Usual Perturbation Technique Research Articles

The nonlinear diffusion equation in cylindrical coordinates

Nonlinear corrections to some classical solutions of the linear diffusion equation in cylindrical coordinates are studied within quadratic approximation. When cylindrical coordinates are used, we try to find a nonlinear correction using quadratic polynomials of Bessel functions whose coefficients are Laurent polynomials of radius. This usual perturbation technique inevitably leads to a series of overdetermined systems of linear algebraic equations for the unknown coefficients (in contrast with the Cartesian coordinates). Using a computer algebra system, we show that all these overdetermined systems become compatible if we formally add one function on radius W(r). Solutions can be constructed as linear combinations of these quadratic polynomials of the Bessel functions and the functions W(r) and W′(r). This gives a series of solutions to the nonlinear diffusion equation; these are found with the same accuracy as the equation is derived.

Separation of variables for the nonlinear wave equation in cylindrical coordinates

Some classical types of nonlinear wave motion in cylindrical coordinates are studied within the quadratic approximation. When cylindrical coordinates are used, the usual perturbation techniques inevitably lead to overdetermined systems of linear algebraic equations for the unknown coefficients (in contrast to the case for Cartesian coordinates). However, we show that these overdetermined systems are compatible for the special case of the nonlinear acoustic wave equation and express the coefficients of the first two harmonics explicitly as polynomials in Bessel functions of the radius and in trigonometric functions of the angle. This gives a series of solutions to the nonlinear acoustic wave equation which are found with the same accuracy as the equation is derived with.

Separation of variables for the nonlinear wave equation in polar coordinates

Some classical types of nonlinear wave motion in polar coordinates are studied within quadratic approximation. When the nonlinear quadratic terms in the wave equation are arbitrary, the usual perturbation techniques used in polar coordinates leads to overdetermined systems of linear algebraic equations for the unknown coefficients. However, we show that these overdetermined systems are compatible with the special case of the nonlinear shallow water equation and express explicitly the coefficients of the first two harmonics as polynomials of the Bessel functions of radius and of the trigonometric functions of angle. It gives a series of solutions to the nonlinear shallow water equation that are periodic in time and found with the same accuracy as the equation is derived.

Nonlinear Wave Equation in Special Coordinates

Some classical types of nonlinear periodic wave motion are studied in special coordinates. In the case of cylinder coordinates, the usual perturbation techniques leads to the overdetermined systems of linear algebraic equations for unknown coefficients whose compatibility is key step of the investigation. Their solutions give solutions to the nonlinear wave equation which are periodic in time and found with the same accuracy as the nonlinear wave equation is derived. Expanding the potential for wave motion in Fourier series, we express explicitly the coefficients of the first two harmonics as quadratic polynomials of Bessel functions. One may speculate that the obtained expressions are only the first two terms of an exact solution to the nonlinear wave equations.

An improved perturbation technique for eigenvalues of continuous systems

AbstractA technique that improves the accuracy of the eigenvalue predictions as computed by a first‐order uniform expansion perturbation scheme for a class of eigenvalue problems associated with linear ordinary and partial differential equations is introduced. As suggested by a comparison with the classical Ritz method, the computational accuracy of a usual perturbation technique such as strained parameters can be effectively increased by simply replacing the usual M‐norms appearing in the denominator of the expressions for the first‐order changes in the eigenvalues by the corresponding MP‐norms, where MP is the perturbed M operator. With increasing sizes of the perturbations of the operators L and M of the class of boundary‐value problems considered, this improved perturbation technique produces results of substantially increased accuracy relative to the usual first‐order uniform expansion perturbation techniques. Numerical examples are presented.

