived for computing all required expansion coefficients from measured turbulence records.
The equilibrium quantum statistics of various anharmonic oscillator systems including relativistic systems is considered within the Wigner phase space formalism. For this purpose the Wigner series expansion for the partition function is generalized to include relativistic corrections. The new series for partition functions and all thermodynamic potentials yield quantum corrections in terms of powers of h(sup 2) and relativistic corrections given by Kelvin functions (modified Hankel functions) K(sub nu)(mc(sup 2)/kT). As applications, the symmetric Toda oscillator, isotonic and singular anharmonic oscillators, and hindered rotators, i.e. oscillators with cosine potential, are addressed.
We present a new algorithm for the numerical evaluation of five-point conformal blocks in d-dimensions, greatly improving the efficiency of their computation. To do this we use an appropriate ansatz for the blocks as a series expansion in radial coordinates, derive a set of recursion relations for the unknown coefficients in the ansatz, and evaluate the series using a Padé approximant to accelerate its convergence. We then study the 〈σσϵσσ〉 correlator in the 3d critical Ising model by truncating the operator product expansion (OPE) and only including operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We approximate the contributions of the operators above the cutoff by the corresponding contributions in a suitable disconnected five-point correlator. Using this approach, we compute a number of OPE coefficients with greater accuracy than previous methods.