The scattering of spherical pulses by slightly rough surfaces

Using a ’’small roughness’’ boundary condition due to Biot, which incorporates the effects of multiple scatter, one can obtain simple, closed-form solutions for the coherent scattering of transient spherical waves. Combining this boundary condition with the normal coordinate technique leads to explicit solutions for impulsive or other point sources above or on a rough, rigid plane, i.e., for both finite and grazing angles of incidence. This approach distinguishes naturally between two types of arrival: body (or volume) waves and boundary waves; the latter incorporate the backscatter. The boundary wave dies off rapidly with distance from the surface. But, parallel to the boundary, in the absence of attenuation, its amplitude decreases only as r−1/2 so that for sufficiently large r, and for source and receiver at or near the surface, the boundary-wave will dominate the direct acoustic pulse arrivals. The methods and results of this paper can be extended to problems of scatter and diffraction of spherical pulses by rigid (or free) plane rough surfaces bounding stratified media or by rough objects of other shapes, e.g., spheres, cylinders, ellipsoids, etc. This theory, valid for wavelengths or pulsewidths long compared to the characteristic roughness dimensions, leads to results that are different from those of the usual perturbation techniques and have been verified experimentally by Medwin et al. in a companion paper.

Stability of laminar flows of micropolar fluids between parallel walls

Numerical results on the stability of laminar flow of micropolar fluids between two parallel walls are presented. A sixth-order stability eigenvalue problem is obtained by the usual perturbation techniques and solved by means of Noumerov difference approximations and the Newton iteration method. As far as the model considered in the present investigation is concerned, the microstructure has a destabilizing effect on two-dimensional flows of micropolar fluids.

The Effect of Cross-Section Curvature on Attenuation in Elliptic Waveguides and a Basic Correction to Previous Formulas

It is shown that a certain asymptotic expansion of the Mathieu function of the fourth kind has almost certainly been inappropriately applied to previous elliptic waveguide calculations. An appropriate expansion is discussed at some length, and leads to a substantial modification of the published formulas. The need for a correction is demanded by a physical requirement to give the limit of the conventional surface impedance expression at the broad face of a very elongated elliptical shape. A curvature effect, the subject of some recent technical discussions, survives the change, and a discussion is given of an unsuccessful attempt to create a generalized form that replaces the ellipse parameters by differentials of the local curvature of the surface. An attempt is also made to justify the reality of such a substantial curvature effect. However, no effect occurs for flat surfaces or those of uniform curvature, confirming the validity of the usual perturbation technique of calculating waveguide attenuation in those cases. An anomalous property of the characteristic numbers of the Mathieu functions is discussed insofar as it relates to the present problem.

Resonant Scattering of Phonons. I. CN-Doped Alkali Halides

The resonant scattering of phonons in KCl:CN and KBr:CN has been investigated. It is found that the ${\ensuremath{\omega}}^{4}$ dependence for the resonant scattering of phonons as obtained by Griffin and Carruthers for a similar resonance process in $n\ensuremath{-}\mathrm{Ge}$ does not explain published phonon-conductivity results in CN-doped alkali halides in the temperature range where the resonant-scattering dips occur. The results can, however, be described very well by the formula $\ensuremath{\tau}_{r}^{}{}_{}{}^{\ensuremath{-}1}\ensuremath{\propto}\frac{{\ensuremath{\omega}}^{2}}{{({\ensuremath{\omega}}^{2}\ensuremath{-}\ensuremath{\omega}_{0}^{}{}_{}{}^{2})}^{2}}$, which has been obtained by Kwok for $n\ensuremath{-}\mathrm{Ge}$ using a Green's-function approach. This approach is well known to be more suitable for frequencies close to the resonant frequencies where the usual perturbation techniques fail because of divergence difficulties.

A New Interpretation of the Anomalous Magnetic and Optical Behavior of Copper Acetate Monohydrate

Interpretation of the extensive magnetic and optical measurements of copper acetate monohydrate on the basis of a C4v symmetry of the complex by different workers are, to some extent, incomplete and often lead to contradictions. Starting with the electronic states of each half of the complex, assumed to have a C2v symmetry as shown by ESR. data, we find that the bonding between the two halves is a mixture of σ- and δ-type as against a pure σ or δ bonding for a C4v symmetry. Operation of H=Hex+HLS+βH(L+2S) on the valence bond states of the whole complex by usual perturbation technique then gives us the expressions for the singlet-triplet separation (J), the spectroscopic splitting factors and the magnetic susceptibilities. It is found that the last two quantities and the optical absorption data can be correlated to a good extent and the ligand field parameters obtained in the process yield a value of − 138 cm−1 for J, which is fair enough under the approximations employed. In this calculation we have found it necessary to consider the dimeric band at 28 000 cm−1 to be a Laporte forbidden type under a strong anisotropic ligand field.